numerical solutions of partial differential...

NUMERICAL SOLUTIONS OF PARTIAL

DIFFERENTIAL EQUATIONS USING B-SPLINE

By

Saima Arshed

A THESIS

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

Supervised By

Prof. Dr. Shahid S. Siddiqi

UNIVERSITY OF THE PUNJAB

QUAID-E-AZAM CAMPUS, LAHORE

SEPTEMBER, 2014

CERTIFICATE

I certify that the research work presented in this thesis is

the original work of Miss. Saima Arshed D/O Muhammad

Arshed Karim and is carried out under my supervision. I

endorse its evaluation for the award of Ph.D. degree through

the official procedure of University of the Punjab.

Prof. Dr. Shahid S. Siddiqi(Supervisor)

ii

DECLARATION

I, Miss. Saima Arshed D/O Muhammad Arshed

Karim, hereby declare that the matter printed in this thesis

is my original work. This thesis does not contain any material

that has been submitted for the award of any other degree in

any university and to the best of my knowledge, neither does

this thesis contain any material published or written previously

by any other person, except due reference is made in the text of

this thesis.

Saima Arshed

iii

Dedications

To my loving Parents, Husband

and Daughter

iv

Table of Contents

Table of Contents v

Abstract xiii

Acknowledgements xiv

1 Introduction 1

1.1 Differential Equations . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Types of Differential Equations . . . . . . . . . . . . 2

1.2 Introduction to Partial Differential Equation . . . . . . . . . 3

1.2.1 Partial Integro-Differential Equation (PIDE) . . . . . 4

1.2.2 Fractional Partial Differential Equation (FPDE) . . . 5

1.3 Applications of PDEs . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Partial Differential Equation in Shock Wave . . . . . 6

1.3.2 Partial Differential Equations in Fluid Mechanics . . 7

1.3.3 Partial Differential Equations in Solar Dynamics . . . 7

1.3.4 Partial Differential Equations in Quantum mechanics 8

1.4 Classification of Linear Second-Order PDEs . . . . . . . . . 8

1.4.1 Parabolic PDEs . . . . . . . . . . . . . . . . . . . . . 9

1.4.2 Hyperbolic PDEs . . . . . . . . . . . . . . . . . . . . 10

1.4.3 Elliptic PDEs . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7.1 Numerical Techniques for Solving PDEs . . . . . . . 13

1.7.2 Discretization . . . . . . . . . . . . . . . . . . . . . . 15

1.7.3 Consistency, Convergence and Stability . . . . . . . . 16

1.7.4 Lax’s Equivalence Theorem . . . . . . . . . . . . . . 17

1.7.5 Fractional Calculus . . . . . . . . . . . . . . . . . . . 17

1.7.6 B-spline . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7.7 B-spline Basis Functions . . . . . . . . . . . . . . . . 19

v

1.7.8 Collocation Method . . . . . . . . . . . . . . . . . . . 21

1.7.9 Collocation Method with B-spline Basis Functions . . 22

1.7.10 Cubic B-spline Collocation Method . . . . . . . . . . 22

1.7.11 Quintic B-spline Collocation Method . . . . . . . . . 24

1.8 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Quintic B-Spline for the Numerical Solution of the Good

Boussinesq Equation 31

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Temporal discretization . . . . . . . . . . . . . . . . . . . . . 33

2.3 Quintic B-spline collocation method . . . . . . . . . . . . . . 35

2.4 The Initial Vector . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 39

2.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Solution of Fourth-Order Partial Differential Equation 44

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Discretization in time . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Quintic B-spline Collocation Method . . . . . . . . . . . . . 47

3.4 The Initial Vector . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 57


3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Numerical Solution of Partial Integro-Differential Equation 63

4.1 Parabolic Integro-Differential Equations . . . . . . . . . . . 64

4.1.1 Temporal Discretization . . . . . . . . . . . . . . . . 65

4.1.2 Cubic B-spline Collocation Method . . . . . . . . . . 68

4.1.3 Numerical Results . . . . . . . . . . . . . . . . . . . 73

4.2 Convection-Diffusion Integro-Differential Equation . . . . . . 80

4.2.1 Temporal Discretization . . . . . . . . . . . . . . . . 82

4.2.2 Discretization in Space . . . . . . . . . . . . . . . . . 84

4.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . 87

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 Numerical Solution of Time-Fractional Fourth-order Partial

Differential Equations 99

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . 102

5.3 Discretization in space . . . . . . . . . . . . . . . . . . . . . 112

vi

5.4 Time-fractional semilinear fourth-order partial differential equa-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115


5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 Numerical Solution of Time-Fractional Convection-Diffusion

Equation 131

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.2 The cubic B-spline . . . . . . . . . . . . . . . . . . . . . . . 134


6.4 Stability and convergence analysis . . . . . . . . . . . . . . . 139



6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7 B-Spline Solution of Time-Fractional Integro Partial Differ-

ential Equation With a Weakly Singular Kernel 152

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 153




7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

References 174

Appendix 182

vii

List of Figures

4.1 The results at M=500 for Example 4.1. . . . . . . . . . . . . 75

4.2 The exact and numerical solutions at M=10 . . . . . . . . . 75


4.4 The exact and numerical solutions at M=10 . . . . . . . . . 78


4.6 The exact and numerical solutions at M=10. . . . . . . . . . 81

4.7 The results at K=500 for Example 4.4. . . . . . . . . . . . . 90

4.8 The exact and numerical solutions at K=10 . . . . . . . . . 90







5.1 The results at M=40, K=1000 and ∆t = 0.00001 for Example

5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2 The exact and numerical solution at K=1000. Dotted line:

numerical solution, Solid line: exact solution . . . . . . . . . 118

5.3 Errors as a function of the time ∆t for α = 0.75 . . . . . . . 119


5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121



viii

ix



5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125




List of Tables

1.1 Coefficients of cubic B-spline and its derivatives at knots xi. 23

1.2 Coefficients of quintic B-spline and its derivatives at knots xi. 24

2.1 Numerical results for Example 2.1 . . . . . . . . . . . . . . . 42

3.1 Coefficients of quintic B-spline and its derivatives at knots xj. 48

3.2 Absolute errors for h = 0.05 at points x = 0.1, x = 0.2, x =

0.3, x = 0.4, x = 0.5 for example 3.1. . . . . . . . . . . . . . . 59

3.3 Absolute errors at midpoints, x = 0.5, for h = 0.05 and

r = 0.5 for example 3.1 . . . . . . . . . . . . . . . . . . . . . 60

3.4 Absolute errors at midpoints, x = 0.5, for h = 0.05 and

r = 2.0 for example 3.1 . . . . . . . . . . . . . . . . . . . . . 60

3.5 Absolute errors at midpoints, x = 0.5, for h = 0.05 for ex-

ample 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.6 Comparison of proposed method with H. Caglar and N. Caglar

[7] in maximum absolute errors for example 3.1 . . . . . . . 61

3.7 Maximum absolute errors with h = 0.05 for example 3.2 . . . 62

4.1 The errors ‖eK‖∞ and ‖eK‖2 when M = 60 and ∆t = 0.0001

for example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 74


for example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 The errors ‖eK‖∞ and ‖eK‖2 when K = 10 and ∆t = 0.0001

for example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 75

x

xi


for example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 77


for example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.6 The errors ‖eK‖∞ and ‖eK‖2 when K = 10 and ∆t = 0.0001

for example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.7 Maximum norm errors ‖eK‖∞ for M = 40 for example 4.3 . 79

4.8 L2 norm errors ‖eK‖2 for M = 40 for example 4.3 . . . . . . 80

4.9 ‖eK‖∞ and ‖eK‖2 for ∆t = 0.0001 for example 4.4 . . . . . . 89

4.10 ‖eK‖∞ and ‖eK‖2 for ∆t = 0.00001 for example 4.4 . . . . . 89





4.15 L2 norm errors ‖eK‖2 for M = 100 for example 4.7 . . . . . 98

4.16 Maximum norm errors ‖eK‖∞ for M = 100 for example 4.7 . 98

5.1 The errors ‖eK‖∞ and ‖eK‖2 for different K taken ∆t=0.00001.117

5.2 The errors ‖eK‖∞ and ‖eK‖2 of different time steps with

M = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3 The errors ‖eK‖∞ and ‖eK‖2 for different M taken ∆t=0.00001.121


M = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122



M = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.7 The errors ‖eK‖∞ and ‖eK‖2 for different M taken ∆t =

0.00001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127


M = 80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.9 The errors ‖eK‖∞ and ‖eK‖2 for different M taken ∆t =

0.00001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xii


M = 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.1 Coefficients of cubic B-spline and its derivatives at knots xi. 135



M = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149



M = 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.1 The errors L∞ and L2 of different time steps with M = 100 . 169

7.2 The errors L∞ and L2 of different time steps with M = 60 . 170

7.3 Results for u with h = 1/50 and T = 1.0 . . . . . . . . . . . 171

7.4 The errors L∞ when ∆t = 0.00001 . . . . . . . . . . . . . . 172

7.5 The errors L∞ when ∆t = 0.000001 . . . . . . . . . . . . . 172

Abstract

The main objective of the thesis is to develop the numerical solution of

partial differential equations, partial integro-differential equations with a

weakly singular kernel, time-fractional partial differential equations and

time-fractional integro partial differential equations.

The numerical solutions of these PDEs have been obtained using cubic

and quintic B-spline collocation method. The time derivatives have been

approximated by finite difference formulas and the time-fractional derivative

has been described in the Caputo sense. The stability and convergence

properties related to the time discretization have been discussed and proven,

theoretically.

The main advantage of B-spline collocation method is that many partial

differential equations which are not simple to solve, can be solved easily

by this method. The collocation method with B-spline as basis functions

represents an economical alternative, since it only requires the evaluation

of the unknown parameters at the grid points.

It has been observed from the numerical results that the presented B-

spline collocation method exhibits high level of efficiency and accuracy.

Moreover, the numerical results approximate the exact solutions very effi-

ciently.

xiii

Acknowledgements

In the name of ALLAH Almighty the most Benevolent, the most Merci-

ful, the Creator of the universe and the Master of Life and Death, who

inculcated His countless blessings upon me to fulfill the requirements of

this dissertation. I offer my extremely humblest, sincerest Darood-O-Salam

to our beloved Prophet Hazrat Muhammad (peace be upon him) who is

forever a symbol of complete guidance in every walk of life for humanity.

I would like to express my sincere gratitude and supreme warm thanks

to my supervisor Prof. Dr. Shahid Saeed Siddiqi for the continuous support

of my Ph.D study and research, for his patience, motivation and immense

knowledge. His guidance helped me in all the time of research and writing

of this thesis. I could not have imagined having a better advisor, overseer

and mentor for my Ph.D study. I warmly thank Dr. Ghazala Akram, for

her valuable advice and friendly help. Her extensive discussions around

my work and interesting explorations in difficult concepts have been very

helpful for this study.

In the first place I would like to record my gratitude to Prof. Dr.

Muhammad Sharif, the Chairman, Department of mathematics, University

of the Punjab, Lahore, Pakistan, for his advices, invaluable and invigo-

rating encouragement and support in various ways. I further categorically

acknowledge to all my honorable teachers without whom I would not be

able to touch this stage of academic zenith.

My special thanks to my research mate Miss Shamaila Rani, and my

colleagues Dr. Uzma Ahmad, Ms. Saira Hameed and Ms. Maasoomah

Sadaf for their constantly varying encouragement and cooperation.

It is not possible for me to name all those who have contributed, directly

or indirectly, towards the completion of my work. I am grateful to all my

well-wishers for their sincere support. I express my apology to all those not

mentioned personally one by one. Words wane in expressing my veneration

for my loving, grateful, and delicate parents and all my family members

xiv

xv

I owe my heartiest gratitude for their assistance and never ending prayers

for my success. I would never have been able to stand today without their

continuous support and generous help.

Lahore Saima Arshed

September, 2014

Chapter 1

Introduction

1.1 Differential Equations

Mathematics is the mother of all sciences and is widely used in physics,

chemistry, biology, statistics, engineering, economics and astronomy etc. In

fact mathematics is tool to study nature and make predictions. But of

course to achieve this goal, the physical situations must first be written

in the language of mathematics. The process of describing physical phe-

nomena using suitable mathematical relations is termed as mathematical

modeling.

The word differential and equations indicate that any equation involv-

ing derivatives. The rate of change of any quantity with respect to other

quantity (quantities) is expressed by a differential equation. Differential

equations play an important role in mathematical modeling of physical phe-

nomena occurring in science, economics, engineering and medicine etc.

Solving various problems in science and engineering requires differential

1

Ch 1: Introduction 2

equations. Many physical, chemical, mathematical models, biological phe-

nomena, economics, financial forecasting, image processing and other fields

are based on differential equation.

1.1.1 Types of Differential Equations

The differential equations are, generally, classified in two types

• Ordinary differential equation (ODE)

It is an equation in which the function depends only on one indepen-

dent variable

dy

dx= y2 − x2 + xy.

• Partial differential equation (PDE)

A partial differential equation is an equation in which the dependent

variable (unknown function) must depends on more than one inde-

pendent variables.

Examples of the PDEs are given by

∂w

∂t= k

∂2w

∂x2(1.1.1)

∂w

∂t= k

(∂2w

∂x2+

∂2w

∂y2

), (1.1.2)

∂w

∂t= k

(∂2w

∂x2+

∂2w

∂y2+

∂2w

∂z2

). (1.1.3)


It may be noted that the heat flow in one dimensional space, two di-

mensional space, and three dimensional space are described by equa-

tions (1.1.1), (1.1.2) and (1.1.3), respectively.

1.2 Introduction to Partial Differential Equa-

tion

Euler’s work in the 18th century marked the beginning of development on

the PDEs. Then d’Alembert, Lagrange, Laplace and Riemann made spec-

tacular progress in the field. Some important PDEs of the 19th century are

Laplace equation, Poisson equation, heat equation, Maxwell’s equations,

Navier-Stoke equations, KdV equation and many other. The mathematical

modeling of many scientific problems involving rates of change w.r.t. two or

more independent variables, like time, length or angle, may lead to a PDE

or to a system of PDEs. Several phenomena that occur in mathematical

physics and different fields of engineering can be modeled by partial differen-

tial equations. In physics for example, the heat flow problems and the wave

propagation phenomena are well formulated by partial differential equa-

tions [4], [16], [25] and [28]. In addition, most physical phenomena of fluid

dynamics, quantum mechanics, electricity, plasma physics, propagation of

shallow water waves and many other models are controlled within its do-

main of validity by partial differential equations [60]. In general, PDEs are,


sometimes, more difficult to solve analytically than the ODEs can be. PDEs

may be solved analytically by methods such as Backlund transformation,

methods of characteristics, Green’s function, integral transform, Lax pair

and separation of variables etc. It may be noted that the analytical meth-

ods, providing exact solutions, are more laborious ones. Analytical methods

become much harder to solve complex problems. Numerical methods have

become very popular among the researchers in the last few decades. There

is a variety of numerical techniques for solving PDEs, such as the finite

difference method [55], finite element method [57], finite volume method

[29], meshfree method [31] and the spectral method [19]. The finite element

method and finite volume method are efficiently used in different branches

of engineering to model complicated problems. The finite difference method

is often regarded as the simplest method [47], the meshfree method is used

to facilitate accurate and stable numerical solutions for PDEs without using

a mesh.

The PDEs can further be classified as under

1.2.1 Partial Integro-Differential Equation (PIDE)

Partial integro-differential equation (PIDE) is an equation that consists of

partial derivatives and integral terms of the unknown function. When the

effects of the memory of the system are considered, the model involves

the integral term containing the unknown function. Following is a simple


example of partial integro-differential equation

∂w

∂t=

∫ t

0

K(t, s)∂2w(x, s)

∂x2ds + g(x, t), x ∈ [a, b], t > 0,

where K(t, s) is the kernel function and g(x, t) is a given smooth function.

Partial integro-differential equations can describe some physical situations

such as fluid dynamics, viscoelasticity, convection-diffusion problems, heat

flow in materials with memory, nuclear reactor dynamics, geophysics and

plasma physics etc.

1.2.2 Fractional Partial Differential Equation (FPDE)

Fractional partial differential equations (FPDEs) are obtained by general-

izing partial differential equations to a fractional order. Following is an

example of non-linear time-fractional PDE

Dαt w(x, t) =

∂2w(x, t)

∂x2+ 6w(x, t)(1− w(x, t)),

where Dαt is a time-fractional differential operator.

Fractional partial differential equations efficiently model various complex

phenomena, which play a vital role in mathematical physics. In the last

few decades, remarkable attention have been made among the researchers

and scientists for finding numerical solutions of FPDEs. FPDEs have mem-

ory and nonlocal relations in time and space variables. Due to this fact,

materials with memory and hereditary effects, fluid flow, diffusive transport,


electrical networks, electromagnetic theory and probability, signal process-

ing and many other physical processes are diverse applications of FPDEs

[40], [41] and [27].

1.3 Applications of PDEs

Partial differential equations are used to describe a wide range of physical

phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow,

elasticity, or quantum mechanics etc. The discussion on the applications of

PDEs are given in the following subsections.

1.3.1 Partial Differential Equation in Shock Wave

A shock wave is a wave which propagates disturbance. It carries energy

as an ordinary wave and can propagate through a medium (solid, liquid,

gas or plasma). A shock wave propagates at a higher speed through most

media than an ordinary wave. The following partial differential equation is

used for this purpose, i.e.

∂w

∂t+ w

∂w

∂x= 0.


1.3.2 Partial Differential Equations in Fluid Mechan-

ics

The Navier-Stokes equations are used to describe the motion of fluid sub-

stances. These equations are also used to model different physical phenom-

ena such as the weather predictions, ocean currents, flow of water in pipe,

and air flow around a wing of an aircraft. Following is the Navier-Stokes

equation

ρ

(∂w

∂t+ w.∇w

)= −∇p +∇.T + g,

where w is the flow velocity, ρ is the fluid density, p is the pressure of the

fluid, T is the component of the total stress tensor and g represents body

forces (per unit volume) acting on the fluid.

1.3.3 Partial Differential Equations in Solar Dynam-

ics

The light is emitted from the sun across the electromagnetic spectrum.

In order to understand solar dynamics, it is important to understand the

radiations of sun. The following wave equation can be used for this purpose,

∂2w

∂t2=

∂2w

∂x2.


1.3.4 Partial Differential Equations in Quantum me-

chanics

The Schrodinger equation is a basic equation of quantum mechanics. It is

a PDE that is used to describe how the quantum state of physical system

vary with time. The general form of time dependent Schrodinger equation,

which gives a description of a system evolving with time can be written as

i~∂Ψ

∂t= JΨ,

where i is the imaginary unit, ~ is the Planck’s constant, Ψ represents the

wave function of the quantum system, and J is the Hamiltonian operator.

1.4 Classification of Linear Second-Order PDEs

An equation of the form

A∂2w

∂x2+ 2B

∂2w

∂x∂y+ C

∂2w

∂y2+ D

∂w

∂x+ E

∂w

∂y+ F = 0,

where A,B, C, D, E, F are constants, is known as

1. Parabolic if, B2 − AC = 0,

2. Hyperbolic if, B2 − AC > 0,

3. Elliptic if, B2 − AC < 0.

The discussion about parabolic, hyperbolic and elliptic PDEs are given in

the following subsections.


1.4.1 Parabolic PDEs

Parabolic PDEs are used for describing broad range of problems, including

heat diffusion, ocean acoustic propagation and mathematical or physical

systems with a time variable.

A simple example of parabolic PDE is the one dimensional heat equation.

It is a parabolic PDE that describes the distribution of heat (or variation

in temperature) in a certain region over time t. Mathematically, the heat

equation has the following form

∂w

∂t= k

∂2w

∂x2,

where w(x, t) is the temperature at time t and at position x, and k is a

positive constant termed as thermal diffusivity.

The convection-diffusion equation is another example of parabolic PDE

which describes a physical phenomena, how particles, energy or other physi-

cal quantities are transferred inside a physical system through two processes:

convection and diffusion. Convection is a process that describes the move-

ment of molecules within fluids, whereas, diffusion describes the spread of

particles from region of higher concentration to region of lower concentra-

tion through random motion of particles. Mathematically, it can be written

as

∂w

∂t+ m

∂w

∂x− b

∂2w

∂x2= g(x, t),


where m > 0 and b > 0 are considered to be positive constants quantifying

the advection(convection) and diffusion processes, respectively. The source

term g(x, t), accounts for an insertion or extraction of mass from the system

as it evolves with time. Specifically, g(x, t) represents the time rate of

change of concentration due to external factors, such as a source or a sink.

1.4.2 Hyperbolic PDEs

Hyperbolic PDEs play an important role in physical problems such as trans-

port process and wave propagation phenomena.

The wave equation is a simple example of a hyperbolic PDE that explains

the description of waves occur in physics, such as light waves, water waves

and sound waves. The transverse vibrations of a string such as a string of

musical instrument is a common example of one dimensional wave equation.

Mathematically, it can be written as

∂2w

∂t2= r2∂2w

∂x2,

where r is a fixed constant termed as wave speed.

“The telegraphers equation (or just telegraph equation) is another ex-

ample of a hyperbolic PDE which describes the flow of current and voltage

in an electrical transmission line with distance and time.

The most general form of telegraph equation in one-space dimension is

∂2w

∂t2= (a + b)

∂w

∂t+ abw = c2∂2w

∂x2+ h(x, t),


where w(x, t) can be voltage or current through the wire at position x and

time t and a = AC

, b = BD

, c2 = 1DC

, where A is conductance of resistor,

B is resistance of resistor, D is inductance of coil and C is capacitance of

capacitor” [17].

1.4.3 Elliptic PDEs

Many real-life problems in physics and engineering can be formulated as

elliptic PDEs.

Poisson’s equation is a simple example of an elliptic PDE, arising in

physical problems such as finding the electric potential of a given charge

distribution. Therefore the potential is related to the charge density by

Poisson’s equation. In a charge-free region of space, the Poisson’s equation

becomes Laplace’s equation.

1.5 Boundary Conditions

In the mathematical models, supplementary information on the boundary

of the domain (according to the time or the spatial dimension) are provided

with the PDE. This information is known as initial conditions (w.r.t. the

time dimension) and as boundary conditions (w.r.t. the space dimension).

Following are the types of boundary conditions associated with PDEs.

1. Dirichlet boundary condition


It is the boundary condition that defines the values of unknown func-

tions on the boundary of the problem domain.

2. Neumann boundary condition

It is the boundary condition that defines the normal derivative on the

boundary of the problem domain.

3. Mixed boundary condition

In mixed boundary condition different boundary conditions are used

in different (disjoint) portions of the boundary of the domain. In a

mixed boundary value problem, the solution is required to satisfy a

Dirichlet or a Neumann boundary condition in a mutually exclusive

way on disjoint parts of the boundary.

4. Robin boundary condition

This boundary condition is defined as a linear combination of the

values of a function and the values of its normal derivatives on the

boundary of the domain. Robin boundary conditions can also be

expressed as a linear combination of Dirichlet boundary conditions

and Neumann boundary conditions.

1.6 Initial Conditions

A large range of PDEs, such as the diffusion problems, heat flow problems

and the wave propagation phenomena, depend on time t. It may be noted


that the initial values of the dependent variable w(x, t) should be given at

the starting time.

In the case of heat flow problems, the initial condition is

w(x, t = 0) = a(x),

In the case of wave propagation problems, the initial conditions are

w(x, t = 0) = b(x),

wt(x, t = 0) = c(x),

where a(x), b(x) and c(x) are unknown functions.

1.7 Preliminaries

1.7.1 Numerical Techniques for Solving PDEs

PDEs can be found in the mathematical modeling of physical, chemical and

biological phenomena. In some situations, the PDEs under consideration

are so complex that finding their analytical solutions are either not possible

or not simple to calculate. This motivates the researchers to construct the

numerical methods for the solution of PDEs.

Numerical methods for obtaining approximate solutions of partial differen-

tial equations may be classified as

• Finite Difference Method (FDM)


In FDM, the PDE is transformed into a system of algebraic equa-

tions by discretization. Firstly, the interval domain is partitioned by

introducing a finite number of nodes. Then the truncated Taylor’s

series is used to express the derivatives involved in partial differential

equation in terms of finite differences. The resulting system of equa-

tions is then, simultaneously solved and the solution of the system of

equations is the required approximation to the exact solution of the

PDE.

The most commonly used finite difference methods are explicit, im-

plicit and Crank-Nicolson finite difference methods.

• Finite Element Method (FEM)

The FEM is another popular method for obtaining the numerical so-

lution of PDEs. “The finite element method replaces the original

function with a function that has some degree of smoothness over the

global domain but is piecewise polynomial on simple cells, such as

small triangles or rectangles” [47].

• Finite Volume Method

The finite-volume method (FVM) is a technique used for calculating

PDEs in the form of algebraic equations. Like FDM or FEM, the

values are computed at discrete points on a meshed geometry. “Fi-

nite volume” means the small volume enclosing each node point on a


mesh. In the finite volume method, the divergence theorem is used

for converting the volume integrals in a partial differential equation

containing a divergence term into surface integrals. These terms are

then evaluated as fluxes at the surfaces of each finite volume. It is

frequently used for discretizing the equations of computational fluid

dynamics.

• Methods of Lines

The method of lines (MOL) is semi-discretizing technique for the nu-

merical solution of PDEs in which only the spatial derivatives have

been discretized.

1.7.2 Discretization

It is the process of transforming continuous models and equations into dis-

crete portions. The following are two types of discretization.

1. Spatial discretization is the process in which the space variables of

the equation under consideration are discretized.

2. Temporal discretization is the process in which the time variables

of the given equation are discretized.


1.7.3 Consistency, Convergence and Stability

Three basic properties that every finite-difference equation of a partial dif-

ferential equation should possess are consistency, convergence and stability.

• Consistency

“Let L(U) = 0 represents the partial differential equation in the in-

dependent variables x and t, with exact solution U , where L is the

differential operator.

Let F (u) = 0 represents the approximating finite-difference equation

with exact solution u, where F is the difference operator.

Let v be a continuous function of x and t with a sufficient number

of continuous derivatives to enable L(v) to be evaluated at the point

(ih, jk).

Then the truncation error Ti,j(v) at the point (ih, jk) is defined by

Ti,j(v) = Fi,j(v)− L(vi,j).

If Ti,j(v) → 0 as h, k → 0, the difference equation is said to be con-

sistent or compatible with the partial differential equation” [55].

• Convergence

“Let U represents the exact solution of a partial differential equa-

tion with independent variables x and t, and u the exact solution of

the difference equation used to approximate the partial differential

equation. The the finite-difference equation is said to be convergent


when u tends to U at a fixed point or along a fixed t−level as h and

k both tend to zero” [55].

• Stability

“Stability means that the errors made at one stage of the calcula-

tions do not cause increasingly large errors as the computations are

continued, but rather will eventually damp out” [23].

1.7.4 Lax’s Equivalence Theorem

“The Lax equivalence theorem is the fundamental theorem in the analysis

of finite difference methods for the numerical solution of partial differential

equations.

It states that given a well posed linear initial-value problem and a linear

finite-difference approximation to it that satisfies the consistency condition,

stability is the necessary and sufficient condition for convergence” [55].

1.7.5 Fractional Calculus

Fractional calculus is a branch of mathematics in which the study of frac-

tional integrals and fractional derivatives has been discussed. The subject

fractional calculus is as old as the conventional calculus, had not been very

popular for a long time. In the last few decades, it has been observed by

applied scientists and mathematicians and engineers that many of the phys-

ical problems such as viscoelastic systems, signal processing and diffusion


processes etc. can be effectively modeled by partial differential equations

with fractional derivatives.

There exists a large literature on different definitions of fractional deriva-

tives. Riemann- Liouville and the Caputo fractional derivatives are com-

monly used.

• Riemann-Liouville Fractional Derivative

“ Dαx (f(x)) =

1

Γ(n− α)

dn

dxn

∫ x

0

(x− t)n−α−1f(t)dt ”[1].

• Caputo Fractional Derivative

“ c0D

αx (f(x)) =

1

Γ(n− α)

∫ x

0

(x− t)n−α−1dnf(t)

dtndt,

n− 1 < α ≤ n ”[1].

1.7.6 B-spline

The basic work on the B-spline (basis spline) was first performed by Schoen-

berg in 1946. After that Cox and de Boor followed the work of Schoenberg

and developed the fundamental algorithms for constructing B-spline basis

functions. B-spline plays a vital role in finding the numerical solution of

ODEs and PDEs and also proved to be very attractive and viable in com-

puter aided geometric design.

B-spline is a generalized form of the Bezier curve. There are two main

disadvantages in Bezier representation.


• The number of control points is directly related to the degree. There-

fore, to increase the complexity of the shape of the curve by adding

control points requires increasing the degree of the curve or satisfying

the continuity conditions between consecutive segments of a compos-

ite curve.

• No local control-only global control.

The entire curve or surface is changed by changing the position on

any control point, making design of specific sections very difficult.

“A B-spline is a spline function that has minimal support with respect to

a given degree, smoothness, and domain partition. A detailed description

of B-splines can be found in (deBoor [9], Prenter [42]).

A fundamental theorem states that every spline function of a given degree,

smoothness, and domain partition can be represented as a linear combina-

tion of B-splines of that same degree and smoothness, and over that same

partition” [22].

1.7.7 B-spline Basis Functions

The B-spline basis function of degree d, denoted Pi,d(t), defined by the knot

vector x0, x1, . . . xm are defined recursively as follows

Pi,0(x) =

1, x ∈ [xi, xi+1[,

0, otherwise


Pi,d(x) =x− xi

xi+d − xi

Pi,d−1(x) +xi+d+1 − x

xi+d+1 − xi+1

Pi+1,d−1(x) (1.7.1)

for i = 0, 1, . . . , n and d ≥ 1.

This formula is known as Cox de- Boor recursion formula.

Zero Degree B-Spline

For degree d = 0, the basis function is just a step function. Thus, the zero

degree B-spline is one of the simplest B-spline basis function and is given

as

Pi,0(x) =

1, x ∈ [xi, xi+1[,

0, otherwise

First Degree B-Spline

The first degree B- spline also known as linear B-spline and can be obtained

using the Cox de- Boor recursion formula given by (1.7.1) by putting d = 1

as

Pi,1(x) =

x−xi

xi+1−xi, x ∈ [xi, xi+1[,

xi+2−xxi+2−xi+1

, x ∈ [xi+1, xi+2[

0, otherwise

Second Degree B-Spline

The second degree B- spline also known as quadratic B-spline and can

be obtained using the Cox de- Boor recursion formula given by (1.7.1) by

putting d = 2 as

Pi,2(x) =

(x−xi)2

(xi−2−xi)(xi+1−xi), x ∈ [xi, xi+1[,

(x−xi)(xi+2−x)(xi+2−xi)(xi+2−xi+1)

+ (x−xi+1)(xi+3−x)(xi+2−xi+1)(xi+3−xi+1)

, x ∈ [xi+1, xi+2[(xi+3−x)2

(xi+3−xi+1)(xi+3−xi+2), x ∈ [xi+2, xi+3[,

0, otherwise


B-Spline Curve

The B-spline curve of degree d (or order d+1) with control points c0, c1, . . . , cn

and knots x0, x1, . . . , xm is defined on the interval [a, b] = [xd, xm−d] by

B(x) =n∑

i=0

ciBi,d(x),

where Bi,d(x) are the B-spline basis functions of degree d.

1.7.8 Collocation Method

Collocation methods were developed for finding the numerical solution of

initial or boundary value problems. Using the collocation methods, the

numerical solution is obtained in the form of linear combination of basis

functions. Using this method, the unknown function is approximated by

passing the piecewise defined polynomial through values of the function at

selected points. The selected points are known as collocation points.

To apply the collocation method, it is first needed to choose a proper basic

functions φ1, φ2, . . . , φM and to subdivide the problem domain [a, b] as a =

x0 < x1 < x2, . . . , xM−1, xM = b. The points x0, x1, . . . , xM−1, xM are called

the nodes or collocation points of the domain [a, b].

The approximate solution can, then, be written in the following form

W (x) =M∑i=0

ciφi(x)

The method requires the approximation to satisfy the given differential

equation at each of the nodes and also satisfy the boundary conditions.


1.7.9 Collocation Method with B-spline Basis Func-

tions

Let ∆∗ = a = x0 < x1 < x2 < · · · < xM = b be the partition of [a, b].

The spacial step length is denoted by h, h = xi − xi−1, i = 1, 2, . . . , M . To

find the numerical solution of partial differential equations using colloca-

tion method with B-spline as basis function, the numerical solution can be

written as a linear combination of basis function

W (x) =i+d−2∑

j=i−d+2

pjPj,d(x), (1.7.2)

where d is the degree of B-spline, i is the number of nodes and pj are the

unknown constants to be determined from the boundary conditions and

collocation form of the partial differential equation.

1.7.10 Cubic B-spline Collocation Method

The basis functions of cubic B-spline Pi,3(x), i = −1, 0, . . . , M + 1 can be

defined, as under

Pi,3(x) =1

h3

(x− xi−2)3, x ∈ [xi−2, xi−1[,

h3 + 3h2(x− xi−1) + 3h(x− xi−1)2 − 3(x− xi−1)

3, x ∈ [xi−1, xi[,

h3 + 3h2(xi+1 − x) + 3h(xi+1 − x)2 − 3(xi+1 − x)3, x ∈ [xi, xi+1[,

(xi+2 − x)3, x ∈ [xi+1, xi+2[,

0, otherwise

The values of successive derivatives P(r)i,3 (x), i = −1, . . . , M + 1; r = 0, 1, 2

at nodes, are listed in Table 1.1.


Table 1.1: Coefficients of cubic B-spline and its derivatives at knots xi.xi−2 xi−1 xi xi+1 xi+2 else

Pi,3(x) 0 1 4 1 0 0P

(1)i,3 (x) 0 3

h 0 −3h 0 0

P(2)i,3 (x) 0 6

h2−12h2

6h2 0 0

The numerical solution can, thus, be obtained by substituting d = 3 in

Eq.(1.7.2),

W (x) =i+1∑

j=i−1

pjPj,3(x), (1.7.3)

Eq.(1.7.3) can be written as

W (xi) = pi−1Pi−1,3(xi) + piPi,3(xi) + pi+1Pi+1,3(xi).

or

W (xi) = pi−1Pi,3(xi+1) + piPi,3(xi) + pi+1Pi,3(xi−1). (1.7.4)

The successive derivatives W (r)(x), r = 1, 2, can be determined in terms of

the time parameter pi, as under

W (1)(xi) = pi−1P(1)i,3 (xi+1) + piP

(1)i,3 (xi) + pi+1P

(1)i,3 (xi−1), (1.7.5)

W (2)(xi) = pi−1P(2)i,3 (xi+1) + piP

(2)i,3 (xi) + pi+1P

(2)i,3 (xi−1). (1.7.6)

Substituting the value of Pi(x) at the nodes from Table 1.1, the Eqs. (1.7.4),

(1.7.5) and (1.7.6) take the following form

W (xi) = pi−1 + 4pi + pi+1,

W (1)(xi) =3

h(pi+1 − pi−1) ,

W (2)(xi) =6

h2(pi−1 − 2pi + pi+1) .


1.7.11 Quintic B-spline Collocation Method

The basis functions of quintic B-spline Pi,5(x), i = −2,−1, . . . ,M + 2 can

be defined, as under

Pi,5(x) =1

h5

(x− xi + 3h)5, x ∈ Ii−3,

(x− xi + 3h)5 − 6(x− xi + 2h)5, x ∈ Ii−2,

(x− xi + 3h)5 − 6(x− xi + 2h)5 + 15(x− xi + h)5, x ∈ Ii−1,

(−x + xi + 3h)5 − 6(−x + xi + 2h)5 + 15(−x + xi + h)5, x ∈ Ii,

(−x + xi + 3h)5 − 6(−x + xi + 2h)5, x ∈ Ii+1,

(−x + xi + 3h)5, x ∈ Ii+2,

0, otherwise

where Ii = [xi, xi+1[. The values of successive derivatives P(r)i,5 (x), i =

−2, . . . ,M + 2; r = 0, 1, 2, 3, 4 at nodes, are listed in Table 1.2.

Table 1.2: Coefficients of quintic B-spline and its derivatives at knots xi.xi−3 xi−2 xi−1 xi xi+1 xi+2 xi+3 else

Pi,5(x) 0 1 26 66 26 1 0 0P

(1)i,5 (x) 0 5

h50h 0 −50

h−5h 0 0

P(2)i,5 (x) 0 20

h240h2

−120h2

40h2

20h2 0 0

P(3)i,5 (x) 0 60

h3−120h3 0 120

h3−60h3 0 0

P(4)i,5 (x) 0 120

h4−480h4

720h4

−480h4

120h4 0 0

The numerical solution can be obtained by substituting d = 5 in Eq.(1.7.2),

W (x) =i+3∑

j=i−3

pjPj,5(x), (1.7.7)

Eq.(1.7.7) can be written as

W (xi) = pi−3Pi−3,5(xi) + pi−2Pi−2,5(xi) + pi−1Pi−1,5(xi) + piPi,5(xi)

+pi+1Pi+1,5(xi) + pi+2Pi+2,5(xi) + pi+3Pi+3,5(xi).


or

W (xi) = pi−3Pi,5(xi+3) + pi−2Pi,5(xi+2) + pi−1Pi,5(xi+1) + piPi,5(xi)

+pi+1Pi,5(xi−1) + pi+2Pi,5(xi−2) + pi+3Pi,5(xi−3). (1.7.8)

The successive derivatives W (r)(x), r = 1, 2, 3, 4, can be determined in terms

of the time parameter pi, as under

W (1)(xi) = pi−3P(1)i,5 (xi+3) + pi−2P

(1)i,5 (xi+2) + pi−1P

(1)i,5 (xi+1) + piP

(1)i,5 (xi)

+pi+1P(1)i,5 (xi−1) + pi+2P

(1)i,5 (xi−2) + pi+3P

(1)i,5 (xi−3), (1.7.9)

W (2)(xi) = pi−3P(2)i,5 (xi+3) + pi−2P

(2)i,5 (xi+2) + pi−1P

(2)i,5 (xi+1) + piP

(2)i,5 (xi)

+pi + 1P(2)i,5 (xi−1) + pi+2P

(2)i,5 (xi−2) + pi+3P

(2)i,5 (xi−3), (1.7.10)

W (3)(xi) = pi−3P(3)i,5 (xi+3) + pi−2P

(3)i,5 (xi+2) + pi−1P

(3)i,5 (xi+1) + piP

(3)i,5 (xi)

+pi+1P(3)i,5 (xi−1) + pi+2P

(3)i,5 (xi−2) + pi+3P

(3)i,5 (xi−3), (1.7.11)

W (4)(xi) = pi−3P(4)i,5 (xi+3) + pi−2P

(4)i,5 (xi+2) + pi−1P

(4)i,5 (xi+1) + piP

(4)i,5 (xi)

+pi+1P(4)i,5 (xi−1) + pi+2P

(4)i,5 (xi−2) + pi+3P

(4)i,5 (xi−3). (1.7.12)

On substituting the value of Pi,5(x) at the nodes from Table 1.2, the Eqs.

(1.7.8) to (1.7.12) take the following form

W (xi) = pi+2 + 26pi+1 + 66pi + 26pi−1 + pi−2,

W (1)(xi) =5

h(pi+2 + 10pi+1 − 10pi−1 − pi−2) ,

W (2)(xi) =20

h2(pi+2 + 2pi+1 − 6pi + 2pi−1 + pi−2) ,

W (3)(xi) =60

h3(pi+2 − 2pi+1 + 2pi−1 − pi−2) ,

W (4)(xi) =120

h4(pi+2 − 4pi+1 + 6pi − 4pi−1 + pi−2) .


1.8 Literature Survey

In 1978, Miller [36] proposed a solution for an integro-differential equation

for heat flow in materials with memory.

In 1990, Jain et al. [24] proposed difference methods for the numerical so-

lution of system of one-dimensional non-linear parabolic PDEs.

In 1991, Evans and Yousif [14] proposed the AGE method for the numerical

solution of the fourth order parabolic equation.

In 1993, T. Tang [58] used Crank-Nicolson approach for the numerical solu-

tion of PIDE with a weakly singular kernel. An error bound is also derived

for the numerical scheme.

In 1992, Chen et al. [8] developed a FEM for the numerical solution of

parabolic integro-differential equation with a weakly singular kernel.

In 1994, Y-q Huang [21] developed a time discretization scheme for the so-

lution of parabolic integro-differential equations.

In 1994, Fairweather [15] proposed spline collocation techniques for the spa-

tial discretization of a class of hyperbolic PIDEs arising in the theory of lin-

ear viscoelasticity. For problems in two space variables, error estimates are

derived for the continuous-time orthogonal spline collocation method and

three discrete-time orthogonal spline collocation methods. The continuous-

time modified cubic spline collocation method for problems in one space

variable is also analyzed.


In 1998, Bratsos [5] presented the method of lines for the approximate so-

lution of the Boussinesq equation.

In 2001, Wazwaz [59] developed modified decomposition method for con-

struction of soliton solutions and periodic solutions of the Boussinesq equa-

tion.

In 2001, Bratsos [6] derived a parametric scheme for the approximate solu-

tion of the Boussinesq equation.

In 2004, Dehghan [10] developed a approximate solution of the three-dimensional

advection-diffusion equation.

In 2004, Dehghan [11] developed finite difference formulas for the approxi-

mate solution of one-dimensional advection-diffusion equation.

In 2005, Aziz et al. [3] proposed a numerical method based on para-

metric quintic spline for space discretization and finite difference formulas

for temporal discretization for the solution of non-homogenous fourth-order

parabolic PDE. Stability analysis of the method has also been carried out.

In 2005, Khan et al. [26] used sextic spline for the numerical solution

of fourth-order parabolic PDE. The approximate solution is obtained by

using a new three-level method based on a sextic spline in space and finite-

difference discretization in time. Stability analysis of the method has also

been carried out.

In 2006, Dehghan [12] used finite difference methods for the approximate

solution of PIDE.


In 2006, Rawashdeh [45] used the collocation spline method to approxi-

mate the solution of semi-differential equation, which was a special kind

of fractional integro-differential equations, then presented the derivation of

the collocation spline method of fractional derivative for fractional integro-

differential equations on polynomial spline space in [44].

In 2007, Mohanty [38] proposed a new two-level implicit difference scheme

for the numerical solution of non-linear singular parabolic partial differen-

tial equations. The problem is solved for appropriate initial and Dirichlet

boundary conditions.

In 2008, Caglar and Caglar [7] developed a approximate method for solving

fourth-order parabolic PDEs using fifth-degree B-spline.

In 2010, Rashidinia et al. [43] proposed a collocation method for the nu-

merical solution of non-linear one-dimensional parabolic PDEs subject to

Dirichlet boundary conditions. The convergence analysis of the method has

also been proven.

In 2010, Wulan Li and Xu Da [61] used finite central difference and finite

element approximations for the approximate solution of parabolic integro-

differential equations.

In 2011, Xuehua Yang et al. [62] proposed a quasi-wavelet based numerical

method for solving fourth-order PIDEs with a weakly singular kernel. The

forward Euler scheme has been used for temporal discretization and the


quasi-wavelet based numerical method has been used for spatial discretiza-

tion. The given problem is solved subject to three boundary conditions, in-

cluding clamped condition, simply supported condition and a transversely

supported condition.

In 2011, Golbabai and Sayevand [18] developed an approach to analysis the

generalized fourth-order diffusion-wave equations.

In 2011, Yasir Nawaz [39] used Variational iteration method and homotopy

perturbation method for solving linear and non-linear fourth-order FIDEs.

The fractional derivatives are described in the Caputo sense. The solutions

of both problems are derived from infinite convergent series.

In 2011, Fadi Awawdeh et al. [2] used homotopy analysis method for the

solution of fractional integro- differential equations.

In 2012, Mittal and Jain [37] developed redefined collocation method with

cubic B-spline as basis functions for the numerical solution of convection-

diffusion equations.

In 2012, Long et al. [33] proposed a quasi-wavelet based numerical method

for a class of PIDE.

In 2013, Zhang et al. [65] proposed a collocation method for the approxi-

mate solution of fourth-order PIDEs with a weakly singular kernel.

In 2013, Zhang and Han [64] proposed a quasi-wavelet based numerical

method for time-dependent FPDE. The second order backward differentia-

tion formula has been used for temporal discretization and the quasi-wavelet


method has been used for spatial discretization. The stability and conver-

gence analysis of temporal discretization has been discussed and proven,

theoretically.

In 2013, Yang et al. [63] developed a new approximate scheme for the

solution of fourth-order PIDE with a weakly singular kernel. In the time

direction, a Crank-Nicolson time-stepping is used to approximate the dif-

ferential term and the product trapezoidal method is employed to treat

the integral term, and the quasi-wavelets numerical method for space dis-

cretization.

In 2014, Jingjun Zhao et al. [66] used piecewise polynomial collocation

techniques for the solution of FIDEs with weakly singular kernels.

In 2014, Eslahchi et al. [13] used Jacobi collocation method for solving

non-linear fractional integro-differential equations.

Chapter 2

Quintic B-Spline for theNumerical Solution of theGood Boussinesq Equation

In this chapter, collocation method using B-spline as basis functions is

applied to develop numerical solution of non-linear Boussinesq equation.

The contents of this chapter have been published in the form of research

paper [48].

2.1 Introduction

An approximation for water waves, particularly for long water waves, is

the Boussinesq approximation which plays a vital role in fluid dynamics.

John Scott Russell made an observation about the wave of translation also

termed as solitary waves or soliton. After that Joseph Boussinesq derived

the equation, now known as the Boussinesq equation (BE), in his research

paper in 1872. The long water waves that are found in different situations

such as waves near sea beaches, rivers and lakes etc. are usually governed

31

Ch 2: Solution of Boussinesq Equation 32

by the Boussinesq equation. The long waves are also modeled by the well

known Korteweg-deVries (KdV) equation. However, the Boussinesq equa-

tion provides a better approximation to long waves. BE possesses soliton

solutions, which are of wide interest in few areas of physical sciences, as, for

example, in plasma physics, fluid dynamics and the study of water waves.

Following is the non-linear Boussinesq equation, describing long water waves

∂2w

∂t2=

∂2h(w)

∂x2+ u

∂4w

∂x4, x ∈ [a, b], 0 < t ≤ T. (2.1.1)

where w = w(x, t), h(w) = w(1+w) and |u| = 1 is a real valued parameter.

u = −1 represents the good Boussinesq equation, while u = 1 represents

the bad Boussinesq equation.

The initial conditions associated with Eq.(2.1.1) are given by

w(x, t0) = f1(x)

and

wt(x, t0) = f2(x)

along with the following boundary conditions

w(a, t) = 0, w(b, t) = 0; 0 < t ≤ T

wxx(a, t) = 0, wxx(b, t) = 0; 0 < t ≤ T.

The exact solution (soliton solution) of the Eq.(2.1.1) is given by [34]

w(x, t) = qB

[sech2

[√B

6

(x− ct + x0

)]

+(β − q

2

)]


where B is the pulse’s amplitude, β is any parameter, x0 is the starting

position of the pulse and c = ±√

2q(β + B3) is the soliton’s velocity.

In the following section, a finite difference approximation is used for tem-

poral discretization of Eq.(2.1.1).

2.2 Temporal discretization

Following is the good Boussinesq equation

∂2w

∂t2=

∂2w

∂x2+

∂2w2

∂x2− ∂4w

∂x4(2.2.1)

The region [a, b] × [0, T ] has been discretized as grid points (xi, tj) where

xi = a + ih, i = 0, 1, 2, . . . ,M and tj = j∆t, j = 0, 1, 2, . . . , K, K∆t = T .

The quantities h and ∆t are the mesh size in the space and time directions,

respectively.

The time derivative has been approximated by the following finite difference

formula

∂2wn

∂t2=

wn+1 − 2wn + wn−1

∆t2(2.2.2)

Substituting Eq.(2.2.2) into Eq.(2.2.1), the following time discretization of

Eq.(2.2.1) is obtained as

wn+1 − 2wn + wn−1

∆t2=

∂2wn

∂x2+

∂2(w2)n

∂x2− ∂4wn

∂x4(2.2.3)


Applying the θ-weighted scheme (0 ≤ θ ≤ 1) to space derivatives to

Eq.(2.2.3), it can take the following form

wn+1 − 2wn + wn−1

∆t2= θ

(∂2wn+1

∂x2+

∂2(w2)n+1

∂x2− ∂4wn+1

∂x4

)

+ (1− θ)

(∂2wn

∂x2+

∂2(w2)n

∂x2− ∂4wn

∂x4

)

where the superscripts n− 1, n, n + 1 denote the adjacent time levels.

Taking θ = 12, the above equation can be rewritten as

wn+1 − 2wn + wn−1

∆t2=

(wn+1xx + wn

xx)

2− (wn+1

xxxx + wnxxxx)

2

+(w2

xx)n+1 + (w2

xx)n

2(2.2.4)

The Taylor expansion is used to linearize the non-linear term in Eq.(2.2.4)

as

(w2xx)

n+1 = 2wnxxw

n+1xx − (w2

xx)n (2.2.5)

Substituting Eq.(2.2.5) into Eq.(2.2.4), Eq.(2.2.4) takes the following form

2wn+1 −∆t2(wn+1

xx − wn+1xxxx + 2wn

xxwn+1xx

)= 4wn − 2wn−1 + ∆t2 (wn

xx − wnxxxx)

(2.2.6)

In the next section, the collocation method based on quintic B-Spline is

used for space discretization.


2.3 Quintic B-spline collocation method

Using

W n+1(x) =M+2∑j=−2

pjPj(x), (2.3.1)

where pj are unknown time dependent quantities to be determined from the

boundary conditions and collocation form of the partial differential equation

as defined in subsection 1.7.11.

The space discretization of Eq.(2.2.6) is carried out, using Eq.(2.3.1), and

the collocation method is applied by identifying the collocation points as

nodes. For i = 0, 1, 2, . . . , M the following relation can be obtained as(

2M+2∑j=−2

pn+1j (t)Pj(xi)−∆t2

M+2∑j=−2

pn+1j (t)P

(2)j (xi) + ∆t2

M+2∑j=−2

pn+1j (t)P

(4)j (xi)

− 2∆t2M+2∑j=−2

pnj (t)P

(2)j (xi)

M+2∑j=−2

pn+1j (t)P

(2)j (xi)

)

=

(4

M+2∑j=−2

pnj (t)Pj(xi)− 2

M+2∑j=−2

pn−1j (t)Pj(xi) + ∆t2

M+2∑j=−2

pnj (t)P

(2)j (xi)

−∆t2M+2∑j=−2

pnj (t)P

(4)j (xi)

)

Simplifying, the above equation yields to the following system of (M+1) lin-

ear equations in (M+5) unknowns pn+1−2 , pn+1

−1 , pn+10 , pn+1

1 , . . . , pn+1M , pn+1

M+1, pn+1M+2,

Ajpn+1j−2 + Bjp

n+1j−1 + Djp

n+1j + Bjp

n+1j+1 + Ajp

n+1j+2 = Hj, n ≥ 1(2.3.2)

where

Hj = Epnj−2 + Fpn

j−1 + Gpnj + Fpn

j+1 + Epnj+2

−2(pn−1

j−2 + 26pn−1j−1 + 66pn−1

j + 26pn−1j+1 + pn−1

j+2

), j = 0, 1, 2, . . . , M,


and

Y j =20

h2

(pn

j−2 + 2pnj−1 − 6pn

j + 2pnj+1 + pn

j+2

), j = 0, 1, 2, 3, . . . , M

Aj = 2− 20∆t2

h2+ 120

∆t2

h4− 40Y j ∆t2

h2, j = 0, 1, 2, 3, . . . ,M

Bj = 52− 40∆t2

h2− 480

∆t2

h4− 80Y j ∆t2

h2, j = 0, 1, 2, 3, . . . , M

Dj = 132 + 120∆t2

h2+ 720

∆t2

h4+ 240Y j ∆t2

h2, j = 0, 1, 2, 3, . . . , M

E = 4 + 20∆t2

h2− 120

∆t2

h4,

F = 104 + 40∆t2

h2+ 480

∆t2

h4,

G = 264− 120∆t2

h2− 720

∆t2

h4.

A unique solution of the above system is obtained by eliminating the para-

meters p−2, p−1, pM+1, pM+2 using the following boundary conditions

w(a, t) = (p−2 + 26p−1 + 66p0 + 26p1 + p2) = 0,

wxx(a, t) =20

h2(p−2 + 2p−1 − 6p0 + 2p1 + p2) = 0,

w(b, t) = (pM−2 + 26pM−1 + 66pM + 26pM+1 + pM+2) = 0,

wxx(b, t) =20

h2(pM−2 + 2pM−1 − 6pM + 2pM+1 + pM+2) = 0.

The values are obtained as under

p−2 = 12p0 − p2, p−1 = −3p0 − p1,

pM+1 = 12pM − pM−2, pM+2 = −3pM − pM−1,

On substituting the values of parameters p−2, p−1, pM+1, pM+2, the system

(2.3.2) reduces to a diagonal system of (M +1) linear equations in (M +1)


unknowns written in the following matrix form

PCn+1 = H,

where

Cn+1 =[pn+1

0 , pn+11 , . . . , pn+1

M

]T,

The coefficient matrix P is given as under

P =

Mj

Nj Oj Bj Aj

Aj Bj Dj Bj Aj

Aj Bj Dj Bj Aj

. . . . . . . . . . . . . . .

Aj Bj Dj Bj Aj

Aj Bj Oj Nj

Mj

,

where

Mj = 12Aj − 3Bj + Dj,

Nj = −3Aj + Bj,

Oj = −Aj + Dj.

2.4 The Initial Vector

The initial vector C0 can be obtained from the initial condition w(x, t0) =

f1(x) which gives (M +1) equations in (M +5) unknowns. For determining


these unknowns, the following relations at the knots are used

wx(x0, t0) = w′(x0), wx(xM , t0) = w

′(xM)

wxx(x0, t0) = w′′(x0), wxx(xM , 0) = w

′′(xM)

which yields a penta-diagonal system of equations written in the following

matrix form

XC0 = Y,

where

X =

54 60 6

1014

1352

1054

1

1 26 66 26 1

1 26 66 26 1. . . . . . . . . . . . . . .

1 26 66 26 1

1 1054

1352

1014

6 60 54

,

C0 =

p00

p01

p02

...

p0M−1

p0M

,


and

Y =

f1(x0) + 3hw′(xi)

5+ 2h2w

′′(xi)

20

f1(x1) + hw′(xi)

40+ h2w

′′(xi)

160

f1(x2)...

f1(xN−2)

f1(xN−1)− hw′(xi)

40+ h2w

′′(xi)

160

f1(xN)− 3hw′(xi)

5+ 2h2w

′′(xi)

20

, i = 0, 1, 2, . . . , M.

In the next section, Von-Neumann stability method has been applied to

examine the stability of the proposed scheme.

2.5 Stability Analysis

Von-Neumann stability method is used for discussing the stability of the

presented method. Von-Neumann stability analysis is a method to investi-

gate the stability of the difference schemes as implemented to linear PDEs.

Since the method is applicable to linear schemes only, the non-linear term

g(w) = w(1 + w) is linearized by taking (1 + w) as a constant value α.

The linearized form of the proposed scheme takes the following form

(epn+1

j−2 + fpn+1j−1 + dpn+1

j + fpn+1j+1 + epn+1

j+2

)

=((

ppnj−2 + spn

j−1 + rpnj + spn

j+1 + ppnj+2

)

−2(pn−1

j−2 + 26pn−1j−1 + 66pn−1

j + 26pn−1j+1 + pn−1

j+2

)), (2.5.1)


where

e = 2− 20α∆t2

h2+ 120

∆t2

h4,

f = 52− 40α∆t2

h2− 480

∆t2

h4,

d = 132 + 120α∆t2

h2+ 720

∆t2

h4,

p = 4 + 20α∆t2

h2− 120

∆t2

h4,

s = 104 + 40α∆t2

h2+ 480

∆t2

h4,

r = 264− 120α∆t2

h2− 720

∆t2

h4.

Putting pnj = zneiβjh, i =

√−1 in Eq.(2.5.1), it can be simplified as

(2e cos(2βh) + 2f cos(βh) + d) z2 − (2p cos(2βh) + 2s cos(βh) + r) z

+2 (2 cos(2βh) + 52 cos(βh) + 66) = 0. (2.5.2)

Let

L = (2e cos(2βh) + 2f cos(βh) + d)

S = (2p cos(2βh) + 2s cos(βh) + r) (2.5.3)

T = 2(2 cos(2βh) + 52 cos(βh) + 66)

Then the Eq.(2.5.2) can be rewritten as

Lz2 − Sz + T = 0 (2.5.4)

Applying the Routh-Hurwitz criterion on Eq.(2.5.4), the necessary and suf-

ficient condition for Eq.(2.5.1) to be stable can be determined as follows:


Using the transformation z = 1+g1−g

and simplifying, Eq.(2.5.4) takes the form

(L + S + T ) g2 + 2 (L− T ) g + (L− S + T ) = 0 (2.5.5)

The necessary and sufficient condition for |z| ≤ 1 leads to

L + S + T ≥ 0, L− T ≥ 0, L− S + T ≥ 0. (2.5.6)

Since,

L + S + T = (2e + 2p + 4) cos(2βh) + (2f + 2s + 104) cos(βh) + (d + r + 132),

L− S + T = (2e− 2p + 4) cos(2βh) + (2f − 2s + 104) cos(βh) + (d− r + 132),

L− T = (2e− 4) cos(2βh) + (2f − 104) cos(βh) + (d− 132). (2.5.7)

Therefore, it can be proved that

L + S + T = 64(15 cos2(φ) + 2 sin4(φ)

) ≥ 0,

L− S + T = 320k2

h4

(12 sin4(φ) + h2 sin2(φ) cos(2φ) + 2h2 sin2(φ)

) ≥ 0,

L− T = 160k2

h4

(12 sin4(φ) + h2 sin2(φ) cos(2φ) + 2h2 sin2(φ)

) ≥ 0,

(2.5.8)

where φ = 12βh.

It may be mentioned, from the above stability analysis procedure, that the

proposed scheme is unconditionally stable. (as there is no restriction on the

space mesh size h).

In the following section, an example is constructed to illustrate the applica-

bility and effectiveness of the method developed.


2.6 Numerical Results

Example 2.1 The good Bousssinesq equation is considered as,

∂2w

∂t2=

∂2

∂x2(w(1 + w))− ∂4w

∂x4, x ∈ [0, 1], 0 < t ≤ T.

subject to the following initial conditions

w(x, 0) = −Bsech2

[√B

6(x + x0)

]−

(β +

1

2

)

wt(x, 0) = −√

2

3B3/2sech2

[√B

6(x + x0)

]tanh

[√B

6(x + x0)

]

along with homogenous boundary conditions

w(0, t) = w(1, t) =∂2w

∂x2(0, t) =

∂2w

∂x2(1, t) = 0, 0 < t ≤ T.

The exact solution of the problem is

w(x, t) = −Bsech2

[√B

6(x− ct + x0)

]−

(β +

1

2

)

This problem is solved using theoretical parameters B = 0.369, β = −12,

velocity c = 0.868 and with initial time t0 = 0. The observed maximum

absolute errors for different values of h and a fixed value of ∆t = 0.05 are

shown in Table 2.1.

Table 2.1: Numerical results for Example 2.1Time Parameters Maximum Absolute Errors

t = 0.5 h = 1/40, x0 = 30 8.2943e− 007t = 1.0 h = 1/60, x0 = 40 7.3326e− 009t = 1.5 h = 1/80, x0 = 50 6.4525e− 011t = 2.0 h = 1/100, x0 = 60 5.2066e− 013


2.7 Conclusion

In this chapter, a numerical method has been presented for the numerical

solution of good Boussinesq equation. The finite difference approximation

is used for temporal discretization and quintic B-spline collocation method

for spatial discretization. The proposed method has been shown to be

unconditionally stable. It is clear from the example that the numerical

solution obtained using the presented method efficiently approximates the

exact solution.

Chapter 3

Solution of Fourth-OrderPartial Differential Equation

In this chapter, a numerical method is developed to solve the fourth-order

parabolic PDE using the collocation technique with quintic B-spline basis

functions. Stability analysis of the method has also been proved.

The contents of this chapter have been published in the form of research

paper [53].

3.1 Introduction

The undamped transverse vibrations of a flexible straight beam, whose sup-

ports do not contribute to the strain energy of the system, can be modeled

by the following fourth-order parabolic partial differential equation

∂2w

∂t2+ m

∂4w

∂x4= h(x, t), x ∈ [0, 1], m > 0, 0 < t ≤ T (3.1.1)

44

Ch 3: Solution of Fourth-order PDE 45

subject to the initial conditions

w(x, 0) = v0(x)

wt(x, 0) = v1(x) (3.1.2)

along with the boundary conditions

w(0, t) = d0(t) w(1, t) = d1(t)

wxx(0, t) = q0(t) wxx(1, t) = q1(t), (3.1.3)

where m is the ratio of flexural rigidity of the beam to its mass per unit

length, w is the transverse displacement of the beam, t and x are the time

and distance variables, respectively. h(x, t) is the dynamic driving force per

unit mass and v0(x), v1(x), d0(t), d1(t), q0(t), q1(t) are continuous functions.

In the following section, time discretization of Eq.(3.1.1) has been carried

out using finite difference approximation.

3.2 Discretization in time

The region [0, 1] × [0, T ] has been discretized as grid points (xi, tj) where

xi = ih, i = 0, 1, 2, . . . ,M and tj = j∆t, j = 0, 1, 2, . . . , K, K∆t = T .


respectively.

The time derivative has been approximated by central difference formula as

∂2wn

∂t2∼= δ2

t wn

∆t2(1 + sδ2t )

, (3.2.1)


where s is a parameter such that the central difference approximation to

the time derivative is O(∆t2) for arbitrary s and O(∆t4) for s = 112

. It may

be mentioned that in Eq.(3.2.1), δ2t w

n = wn+1− 2wn + wn−1, wn = w(x, tn)

and w0 = w(x, 0) = v0(x).

Substituting Eq.(3.2.1) in Eq.(3.1.1), the temporal discretization of the

given problem takes the following form

δ2t w

n

∆t2(1 + sδ2t )

+ mwnxxxx = h(x, tn)

The above equation can be rewritten as

wn+1 + ms∆t2wn+1xxxx

=(2wn − wn−1 −m∆t2wn

xxxx + 2ms∆t2wnxxxx −ms∆t2wn−1

xxxx

+∆t2h(x, tn) + ∆t2s (h(x, tn+1)− 2h(x, tn) + h(x, tn−1)))(3.2.2)

In each time-level, there is an ordinary differential equation in the form of

Eq.(3.2.2) along with the boundary conditions Eq.(3.1.3), which is to be

solved by collocation method using quintic B-spline as basis functions.

The proposed scheme Eq.(3.2.2) is a three time-level scheme. For imple-

mentation of the presented scheme, it is first needed to have the values of

w at the nodal points at the zeroth w0 and first w1 time-levels.

To compute w1, the initial conditions w(x, 0) = v0(x) and wt(x, 0) = v1(x),

are used. Since w0 = w(x, 0) = v0(x) is the value of w at the zeroth level

time, therefore using Taylor series, w1 has been computed as

w1 = w0 + ∆tw0t +

∆t2

2!w0

tt +∆t3

3!w0

ttt +∆t4

4!w0

tttt + O(∆t5) (3.2.3)


w0 and w0t are known from the initial conditions. This implies that all

successive tangential derivatives are known at initial time t = 0, which

further shows that w, wx, wxx,. . . ,wt, wtx, wtxx,. . . are known at t = 0.

In the following section, quintic B-spline collocation method has been used

for space discretization of Eq.(3.2.2)

3.3 Quintic B-spline Collocation Method

The interval [0, 1] of domain has been subdivided as

0 = x0 < x1 < x2 < · · · < xM = 1

The basis functions Pj(x), j = −2, . . . ,M +2 of quintic B-spline are defined

as

Pj(x) =1

h5

(x− xj + 3h)5, x ∈ Ij−3,

(x− xj + 3h)5 − 6(x− xj + 2h)5, x ∈ Ij−2,

(x− xj + 3h)5 − 6(x− xj + 2h)5 + 15(x− xj + h)5, x ∈ Ij−1,

(−x + xj + 3h)5 − 6(−x + xj + 2h)5 + 15(−x + xj + h)5, x ∈ Ij,

(−x + xj + 3h)5 − 6(−x + xj + 2h)5, x ∈ Ij+1,

(−x + xj + 3h)5, x ∈ Ij+2,

0, otherwise

where Ij = [xj, xj+1).

The values of successive derivatives P(r)j (x), j = −2, . . . , M + 2; r =

0, 1, 2, 3, 4 at nodes, are given in Table 3.1.

The numerical treatment for fourth-order PDE using B-spline collocation

method is to find an approximate solution W (x, t) to the exact solution


Table 3.1: Coefficients of quintic B-spline and its derivatives at knots xj.xj−3 xj−2 xj−1 xj xj+1 xj+2 xj+3 else

Pj(x) 0 1 26 66 26 1 0 0P ′

j(x) 0 5h

50h 0 −50

h−5h 0 0

P ′′j (x) 0 20

h240h2

−120h2

40h2

20h2 0 0

P ′′′j (x) 0 60

h3−120h3 0 120

h3−60h3 0 0

P(4)j (x) 0 120

h4−480h4

720h4

−480h4

120h4 0 0

w(x, t) in the following form

W (x, t) =M+2∑j=−2

pjPj(x), (3.3.1)

where pj are time dependent parameters which will be determined for each

time level and Pj are quintic B-spline basis functions.

Using Eq.(3.3.1) and quintic B-spline basis functions Pj, the numerical val-

ues at the knots of W (x) and its successive derivatives W (r)(x), r = 1, 2, 3, 4,

have been determined in terms of the time parameter pj as

Wj = pj+2 + 26pj+1 + 66pj + 26pj−1 + pj−2,

hW′j = 5 (pj+2 + 10pj+1 − 10pj−1 − pj−2) ,

h2W′′j = 20 (pj+2 + 2pj+1 − 6pj + 2pj−1 + pj−2) ,

h3W′′′j = 60 (pj+2 − 2pj+1 + 2pj−1 − pj−2) ,

h4W(4)j = 120 (pj+2 − 4pj+1 + 6pj − 4pj−1 + pj−2) .


Substituting Eq.(3.3.1) into Eq.(3.2.2) yields the following equation

((pn+1

j+2 + 26pn+1j+1 + 66pn+1

j + 26pn+1j−1 + pn+1

j−2

)

+120ms∆t2

h4

(pn+1

j+2 − 4pn+1j+1 + 6pn+1

j − 4pn+1j−1 + pn+1

j−2

))

=(2(pn

j+2 + 26pnj+1 + 66pn

j + 26pnj−1 + pn

j−2

)

− (pn−1

j+2 + 26pn−1j+1 + 66pn−1

j + 26pn−1j−1 + pn−1

j−2

)

−120m∆t2

h4

(pn

j+2 − 4pnj+1 + 6pn

j − 4pnj−1 + pn

j−2

)

+240ms∆t2

h4

(pn

j+2 − 4pnj+1 + 6pn

j − 4pnj−1 + pn

j−2

)

−120ms∆t2

h4

(pn−1

j+2 − 4pn−1j+1 + 6pn−1

j − 4pn−1j−1 + pn−1

j−2

)

+∆t2hnj + ∆t2s

[hn+1

j − 2hnj + hn−1

j

]).


Simplifying, the above equation yields the following system of (M + 1)

equations in (M + 5) unknowns pn+1−2 , pn+1

−1 , pn+10 , pn+1

1 , ..., pn+1M , pn+1

M+1, pn+1M+2

(pn+1

j+2

(1 +

120ms∆t2

h4

)+ pn+1

j+1

(26− 480ms∆t2

h4

)+ pn+1

j

(66 +

720ms∆t2

h4

)

+pn+1j−1

(26− 480ms∆t2

h4

)+ pn+1

j−2

(1 +

120ms∆t2

h4

))

=

(pn

j+2

(2− 120m∆t2

h4+

240ms∆t2

h4

)+ pn

j+1

(52 +

480m∆t2

h4− 960ms∆t2

h4

)

+pnj

(132− 720m∆t2

h4+

1440ms∆t2

h4

)+ pn

j−1

(52 +

480m∆t2

h4− 960ms∆t2

h4

)

+pnj−2

(2− 120m∆t2

h4+

240ms∆t2

h4

)

+pn−1j+2

(−1− 120ms∆t2

h4

)+ pn−1

j+1

(−26 +

480ms∆t2

h4

)

+pn−1j

(−66− 720ms∆t2

h4

)+ pn−1

j−1

(−26 +

480ms∆t2

h4

)

+pn−1j−2

(−1− 120ms∆t2

h4

)

+∆t2hnj + ∆t2s

(hn+1

j − 2hnj + hn−1

j

)), 0 ≤ i ≤ M. (3.3.2)

A unique solution of the above system is obtained by eliminating the para-

meters p−2, p−1, pM+1, pM+2 using the following boundary conditions

w(a, t) = (p−2 + 26p−1 + 66p0 + 26p1 + p2) = d0(t),

wxx(a, t) =20

h2(p−2 + 2p−1 − 6p0 + 2p1 + p2) = q0(t),

w(b, t) = (pM−2 + 26pM−1 + 66pM + 26pM+1 + pM+2) = d1(t),

wxx(b, t) =20

h2(pM−2 + 2pM−1 − 6pM + 2pM+1 + pM+2) = q1(t).


Solving above equations give

p−2 =1

240

(−20d0 + 13q0h2 + 2880p0 − 240p2

),

p−1 =d0

24− q0h

2

480− 3p0 − p1,

pM+1 =1

240

(−20d1 + 13q1h2 + 2880pM − 240pM−2

),

pM+2 =d1

24− q1h

2

480− 3pM − pM−1.

On substituting the values of parameters p−2, p−1, pM+1, pM+2, the system

(3.3.2) reduces to a diagonal system of (M +1) linear equations in (M +1)

unknowns written in the following matrix form

ACn+1 = h4(BCn + DCn−1 + H

), (3.3.3)

where

A =

7200∆t2ms2

p q r s

a b c b a

a b c b a. . . . . . . . . . . . . . .

a b c b a

s r q p

7200∆t2ms2

,


where

a =(h4 + 120∆t2ms

),

b =(26h4 − 480∆t2ms

),

c =(66h4 + 720∆t2ms

),

p =(23h4 − 840∆t2ms

),

q =(65h4 + 600∆t2ms

),

r =(26h4 − 480∆t2ms

),

s =(h4 + 120∆t2ms

).

B =

3600∆t2m(−1+2s)h4 0 0 0 0

x y z v 0

T R Z R T

T R Z R T. . . . . . . . . . . . . . .

T R Z R T

0 v z y x

0 0 0 0 3600∆t2m(−1+2s)h4

,


where

x = (46 +840m∆t2

h4− 1680ms∆t2

h4,

y = (130− 600m∆t2

h4+

1200ms∆t2

h4),

z = (52 +480m∆t2

h4− 960ms∆t2

h4),

v = (2− 120m∆t2

h4+

240ms∆t2

h4),

Y = (26− 480ms∆t2

h4),

Z = (132− 720m∆t2

h4+

1440ms∆t2

h4),

R = (52 +480m∆t2

h4− 960ms∆t2

h4),

T = (2− 120m∆t2

h4+

240ms∆t2

h4).

D =

−3600∆t2msh4

l m n o

L M N M L

L M N M L. . . . . . . . . . . . . . .

L M N M L

o n m l

−3600∆t2msh4

,


where

l =

(−23 +

840ms∆t2

h4

),

m =

(−65− 600ms∆t2

h4

),

n =

(−26 +

480ms∆t2

h4

),

o =

(−1− 120ms∆t2

h4

),

L =

(−1− 120ms∆t2

h4

),

M =

(−26 +

480ms∆t2

h4

),

N =

(−66− 720ms∆t2

h4

).

Cn =

pn0

pn1

pn2

...

pnM−1

pnM

,

H =

∆t2dn0 + ∆t2s

(dn+1

0 − 2dn0 + dn−1

0

)− 15∆t2m(−4d0+h2q0)2h4

∆t2dn1 + ∆t2s

(dn+1

1 − 2dn1 + dn−1

1

)+

∆t2ms(−20d0+h2q0)4h4

+120m∆t2

h4 (d0

24− h2q0

480) (−1 + s)

∆t2dn2 + ∆t2s

(dn+1

2 − 2dn2 + dn−1

2

)...

∆t2dnM−2 + ∆t2s

(dn+1

M−2 − 2dnM−2 + dn−1

M−2

)

∆t2dnM−1 + ∆t2s

(dn+1

M−1 − 2dnM−1 + dn−1

M−1

)+

∆t2ms(−20d1+h2q1)4h4

+120m∆t2

h4 (d1

24− h2q1

480) (−1 + s)

∆t2dnN + ∆t2s

(dn+1

M − 2dnM + dn−1

M

)− 15∆t2m(−4d1+h2q1)2h4

.


Here A, B and D are (M + 1) × (M + 1) penta-diagonal matrices, and H

is an (M + 1) column vector.

The time evolution of the numerical solution W (x, t) is found by the time

evolution of the vector Cn. This is determined by repeatedly solving the

recurrence relationship, once the initial vector C0 has been obtained from

the given initial conditions.

3.4 The Initial Vector

The initial condition w(x, 0) = v0(x) has been used to find the initial vector

C0, which gives (M + 1) equations in (M + 5) unknowns. The following

relations has been used to find these unknowns as

wx(x0, 0) = w′(x0), wx(xM , 0) = w

′(xM)

wxx(x0, 0) = w′′(x0), wxx(xM , 0) = w

′′(xM),

which leads to a penta-diagonal system of equations written in the following

matrix form

GC0 = E, (3.4.1)


where

G =

54 60 6

1014

1352

1054

1

1 26 66 26 1

1 26 66 26 1. . . . . . . . . . . . . . .

1 26 66 26 1

1 1054

1352

1014

6 60 54

,

C0 =

p00

p01

p02

...

p0M−1

p0M

,

and

E =

v0(x0) + 3hw′(xm)5

+ 2h2w′′(xm)

20

v0(x1) + hw′(xm)40

+ h2w′′(xm)

160

v0(x2)...

v0(xM−2)

v0(xM−1)− hw′(xm)40

+ h2w′′(xm)

160

v0(xM)− 3hw′(xm)5

+ 2h2w′′(xm)

20

,m = 0, 1, 2, ..., M.

In the next section, Von-Neumann stability method has been applied to

examine the stability of the presented scheme.


3.5 Stability Analysis

The stability of the proposed scheme (3.3.2) is investigated by Von Neu-

mann method stability method. For investigating stability, considering

h(x, t) = 0 in Eq.(3.3.2).

Substitute pnj = zneiβjh, i =

√−1 in Eq.(3.3.2), after simplifying, it can be

written as

z2 − 2γz + 1 = 0, (3.5.1)

where

γ = 1− 120mr2sin4ψ

(240msr2 + 16)sin4ψ − 210sin2ψ + 120

ψ = 12θh, θ is the variable in the Fourier expansion and r = ∆t

h2 . Apply-

ing the Routh-Hurwitz criterion to Eq.(3.5.1), the following necessary and

sufficient conditions for (3.3.2) to be stable are obtained as

−1 ≤ 1− 120mr2 sin4 ψ

(240msr2 + 16) sin4 ψ − 210 sin2 ψ + 120≤ 1

Simplifying, the left inequality is obtained as

(16 + 60mr2(2s− 1)

)sin4 ψ − 210 sin2 ψ + 120 ≥ 0 (3.5.2)

which shows that the scheme (3.3.2) is unconditionally stable if s ≥ 12, and

conditionally stable if

s <1

2, r ≤ 2√

15(1− 2s).

In the following section, two examples are constructed to illustrate the

applicability and effectiveness of the method developed.



The maximum absolute errors and absolute errors at the points x = 0.1,

x = 0.2, x = 0.3, x = 0.4 and x = 0.5 are calculated by the proposed

method. The numerical results are compared with Evans and Yousif [14],

Aziz et al. [3] and H. Caglar and N. Caglar [7]. Following Tables 3.2, 3.3,

3.4 and 3.6 show that the proposed method gives better results.

Example 3.1

The following fourth-order parabolic PDE is considered as

∂2w

∂t2+

∂4w

∂x4= (π4 − 1) sin πx cos t, x ∈ [0, 1], 0 < t ≤ T,

with the initial conditions

w(x, 0) = sin πx, wt(x, 0) = 0, x ∈ [0, 1],


w(0, t) = w(1, t) =∂2w

∂x2(0, t) =

∂2w

∂x2(1, t) = 0, 0 < t ≤ T.


w(x, t) = sin πx cos t

This problem is solved with s = 14, h = 0.05, ∆t = 0.005, giving r = 2, and

with h = 0.05, ∆t = 0.00125, giving r = 0.5.

The absolute errors at points x = 0.1, x = 0.2, x = 0.3, x = 0.4, x = 0.5,

computed by the proposed method are compared with Evans [14] and Aziz


et al. [3] for K = 10 and K = 16 time steps and are presented in Table 3.2.

The absolute errors for h = 0.05 and r = 0.5 at the mid point x = 0.5

are compared with Evans [14] and Aziz et al. [3] for K = 32, K = 48 and

K = 64 time steps and are presented in Table 3.3.

The absolute errors for h = 0.05 and r = 2.0 at the mid point x = 0.5

are compared with Evans [14] and Aziz et al. [3] for K = 25, K = 75 and

K = 100 time steps and are presented in Table 3.4.

From Tables 3.2, 3.3 and 3.4, it is clear that the proposed method is better

than Evans [14] and Aziz et al. [3]. Moreover, the same problem is solved

with different values of r and the computations are carried out for different

time steps. The absolute errors at x = 0.5 for r =√

16

and r =√

184

are

given in Table 3.5 for h = 0.05.

The maximum absolute errors for different values of M and for a fixed value

of ∆t = 0.005 are compared with H. Caglar and N. Caglar [7], are given in

Table 3.6.

Table 3.2: Absolute errors for h = 0.05 at points x = 0.1, x = 0.2, x =0.3, x = 0.4, x = 0.5 for example 3.1.

r K x = 0.1 x = 0.2 x = 0.3 x = 0.4 x = 0.5ProposedMethods = 1

4

2.0 10 1.23× 10−4 2.35× 10−4 3.24× 10−4 3.81× 10−4 4.0× 10−4

0.5 16 1.70× 10−5 3.23× 10−5 4.45× 10−5 5.23× 10−5 5.5× 10−5

Evans [14] 2.0 10 2.2× 10−4 4.1× 10−4 5.4× 10−4 6.2× 10−4 6.5× 10−4

0.5 16 2.5× 10−5 4.7× 10−5 6.6× 10−5 7.8× 10−5 8.2× 10−5

[3](0,0,1) 2.0 10 1.5× 10−4 2.8× 10−4 3.7× 10−4 4.2× 10−4 4.4× 10−4

0.5 16 3.2× 10−5 5.1× 10−5 6.2× 10−5 6.9× 10−5 7.2× 10−5


Table 3.3: Absolute errors at midpoints, x = 0.5, for h = 0.05 and r = 0.5for example 3.1

32 Time steps 48 Time steps 64 Time stepsProposed Method (s = 1

4) 1.889× 10−4 3.9723× 10−4 6.7281× 10−4

Evans [14] 3.1× 10−4 6.9× 10−4 1.2× 10−3

[3](0,0,1) (s = 14) 3.0× 10−4 7.0× 10−4 1.2× 10−3

Table 3.4: Absolute errors at midpoints, x = 0.5, for h = 0.05 and r = 2.0for example 3.1

25 Time steps 75 Time steps 100 Time stepsProposed Method (s = 1

4) 1.8155× 10−3 5.6442× 10−3 3.7583× 10−3

Evans [14] 3.3× 10−3 4.1× 10−3 3.9× 10−3

[3](0,0,1) (s = 14) 2.7× 10−3 7.8× 10−3 3.0× 10−3

Example 3.2

The following fourth-order parabolic PDE is considered as

∂2w

∂t2+

∂4w

∂x4= (1 + π4)et sin πx, x ∈ [0, 1], 0 < t ≤ T,

with the initial conditions

w(x, 0) = sin πx, wt(x, 0) = sin πx, x ∈ [0, 1]


w(0, t) = w(1, t) =∂2w

∂x2(0, t) =

∂2w

∂x2(1, t) = 0, 0 < t ≤ T.


w(x, t) = et sin πx

The maximum absolute errors for r =√

16

and r =√

184

are given in Table

3.7 for h = 0.05. The computations are carried out for larger time steps K.


Table 3.5: Absolute errors at midpoints, x = 0.5, for h = 0.05 for example3.1

r 10 Time steps 20 Time steps 30 Time steps

Proposed Method (s = 14)

√16 1.603× 10−5 5.2041× 10−5 1.0777× 10−4

s = 14

√184 1.1686× 10−6 3.7986× 10−6 7.8889× 10−6

Table 3.6: Comparison of proposed method with H. Caglar and N. Caglar[7] in maximum absolute errors for example 3.1M ∆t ProposedMethod H.CaglarandN.Caglar[7]121 0.005 1.2669× 10−5 9.3252839× 10−5

191 0.005 1.0393× 10−6 1.0624582× 10−6

3.7 Conclusion

Quintic B-spline collocation method is developed for the numerical solu-

tion of fourth-order parabolic partial differential equation. The numerical

solution is obtained using new three time-level scheme based on a quintic

B-spline for space derivatives and central difference approximation for time

derivatives. The proposed method is shown to be unconditionally stable for

s ≥ 12

and conditionally stable for s < 12

and r ≤ 2√15(1−2s)

. It is observed

from the Tables 3.2, 3.3, 3.4, 3.6 that the proposed method gives better

results as compared with [7], [14] and [3].


Table 3.7: Maximum absolute errors with h = 0.05 for example 3.2r 10 Time steps 50 Time steps 100 Time steps

Proposed Method (s = 14)

√16 9.94× 10−4 7.725× 10−4 9.7375× 10−5

s = 14

√184 2.9946× 10−4 2.792× 10−4 2.133× 10−4

Chapter 4

Numerical Solution of PartialIntegro-Differential Equation

This chapter presents the numerical solution of partial integro-differential

equations. Many mathematical formulations of physical phenomena contain

partial integro-differential equations (PIDE). Partial integro-differential equa-

tions can describe some physical situations such as viscoelasticity, convection-

diffusion problems, heat flow in materials with memory, nuclear reactor dy-

namics, geophysics and plasma physics etc.

This chapter is divided into two sections. The numerical solution of par-

abolic integro-differential equation with a weakly singular kernel has been

presented in section 4.1. In section 4.2, the numerical solution of convection-

diffusion integro-differential equation with a weakly singular kernel has been

proposed.

To solve the PIDEs, the methods developed in this chapter, are based on

the cubic B-spline collocation technique.

The contents of this chapter have been published in the form of two research

63

Ch 4: Solution of Integro-PDE 64

papers [52] and [54].

4.1 Parabolic Integro-Differential Equations

The following parabolic integro-differential equation with a weakly singular

kernel, has been considered

∫ t

0

β(t− s)wt(x, s)ds− wxx(x, t) = f(x, t), x ∈ [a, b], t > 0, (4.1.1)

subject to the following initial condition

w(x, 0) = g0(x), a ≤ x ≤ b (4.1.2)


w(a, t) = f0(t), w(b, t) = f1(t), t ≥ 0 Dirichlet conditions

or

wx(a, t) = r0(t), wx(b, t) = r1(t), t ≥ 0 Neumann conditions

(4.1.3)

The kernel function is defined as

β(t) =tα−1

Γ(α), 0 < α < 1

The kernel function β(t) is a singular t = 0.

Γ represents the gamma function, g0(x), f0(t), f1(t), r0(t), r1(t) are known

functions, f(x, t) is a given smooth function and the function w(x, t) is un-

known.

The applications of the parabolic integro-differential Eq.(4.1.1) along with


the constraints (4.1.2) and (4.1.3) can be found in heat conduction in ma-

terial with memory [36], compression of poroviscoelastic media, population

dynamics, nuclear reactor dynamics etc.

It can be observed that in Eq.(4.1.1), the kernel function has a weak singu-

larity at the origin [58]. This is mostly interesting in viscoelasticity, because

it can smooth the solution when the boundary data is discontinuous [8].

Solution of integro-partial differential equations has recently attracted much

attention of research.

In the following section, time discretization of Eq.(4.1.1) has been carried

out using finite difference approximation.

4.1.1 Temporal Discretization

The region [a, b] × [0, T ] has been discretized as grid points (xi, tj) where

xi = ih, i = 0, 1, 2, . . . , M and tj = j∆t, j = 0, 1, 2, . . . , K, K∆t = T . The

quantities h and ∆t are the mesh sizes in the space and time directions,

respectively.

A finite difference approximation is used to discretize the time derivative


involved in Eq.(4.1.1) at time point t = tj+1 as

∫ tj+1

0

(tj+1 − s)α−1

Γ(α)

∂w(x, s)

∂sds

=

∫ t1

t0

w(x, t1)− w(x, t0)

∆t

(tj+1 − s)α−1

Γ(α)ds

+

j∑r=1

∫ tr+1

tr

(tj+1 − s)α−1

Γ(α)

w(x, tr+1)− w(x, tr−1)

2∆tds

=w(x, t1)− w(x, t0)

Γ(α)∆t

∫ t1

t0

1

(tj+1 − s)1−αds

+

j∑r=1

w(x, tr+1)− w(x, tr−1)

2∆tΓ(α)

∫ tr+1

tr

1

(tj+1 − s)1−αds

=w(x, t1)− w(x, t0)

αΓ(α)∆t1−α[(j + 1)α − jα]

+

j−1∑r=0

w(x, tj+1−r)− w(x, tj−r−1)

2αΓ(α)∆t1−α[(r + 1)α − rα]

= bjw(x, t1)− w(x, t0)

Γ(α + 1)∆t1−α+

1

2Γ(α + 1)

j−1∑r=0

brw(x, tj+1−r)− w(x, tj−r−1)

∆t1−α,

(4.1.4)

where br = (r + 1)α − rα, r = 0, 1, 2, . . . , j.

The discrete differential operator Lαt can be introduced as

Lαt w(x, tj+1) = bj

w(x, t1)− w(x, t0)

Γ(α + 1)∆t1−α+

1

2Γ(α + 1)

j−1∑r=0

brw(x, tj+1−r)− w(x, tj−r−1)

∆t1−α

Eq.(4.1.4) can, then, be rewritten as

∫ tj+1

0

(tj+1 − s)α−1

Γ(α)

∂w(x, s)

∂sds = Lα

t w(x, tj+1)

Approximating∫ tj+1

0

(tj+1−s)α−1

Γ(α)∂w(x,s)

∂sds in Eq.(4.1.1) by Lα

t w(x, tj+1) leads

to the following difference scheme

Lαt w(x, tj+1)− wxx(x, tj+1) ∼= f(x, tj+1)


Substituting the value of Lαt w(x, tj+1), the above equation can be rewritten

as

(bj

Γ(α + 1)

w(x, t1)− w(x, t0)

∆t1−α+

1

2Γ(α + 1)

j−1∑r=0

brw(x, tj−r+1)− w(x, tj−r−1)

∆t1−α

−wxx(x, tj+1))

∼= f(x, tj+1), (4.1.5)

which leads to the following finite difference scheme

(b0w

j+1(x)− 2Γ(α + 1)∆t1−α ∂wj+1

∂x2

)

=

(b0w

j−1(x)−j−1∑r=1

br

(wj−r+1(x)− wj−r−1(x)

)

−2bj

(w1(x)− w0(x)

)+ 2Γ(1 + α)∆t1−αf j+1(x)

), (4.1.6)

where wj+1(x) = w(x, tj+1), br = (r + 1)α − rα, r = 0, 1, 2, . . . , j, which

shows that b0 = 1. Taking a0 = 2Γ(α + 1)∆t1−α, then the Eq.(4.1.6) can

be reformulated as

(wj+1(x)− a0

∂2wj+1

∂x2

)

=

(−b1w

j(x) +

j−1∑r=1

(br−1 − br+1) wj−r(x)

−bjw1(x) + (bj−1 + 2bj) w0(x) + a0f

j+1(x)), j ≥ 1. (4.1.7)

In each time level, there is an ordinary differential equation in the form of

Eq.(4.1.7) with the boundary conditions, which is solved by cubic B-spline

collocation method.

To apply the presented scheme (4.1.7), it is needed to have the values of w


at the nodal points at the zeroth (w0) and first (w1) level times.

Taking j = 0 (the special case), in Eq.(4.1.5), w1 can be computed as

w1(x)− 1

2a0

∂2w1

∂x2= w0(x) +

1

2a0f

1(x), (4.1.8)

where w0 = w(x, 0) = g0(x) is the value of w at the zeroth level time (the

initial condition).

In the next section, the collocation method based on cubic B-Spline is used

for space discretization.

4.1.2 Cubic B-spline Collocation Method

Using

W j+1(x) =M+1∑i=−1

piPi(x), (4.1.9)

where pi are unknown time dependent quantities to be determined from the

boundary conditions and collocation form of the partial integro-differential

equation as defined in section 1.7.10.

The space discretization of Eq.(4.1.7) has been carried out using Eq.(4.1.9)

and the collocation method is implemented by identifying the collocation


points as nodes. For i = 0, 1, 2, . . . , M , the following relation can be ob-

tained as

((pj+1

i−1 + 4pj+1i + pj+1

i+1

)− a06

h2

(pj+1

i−1 − 2pj+1i + pj+1

i+1

))

=

(−b1

(pj

i−1 + 4pji + pj

i+1

)+

j−1∑r=1

(br−1 − br+1)(pj−r

i−1 + 4pj−ri + pj−r

i+1

)

−bj

(p1

i−1 + 4p1i + p1

i+1

)+ (bj−1 + 2bj)

(p0

i−1 + 4p0i + p0

i+1

)+ a0f

j+1i

).

(4.1.10)

Simplifying, the above relation yields the following system of (M +1) linear

equations in (M + 3) unknowns pj+1−1 , pj+1

0 , pj+11 , . . . , pj+1

M , pj+1M+1,

(1− a0

6

h2

)pj+1

i−1 +

(4 + a0

12

h2

)pj+1

i +

(1− a0

6

h2

)pj+1

i+1 = Fi,

j ≥ 1, i = 0, 1, 2, . . . , M, (4.1.11)

where

Fi = −b1

(pj

i−1 + 4pji + pj

i+1

)+

j−1∑r=1

(br−1 − br+1)(pj−r

i−1 + 4pj−ri + pj−r

i+1

)

−bj

(p1

i−1 + 4p1i + p1

i+1

)+ (bj−1 + 2bj)

(p0

i−1 + 4p0i + p0

i+1

)+ a0f

j+1i

The unique solution of the system (4.1.11) is obtained by eliminating the

parameters p−1 and pM+1 using boundary conditions.

First, Dirichlet boundary conditions are used to eliminate p−1 and pM+1,

as under

w(a, t) = (p−1 + 4p0 + p1) = f0(t),

w(b, t) = (pM−1 + 4pM + pM+1) = f1(t).


p−1 = −4p0 − p1 + f0(t),

pM+1 = −4pM − pM−1 + f1(t).

After eliminating p−1 and pM+1, the system (4.1.11) is reduced to a tri-

diagonal system of (M + 1) linear equations in (M + 1) unknowns. This

system can be rewritten in matrix form as

ACj+1 = G, j = 1, 2, 3, . . . , K (4.1.12)

where

Cj+1 =[pj+1

0 , pj+11 , . . . , pj+1

M

]T, j = 1, 2, 3, . . . , K.

The coefficient matrix A is given as under

A =

a036h2

α β α

α β α. . . . . . . . .

α β α

a036h2

,

where

α =

(1− a0

6

h2

),

β =

(4 + a0

12

h2

).

p−1 and pM+1 can also be eliminated from Eq.(4.1.11) using Neumann

boundary conditions, as under

wx(a, t) =3

h(−p−1 + p1) = r0(t),

wx(b, t) =3

h(−pM−1 + pM+1) = r1(t).


p−1 = p1 − h

3r0(t),

pM+1 = pM−1 +h

3r1(t).

On substituting the values of parameters p−1 and pM+1, the system (4.1.11)

reduces to a diagonal system of (M + 1) equations in (M + 1) unknowns

written in the following matrix form

BCj+1 = H, j = 1, 2, 3, . . . , K (4.1.13)

where

Cj+1 =[pj+1

0 , pj+11 , . . . , pj+1

M

]T, j = 1, 2, 3, . . . , K

The coefficient matrix B is given as under

B =

(4 + a0

12h2

) (2− a0

12h2

)

γ δ γ

γ δ γ. . . . . . . . .

γ δ γ(2− a0

12h2

) (4 + a0

12h2

)

,

where

γ =

(1− a0

6

h2

),

δ =

(4 + a0

12

h2

).

Using the system (4.1.11), for j = 1, following is the system of (M + 1)

linear equations in (M + 3) unknowns p2−1, p

20, . . . , p

2M , p2

M+1,

(1− a0

6

h2

)p2

i−1 +

(4 + a0

12

h2

)p2

i +

(1− a0

6

h2

)p2

i+1 = Fi, i = 0, 1, 2, . . . , M,

(4.1.14)


where

Fi = −2b1

(p1

i−1 + 4p1i + p1

i+1

)+ (b0 + 2b1)

(p0

i−1 + 4p0i + p0

i+1

)+ a0f

2i .

To find the value of C2 = [p20, p

21, . . . , p

2M ]

T, it is first needed to find the value

of C1 = [p10, p

11, . . . , p

1M ]

T. The value of C1 is obtained, solving Eq.(4.1.8)

using cubic B-spline collocation method, as

(1− a0

3

h2

)p1

i−1 +

(4 + a0

6

h2

)p1

i +

(1− a0

3

h2

)p1

i+1 = Fi , i = 0, 1, 2, . . . ,M,

(4.1.15)

where

Fi =(p0

i−1 + 4p0i + p0

i+1

)+

1

2a0f

1i .

The above Eq.(4.1.15) is a system of (M + 1) linear equations in (M +

3) unknowns p1−1, p

10, . . . , p

1M , p1

M+1. The unique solution of this system is

obtained by eliminating the parameters p−1 and pM+1 using Dirichlet and

Neumann boundary conditions.

The time evolution of the numerical solution W j+1 is determined by the

time evolution of the vector Cj+1. This is found by repeatedly solving the

recurrence relationship, after the initial vector C0 = [p00, p

01, . . . , p

0M ]

Thas

been computed from the initial condition.

To test the accuracy of the presented method, three examples are considered

in the following section.


4.1.3 Numerical Results

Let tj = j∆t, j = 0, 1, 2, . . . , K, h = 1M

, where K denotes the final time level

tK and M +1 is the number of nodes. In order to check the accuracy of the

proposed method, the maximum norm errors and L2 norm errors between

numerical and exact solution are given with the following definitions

Maximum norm error : ‖eK‖∞ = max0≤i≤M

∣∣u (xi, tK)− UKi

∣∣

L2 norm error : ‖eK‖2 =1

M

(M∑i=0

∣∣u (xi, tK)− UKi

∣∣2) 1

2

.

Example 4.1

Following is the second order parabolic integro-differential equation

∫ t

0

(t− s)−0.5

Γ(α)wt(x, s)ds− wxx(x, t) = f(x, t), x ∈ [0, 1],

with the initial condition

w(x, 0) = sin πx, x ∈ [0, 1],


w(0, t) = w(1, t) = 0, t ≥ 0


w(x, t) = (t + 1) sin πx.

The numerical solutions at M = 60, ∆t = 0.0001 and ∆t = 0.001, with

different time levels K, are tabulated in Table 4.1 and 4.2, respectively. The


numerical solutions at K = 10 and ∆t = 0.0001 for different values of M

are tabulated in Table 4.3. The accuracy of the presented method is also

tested by varying the values of the parameters ∆t, h = 1M

and K, which

shows that the presented method is efficient.

To show the effect of the presented method for large value of K, the exact

and the numerical solutions are plotted using M = 100, K = 500 and

∆t = 0.0001 as shown in Fig.4.1. When M = 100, ∆t = 0.0001 and

K = 10 the exact solution and the numerical solution at the K time level

are shown in Fig.4.2. It can be observed from the Tables 4.1, 4.2 and 4.3

and Figures 4.1 and 4.2, that the presented method approximates the exact

solution efficiently.

Table 4.1: The errors ‖eK‖∞ and ‖eK‖2 when M = 60 and ∆t = 0.0001 forexample 4.1

K ‖eK‖∞ ‖eK‖2

10 4.1248× 10−5 3.7654× 10−6

20 2.3848× 10−5 2.1770× 10−6

30 1.2071× 10−5 1.1019× 10−6

40 3.1259× 10−6 2.8536× 10−7

50 4.0669× 10−6 3.7126× 10−7


K ‖eK‖∞ ‖eK‖2

10 8.7764× 10−4 8.0117× 10−5

20 8.5436× 10−4 7.7992× 10−5

30 8.4063× 10−4 7.6738× 10−5

40 8.3093× 10−4 7.5853× 10−5

50 8.2337× 10−4 7.5163× 10−5


Table 4.3: The errors ‖eK‖∞ and ‖eK‖2 when K = 10 and ∆t = 0.0001 forexample 4.1

M ‖eK‖∞ ‖eK‖2

10 2.0176× 10−3 4.5115× 10−4

20 4.2892× 10−4 6.7818× 10−5

30 1.3503× 10−4 1.7433× 10−5

40 3.2197× 10−5 3.5998× 10−6

50 1.5396× 10−5 1.5396× 10−6

Numerical solution1.0

1.5

2.0

t

0 50 100x

0.0

0.5

1.0

1.5

u

,

Exact solution1.0

1.5

2.0

t

0 50 100x

0.0

0.5

1.0

1.5

u

Figure 4.1: The results at M=500 for Example 4.1.

Example 4.2

Following is the parabolic integro-differential equation

∫ t

0

(t− s)−0.5

Γ(α)wt(x, s)ds− wxx(x, t) = f(x, t), x ∈ [0, 1],

The M-th exact solution

The M-th numerical

solution

0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

u

Figure 4.2: The exact and numerical solutions at M=10


with the following initial condition

w(x, 0) = cos πx, x ∈ [0, 1],

along with the Dirichlet boundary conditions

w(0, t) = (t + 1) ,

w(1, t) = (t + 1) cos (π) , t ≥ 0.


w(x, t) = (t + 1) cos πx.

The numerical solutions at M = 60, ∆t = 0.0001 and ∆t = 0.001 with

different time levels K are tabulated in Table 4.4 and 4.5, respectively. The

numerical solutions at K = 10 and ∆t = 0.0001 for different values of M

are tabulated in Table 4.6. The accuracy of the presented method is also

tested by varying the values of the parameters ∆t, h = 1M

and K, which

shows that the presented method is efficient.





are shown in Fig.4.4. It can be observed from the Tables 4.4, 4.5 and 4.6

and Figures 4.3 and 4.4, that the developed method approximates the exact




K ‖eK‖∞ ‖eK‖2

10 9.1411× 10−5 6.8482× 10−6

20 9.1003× 10−5 5.3983× 10−6

30 9.0654× 10−5 5.5083× 10−6

40 8.9746× 10−5 5.5048× 10−6

50 8.8599× 10−5 5.4441× 10−6


K ‖eK‖∞ ‖eK‖2

10 9.9041× 10−4 8.5887× 10−5

20 9.8809× 10−4 8.5793× 10−5

30 9.8464× 10−4 8.5511× 10−5

40 9.8077× 10−4 8.5187× 10−5

50 9.7666× 10−4 8.4844× 10−5

Example 4.3

Following is the parabolic integro-differential equation

∫ t

0

(t− s)−0.5

Γ(α)wt(x, s)ds− wxx(x, t) = f(x, t), x ∈ [−1, 1],


w(x, 0) = sin πx, x ∈ [−1, 1],

Table 4.6: The errors ‖eK‖∞ and ‖eK‖2 when K = 10 and ∆t = 0.0001 forexample 4.2

M ‖eK‖∞ ‖eK‖2

10 8.5855× 10−4 1.8436× 10−4

20 1.6181× 10−4 2.0645× 10−5

30 4.0921× 10−5 2.9628× 10−6

40 1.3215× 10−5 1.162× 10−6


Exact solution

1.0

1.5

2.0

t

0

50

100

x

-1.0

-0.5

0.0

0.5

1.0

u

,

Numerical solution

1.0

1.5

2.0

t

0

50

100

x

-1.0

-0.5

0.0

0.5

1.0

u


The M-th exact solution

The M-th numerical solution

0.2 0.4 0.6 0.8 1.0x

-1.0

-0.5

0.5

1.0

u

Figure 4.4: The exact and numerical solutions at M=10


along with Neumann boundary conditions

wx(−1, t) = π (t + 1)2 cos π,

wx(1, t) = π (t + 1)2 cos π, t ≥ 0.


w(x, t) = (t + 1)2 sin πx.


different time levels K are tabulated in Table 4.7 and 4.8, respectively. The

accuracy of the presented method is also tested by varying the values of the

parameters ∆t and K, which shows that the presented method is efficient.





are shown in Fig.4.6. It can be observed from the Tables 4.7 and 4.8 and

Figures 4.5 and 4.6, that the presented method approximates the exact


Table 4.7: Maximum norm errors ‖eK‖∞ for M = 40 for example 4.3M K ∆∆t = 0.001 ‖eK‖∞ ∆t = 0.00125 ‖eK‖∞40 10 5.9948× 10−4 1.0018× 10−3

20 4.3331× 10−4 7.2357× 10−4

30 7.0620× 10−4 1.1162× 10−4

40 1.3169× 10−3 1.9388× 10−3

50 2.0042× 10−3 2.8735× 10−3


Table 4.8: L2 norm errors ‖eK‖2 for M = 40 for example 4.3M K ∆t = 0.001 ‖eK‖2 ∆t = 0.00125 ‖eK‖2

40 10 6.2565× 10−5 1.0570× 10−4

20 4.6637× 10−5 7.7196× 10−5

30 3.3271× 10−5 5.7804× 10−5

40 8.9371× 10−5 1.2795× 10−4

50 1.6722× 10−4 2.3443× 10−4

Exact solution

1.0

1.5

2.0

t10

20

30

40

x

-1.0

-0.5

0.0

0.5

1.0

u

,

Numerical solution

1.0

1.5

2.0

t10

20

30

40

x

-1.0

-0.5

0.0

0.5

1.0

u


In the following section, the convection-diffusion integro-differential equa-

tion has been considered.

4.2 Convection-Diffusion Integro-Differential

Equation

The following convection-diffusion integro-differential equation has been

considered

wt(x, t) + mwx(x, t)− bwxx(x, t)−∫ t

0

S(x, t, s)w(x, s)ds = f(x, t),

x ∈ [0, L], 0 < t ≤ T,


The M-th exact

solutionThe M-th numerical

solution

-1.0 -0.5 0.5 1.0x

-1.0

-0.5

0.5

1.0

u

Figure 4.6: The exact and numerical solutions at M=10.

where m > 0 and b > 0 are considered to be positive constants quantifying

the advection(convection) and diffusion processes, respectively. The integral

is called memory term, S(x, t, s) is the kernel function satisfying

maxx∈[0,L]

|S(x, t, s)| ≤ C|A(t, s)Sα(t− s)|,

where A is sufficiently smooth in t and s, and the Hammerstein kernel

Sα(t− s) =

(t− s)−α, 0 < α < 1,

S(t− s), otherwise

S is smooth function, Sα(t − s) = (t − s)−α is said to be weakly singular

kernel.

The convection-diffusion integro-differential equation with a weakly singular

kernel of the following form has been considered in this section


0

(t− s)−αw(x, s)ds = f(x, t),

x ∈ [0, L], 0 < t ≤ T,(4.2.1)

subject to the initial condition

w(x, 0) = g0(x), 0 ≤ x ≤ L, (4.2.2)



w(0, t) = f0(t), w(L, t) = f1(t), 0 < t ≤ T, (4.2.3)

where g0(x), f0(t), f1(t) are known functions, f(x, t) ia a given smooth

function.

If the memory term is zero, the Eq.(4.2.1) reduces to the more general

inhomogeneous convection-diffusion equation given by

wt(x, t) + mwx(x, t)− bwxx(x, t) = f(x, t),

x ∈ [0, L], 0 < t ≤ T. (4.2.4)

The source term f(x, t), accounts for an insertion or extraction of mass from

the system as it evolves with time. Specifically, f(x, t) represents the time

rate of change of concentration due to external factors, such as a source or

a sink.

The integro-differential Eq.(4.2.1) along with the constraints (4.2.2) and

(4.2.3) is of fundamental importance in many physical systems, especially

those involving fluid flow. Eq.(4.2.1) is the one dimensional version of the

PIDEs which describe convection-diffusion of quantities such as mass, heat,

energy, vorticity, and so forth [11],[56],[10] and [58].

4.2.1 Temporal Discretization

The time derivative is discretized by the first-order backward Euler scheme.

Let tj = j∆t, where ∆t is the time step, wj(x) is an approximation to the


value of w(x, t) at a time point t = tj, j = 0, 1, . . . K.

Considering the temporal discrete process of Eq.(4.2.1) at time point t =

tj+1, the first expression in left side of Eq.(4.2.1) is approximated by

wt(x, tj+1) ≈ w(x, tj+1)− w(x, tj)

∆t(4.2.5)

Substituting Eq.(4.2.5) in Eq.(4.2.1), it can be written as

(w(x, tj+1)− w(x, tj)

∆t+ mwx(x, tj+1)− bwxx(x, tj+1)

−∫ tj+1

0

(tj+1 − s)−α w(x, s)ds

)

= f(x, tj+1) (4.2.6)

The integral term in the above equation, can be considered as

∫ tj+1

0

(tj+1 − s)−α w(x, s)ds =

∫ tj+1

0

s−αw(x, tj+1 − s)ds

=

j∑

k=0

∫ tk+1

tk

s−αw (x, tj+1 − s) ds

≈j∑

k=0

w (x, tj−k+1)

∫ tk+1

tk

s−αds

=∆t1−α

1− α

j∑

k=0

w (x, tj−k+1)((k + 1)1−α − k1−α

)

(4.2.7)

Substituting Eq.(4.2.7) in Eq.(4.2.6), it can takes the following form

(w(x, tj+1)− w(x, tj)

∆t+ mwx(x, tj+1)− bwxx(x, tj+1)

−∆t1−α

1− α

j∑

k=0

w (x, tj−k+1)((k + 1)1−α − k1−α

))

= f j+1(x) (4.2.8)



(wj+1(x) + m∆twj+1

x (x)− b∆twj+1xx (x)− ∆t2−α

1− αwj+1(x)

)

=

(wj(x) +

∆t2−α

1− α

j∑

k=1

bkwj−k+1(x) + ∆tf j+1(x)

), j ≥ 1 (4.2.9)

where wj+1(x) = w(x, tj+1), f(x, tj+1) = f j+1(x),

bk = (k + 1)α − kα, k = 1, 2, . . . , j.

For j = 0, the presented scheme can takes the following form

(w1(x) + m∆tw1

x(x)− b∆tw1xx(x)− ∆t2−α

1− αw1(x)

)

=(w0(x) + ∆tf 1(x)

)(4.2.10)

4.2.2 Discretization in Space

Consider a uniform mesh ∆ with the grid points λij to discretize the region

Ω = [0, L] × [0, T ]. Each λij is the grid point (xi, tj) where xi = ih, i =

0, 1, 2, . . . , M and tj = j∆t, j = 0, 1, 2, . . . , K, K∆t = T . The quantities h

and ∆t are the mesh sizes in the space and time directions, respectively.

Using

W j+1(x) =M+1∑i=−1

piPi(x), (4.2.11)

where pi are unknown time dependent quantities to be determined from the

boundary conditions and collocation form of the partial integro-differential

equation as defined in subsection 1.7.10.

The space discretization of Eq.(4.2.9) is carried out using Eq.(4.2.11) and


the collocation method is implemented by identifying the collocation points

as nodes. For i = 0, 1, 2, . . . ,M , the following relation can be obtained as

((pj+1

i−1 + 4pj+1i + pj+1

i+1

)+ m∆t

3

h

(−pj+1i−1 + pj+1

i+1

)− b∆t6

h2

(pj+1

i−1 − 2pj+1i + pj+1

i+1

)

−∆t2−α

1− α

(pj+1

i−1 + 4pj+1i + pj+1

i+1

))

=

((pj

i−1 + 4pji + pj

i+1

)+

∆t2−α

1− α

j∑

k=1

bj

(pj+1−k

i−1 + 4pj+1−ki + pj+1−k

i+1

)+ ∆tf j+1

i

)

Simplifying, the above relation yields to the following system of (M + 1)

linear equations in (M + 3) unknowns pj+1−1 , pj+1

0 , pj+11 , . . . , pj+1

M , pj+1M+1,

(pj+1

i−1

(1−m∆t

3

h− b∆t

6

h2− ∆t2−α

1− α

)+ pj+1

i

(4 + b∆t

12

h2− 4

∆t2−α

1− α

)

+pj+1i+1

(1 + m∆t

3

h− b∆t

6

h2− ∆t2−α

1− α

))

= Fi i = 0, 1, 2, . . . , M, (4.2.12)

where

Fi =(pj

i−1 + 4pji + pj

i+1

)+

∆t2−α

1− α

j∑

k=1

bk

(pj+1−k

i−1 + 4pj+1−ki + pj+1−k

i+1

)+ ∆tf j+1

i .


parameters p−1 and pM+1 using the following boundary conditions

w(x0, t) = (p−1 + 4p0 + p1) = f0(t),

w(xM , t) = (pM−1 + 4pM + pM+1) = f1(t).

p−1 = −4p0 − p1 + f0(t),

pM+1 = −4pM − pM−1 + f1(t).


On substituting the values of parameters p−1 and pM+1, the system (4.2.12)

reduces to a diagonal system of (M + 1) equations in (M + 1) unknowns


ACj+1 = F, j = 1, 2, 3, . . . , K,

where

Cj+1 =[pj+1

0 , pj+11 , . . . , pj+1

M

]T, j = 1, 2, 3, . . . , K.


A =

(12m∆t

h+ 36b∆t

h2

)6m∆t

h

p q r

p q r. . . . . . . . .

p q r

6m∆th

(12m∆t

h+ 36b∆t

h2

)

,

where

p =

(1−m∆t

3

h− b∆t

6

h2− ∆t2−α

1− α

),

q =

(4 + b∆t

12

h2− 4

∆t2−α

1− α

),

r =

(1 + m∆t

3

h− b∆t

6

h2− ∆t2−α

1− α

).


21, . . . , p

2M ]

T, firstly, it is necessary to find

the value of C1 = [p10, p

11, . . . , p

1M ]

T.The value of C1 is obtained, solving


Eq.(4.2.10) using cubic B-spline collocation method, as

(p1

i−1

(1−m∆t

3

h− b∆t

6

h2− ∆t2−α

1− α

)+ p1

i

(4 + b∆t

12

h2− 4

∆t2−α

1− α

)

+p1i+1

(1 + m∆t

3

h− b∆t

6

h2− ∆t2−α

1− α

))

= Fi i = 0, 1, 2, . . . , M, (4.2.13)

where

Fi =(p0

i−1 + 4p0i + p0

i+1

)+ ∆tf 1

i .

The above Eq.(4.2.13) is a system of (M + 1) linear equations in (M + 3)

unknowns p1−1, p

10, . . . , p

1M , p1

M+1. The unique solution of this system is ob-

tained by eliminating the parameters p−1 and pM+1 using boundary condi-

tions.

The time evolution of the numerical solution W j+1 is determined by the

time evolution of the vector Cj+1. This is obtained by repeatedly solving

the recurrence relationship, once the initial vector C0 = [p00, p

01, . . . , p

0M ]

T,

has been computed from the initial condition.

To test the accuracy of the presented method, four examples are considered


4.2.3 Numerical Results

The accuracy of the presented method has been discussed by calculating

maximum norm errors and L2 norm errors between numerical and exact

solutions. The accuracy of the presented method is also tested by varying


the values of parameters h, ∆t, m and b.

Some important non-dimensional parameters in numerical analysis are de-

fined as follows:

Courant number : Cr = m∆t

h

Diffusion number : S = b∆t

h2

Grid Peclet number : Pe =Cr

S=

m

bh

When the P eclet number is high, the convection term dominates and when

the P eclet number is low, the diffusion term dominates.

Example 4.4

The following convection-diffusion integro-differential equation is considered


0

(t− s)−αw(x, s)ds = f(x, t),

x ∈ [0, 1], α =1

2, t > 0,

with m = 0.05, b = 0.4 and the following initial condition

w(x, 0) = sin πx, 0 ≤ x ≤ 1,


w(0, t) = w(1, t) = 0, 0 ≤ t ≤ T.


w(x, t) = (t + 1)2 sin πx.


The numerical solutions for M = 100 and M = 50 for ∆t = 0.0001 and

∆t = 0.00001, with different time levels K, are presented in Table 4.9 and

4.10, respectively. Pe = 0.00125 and Pe = 0.0025 for h = 0.01 and h = 0.02

respectively. Here, P eclet number Pe is low, which shows that the diffusion

term dominates. The accuracy of the presented method is also tested by

varying the values of the parameters ∆t, h = 1M

and K, which shows that

the presented method is efficient. It can be observed from the Tables 4.9

and 4.10, that the developed method approximates the exact solution very

efficiently.

Table 4.9: ‖eK‖∞ and ‖eK‖2 for ∆t = 0.0001 for example 4.4h K Pe ‖eK‖∞ ‖eK‖2

0.01 10 0.00125 9.4351× 10−6 1.3253× 10−7

50 1.1446× 10−5 3.3175× 10−7

100 1.1943× 10−5 5.7312× 10−7

500 3.5394× 10−5 2.4398× 10−6

0.02 10 0.0025 2.6417× 10−5 5.6276× 10−7

50 3.9890× 10−5 1.0863× 10−6

100 4.4030× 10−5 1.3976× 10−6

500 5.3632× 10−5 2.8654× 10−6


0.01 10 0.00125 4.3771× 10−6 4.4106× 10−8

50 8.2580× 10−6 9.5928× 10−8

100 9.5573× 10−6 1.2493× 10−7

500 1.1579× 10−5 2.2516× 10−7

0.02 10 0.0025 6.7929× 10−6 1.3756× 10−7

50 2.004× 10−5 4.1125× 10−7

100 2.6950× 10−5 5.8715× 10−7

500 4.0214× 10−5 1.2581× 10−6


Exact solution

1.0

1.5

2.0

t

0

50

100

x

0.0

0.5

1.0

u

,

Numerical solution

1.0

1.5

2.0 0

50

1000.0

0.5

1.0

Figure 4.7: The results at K=500 for Example 4.4.

0.2 0.4 0.6 0.8 1.0x

0.2

0.4

0.6

0.8

1.0

u

Figure 4.8: The exact and numerical solutions at K=10


Example 4.5



0

(t− s)−αw(x, s)ds = f(x, t),

x ∈ [0, 4π], α =1

4, t > 0,


w(x, 0) = 2 sin2 x, 0 ≤ x ≤ 4π,


w(0, t) = w(4π, t) = 0, 0 ≤ t ≤ T.


w(x, t) = 2 (t + 1) sin2 x.

The numerical solutions for two different grid sizes M = 10 and M = 50 for

∆t = 0.0001 and ∆t = 0.00001, with different time levels K, are presented

in Table 4.11 and Table 4.12, respectively. Pe = 1.25663 and Pe = 0.25132

for h = 4π10

and h = 4π50

respectively. Here, P eclet number Pe corresponds

to h = 4π10

is high, which indicates that the convection term dominates.

Pe corresponds to h = 4π50

is low, which indicates that the diffusion term

dominates. The accuracy of the presented method is also tested by varying

the values of the parameters ∆t, h = 1M

and K, which shows that the

presented method is efficient. It can be observed from the Tables 4.11 and


Exact solution

1.0

1.5

2.0t

0

20

40x

0.0

0.5

1.0

1.5

2.0

u

,

Numerical solution

1.0

1.5

2.0t

0

2040x

0.0

0.5

1.0

1.5

2.0

u


4.12, that the developed method approximates the exact solution efficiently.

Table 4.11: ‖eK‖∞ and ‖eK‖2 for ∆t = 0.0001 for example 4.5M K Pe ‖eK‖∞ ‖eK‖2

10 10 1.25663 1.8012× 10−4 3.6509× 10−5

50 8.1725× 10−4 1.6629× 10−4

100 1.5025× 10−3 3.0190× 10−4

50 10 0.25132 6.3946× 10−6 7.5264× 10−7

50 1.7846× 10−4 1.4880× 10−5

100 6.8299× 10−4 5.6108× 10−5

Table 4.12: ‖eK‖∞ and ‖eK‖2 for ∆t = 0.00001 for example 4.5M K Pe ‖eK‖∞ ‖eK‖2

10 10 1.25663 1.9055× 10−5 3.7849× 10−6

50 9.3283× 10−5 1.8669× 10−5

100 1.8239× 10−4 3.6800× 10−5

50 10 0.25132 9.4661× 10−7 7.2971× 10−8

50 4.2270× 10−6 2.8923× 10−7

100 7.1358× 10−6 6.3521× 10−7


10 20 30 40 50x

0.5

1.0

1.5

2.0

u


Example 4.6



0

(t− s)−αw(x, s)ds = f(x, t),

x ∈ [0, 1], α =1

3, t > 0,


w(x, 0) = 1− cos 2πx + 2π2x(1− x), 0 ≤ x ≤ 1,

along with appropriate boundary conditions

w(0, t) = w(1, t) = 0, 0 ≤ t ≤ T.


w(x, t) = (t + 1)2 (1− cos 2πx + 2π2x(1− x)

).

The numerical solutions for two different grid sizes M = 50 and M = 100 for

∆t = 0.0001 and ∆t = 0.00001, with different time levels K, are presented

in Table 4.13 and Table 4.14 respectively. Pe = 0.0002 and Pe = 0.0001


for h = 0.02 and h = 0.01 respectively. Here, P eclet number Pe is low,

which indicates that the diffusion term dominates. The accuracy of the

presented method is also tested by varying the values of the parameters

∆t, h = 1M

and K, which shows that the presented method is efficient. It

can be observed from the Tables 4.13 and 4.14, that the developed method

approximates the exact solution very well.


0.02 10 0.0002 2.4282× 10−5 2.007× 10−6

50 1.009× 10−4 8.8800× 10−6

100 1.7172× 10−4 1.5778× 10−5

500 4.6998× 10−4 4.1795× 10−5

0.01 10 0.0001 6.6290× 10−6 5.0793× 10−7

50 3.1392× 10−5 2.4155× 10−6

100 5.9245× 10−5 4.6368× 10−6

500 2.2874× 10−4 1.9337× 10−5


0.02 10 0.0002 2.9281× 10−6 2.4829× 10−7

50 1.2548× 10−5 1.1989× 10−6

100 2.4765× 10−5 2.3503× 10−6

500 1.1947× 10−4 1.0652× 10−5

0.01 10 0.0001 1.9020× 10−6 4.5422× 10−8

50 3.0941× 10−6 2.0137× 10−7

100 5.9627× 10−6 3.9130× 10−7

500 2.6198× 10−5 1.7531× 10−6


Exact solution

1.0

1.5

2.0

t

0

50

100

x

0

2

4

6

u

,

Numerical solution

1.0

1.5

2.0

t

0

50

100

x

0

2

4

6

u


20 40 60 80 100x

1

2

3

4

5

6

7

u



Example 4.7



0

(t− s)−αw(x, s)ds = f(x, t),

x ∈ [0, 1], α =1

10, t > 0,


w(x, 0) = cos πx, 0 ≤ x ≤ 1

and along with the boundary conditions

w(a, t) = (t + 1) , w(b, t) = − (t + 1) , t ≥ 0.


w(x, t) = (t + 1) cos πx


different time levels K are presented in Table 4.15 and 4.16 respectively.

The accuracy of the presented method is also tested by varying the values

of the parameters ∆t and K, which shows that the presented method is

efficient. Pe = 1 for h = 0.01. Here, P eclet number Pe is high, which shows

that the convection term dominates.

4.3 Conclusion

The numerical solutions of parabolic integro-differential equation and convection-

diffusion integro-differential equation have been developed using collocation


Exact solution

1.0

1.5

2.0

t5

10

x

-1.0

-0.5

0.0

0.5

1.0

u

,

Numerical solution

1.0

1.5

2.0t

5

10x

-1.0-0.50.0

0.5

1.0

u


20 40 60 80 100x

-1.0

-0.5

0.5

1.0

u



Table 4.15: L2 norm errors ‖eK‖2 for M = 100 for example 4.7h K Pe ∆t = 0.0001 ‖eK‖2 ∆t = 0.00001 ‖eK‖2

0.01 10 1 1.1709× 10−6 1.4690× 10−8

50 2.0104× 10−5 2.5430× 10−7

100 7.2658× 10−5 9.2099× 10−7

500 1.5005× 10−3 1.9062× 10−5

Table 4.16: Maximum norm errors ‖eK‖∞ for M = 100 for example 4.7h K Pe ∆t = 0.0001 ‖eK‖∞ ∆t = 0.00001 ‖eK‖∞

0.01 10 1 3.0131× 10−5 3.7552× 10−7

50 3.3049× 10−4 5.006× 10−6

100 1.0609× 10−3 1.6837× 10−5

500 2.1712× 10−2 2.8412× 10−4

method. In parabolic integro-differential equation, the finite central differ-

ence formula has been used for temporal discretization and cubic B-spline

collocation method has been used for spatial discretization. In convection-

diffusion integro-differential equation, backward Euler formula has been

used for temporal discretization and the cubic B-spline collocation method

has been used for spatial discretization. The efficiency of the presented

method has been tested by varying the values of parameters h, ∆t and K.

It has been observed from the numerical results, that the methods developed

in section 4.1 and 4.2, exhibit high level of accuracy and efficiency.

Chapter 5

Numerical Solution ofTime-Fractional Fourth-orderPartial Differential Equations

The time-fractional fourth-order PDEs found in many applications in real

life problems such as modeling of thin beams and plates, strain gradient

elasticity and phase separation in binary mixtures, which are basic ele-

ments in engineering structures and are of great practical significance to

civil, mechanical and aerospace engineering.

A collocation method with quintic B-spline basis functions is applied to

develop numerical solution of time-fractional fourth-order PDE. The time-

fractional derivative is described in the Caputo sense. Backward Euler

formula is used for temporal discretization and the quintic B-spline colloca-

tion method is used for spatial discretization. The stability and convergence

properties related to the time discretization are discussed and proven, theo-

retically. The given problem is solved with three different types of boundary

99

Ch 5: Solution of Time-Fractional Fourth-order PDE 100

conditions, including clamped-type condition, simply supported-type con-

dition and a transversely supported-type condition. Numerical results are

discussed to investigate the accuracy and efficiency of the presented method.

The contents of this chapter have been published in the form of a research

paper [51].

5.1 Introduction

Fractional calculus is a branch of mathematics in which the study of frac-

tional integrals and fractional derivatives has been discussed. Fractional

calculus is as old as the conventional calculus, but has not been admired for

a long time. In the last few decades, it has been observed by applied scien-

tists and mathematicians and engineers that many of the physical problems

such as viscoelastic systems, signal processing and diffusion processes etc.

can be effectively modeled by partial differential equations with fractional

derivatives. Oldham and Spanier [40], Podlubny [41] and Miller and Ross

[35] provided the history and a comprehensive treatment of this subject.

The fourth-order problems play a vital role in different branches of engineer-

ing and science. For example, bridge slabs, floor systems, window glasses

and airplane wings can be modeled as plates with various boundary sup-

ports which are governed by fourth-order PDEs.


Following is the time-fractional fourth-order PDE with a fractional deriva-

tive of order α, (0 < α < 1)

∂αw

∂tα+ µ

∂4w

∂x4= h(x, t), x ∈ Ω = [0, L], 0 < t ≤ T, (5.1.1)


w(x, 0) = g0(x), 0 ≤ x ≤ L,

where µ is the ratio of flexural rigidity of the beam to its mass per unit

length, w is the transverse displacement of the beam, t and x are the time

and distance variables, respectively. h(x, t) is the dynamic driving force per

unit mass and g0(x) is continuous function.

When α = 1, Eq.(5.1.1) becomes fourth-order PDE

∂w

∂t+ µ

∂4w

∂x4= f(x, t), x ∈ Ω = [0, L], 0 < t ≤ T. (5.1.2)

The three different boundary conditions are considered as under

• Clamped boundary conditions

It is obtained by casting the end of a beam into concrete. It has zero

displacement and also zero slope. The clamped boundary conditions

take the following form

(I)

w(0, t) = w(L, t) = 0,

wx(0, t) = wx(L, t) = 0, 0 ≤ t ≤ T.

• Simply-supported boundary conditions

Simply-supported boundary conditions are

(II)

w(0, t) = w(L, t) = 0,

wxx(0, t) = wxx(L, t) = 0, 0 ≤ t ≤ T.


• Transversely supported boundary conditions

Transversely supported boundary conditions are

(III)

w(0, t) = w(L, t) = 0,

wxx(0, t)− Pwx(0, t) = wxx(L, t)− Pwx(L, t) = 0, 0 ≤ t ≤ T,

where P is a constant.

The time-fractional derivative ∂αw∂tα

involved in Eq.(5.1.1) is described by the

Caputo fractional derivative of order α, given by

∂αw(x, t)

∂tα=

1

Γ(1−α)

∫ t

0∂w(x,s)

∂sds

(t−s)α , 0 < α < 1,∂w(x,t)

∂t, α = 1.

5.2 Temporal Discretization

The time-fractional derivative ∂αw(x,t)∂tα

has been discretized using first-order

backward Euler formula. Let tn = n∆t, n = 0, 1, 2, . . . , K, in which ∆t = TK

is the time step size. wn(x) is an approximation to the value of w(x, t) at

a time point t = tn, n = 0, 1, . . . , K − 1.

The time-fractional derivative in Eq.(5.1.1) at time point t = tn+1, can be


approximated as

∂αw(x, tn+1)

∂tα=

1

Γ(1− α)

∫ tn+1

0

∂w(x, s)

∂s

ds

(tn+1 − s)α

=1

Γ(1− α)

n∑j=0

∫ tj+1

tj

∂w(x, s)

∂s

ds

(tn+1 − s)α

=1

Γ(1− α)

n∑j=0

w(x, tj+1)− w(x, tj)

∆t

∫ tj+1

tj

ds

(tn+1 − s)α

+rn+1∆t

=1

Γ(1− α)

n∑j=0


∆t

∫ tn+1−j

tn−j

dτ

τα+ rn+1

∆t

=1

Γ(1− α)

n∑j=0

w(x, tn+1−j)− w(x, tn−j)

∆t

∫ tj+1

tj

dτ

τα+ rn+1

∆t

=1

Γ(2− α)

n∑j=0


∆tα((j + 1)1−α − j1−α

)

+rn+1∆t

=1

Γ(2− α)

n∑j=0

bjw(x, tn+1−j)− w(x, tn−j)

∆tα+ rn+1

∆t , (5.2.1)

where bj = (j + 1)1−α − j1−α and τ = (tn+1 − s). The coefficients bj have

the following properties

• bj > 0, j = 0, 1, 2, . . . , n,

• 1 = b0 > b1 > b2 > . . . > bn, bn → 0 as n →∞,

• ∑nj=0(bj − bj+1) + bn+1 = (1− b1) +

∑n−1j=1 (bj − bj+1) + bn = 1.

The discrete differential operator Lαt is defined as

Lαt u(x, tn+1) :=

1

Γ(2− α)

n∑j=0


∆tα.


Then Eq.(5.2.1) takes the following form

∂αw(x, tn+1)

∂tα= Lα

t w(x, tn+1) + rn+1∆t . (5.2.2)

The truncation error between Lαt w(x, tn+1) and ∂αw(x,tn+1)

∂tαin [30] is denoted

by rn+1∆t , and

rn+1∆t ≤ Cw∆t2−α, (5.2.3)

where Cw is a constant only related to w.

Using Lαt w(x, tn+1) as an approximation of 1

Γ(1−α)

∫ tn+1

0∂w(x,s)

∂sds

(tn+1−s)α , the

finite difference scheme corresponding to (5.1.1) takes the following form

Lαt w(x, tn+1) + µ

∂4w(x, tn+1)

∂x4= h(x, tn+1),

Substituting the value of Lαt w(x, tn+1), the above equation takes the follow-

ing form

1

Γ(2− α)

n∑j=0


∆tα+ µ

∂4w(x, tn+1)

∂x4= h(x, tn+1).


(wn+1(x) + α0 µ

∂4wn+1

∂x4

)

=

((1− b1)w

n(x) +n−1∑j=1

(bj − bj+1)wn−j(x) + bnw

0(x) + α0hn+1(x)

),

n = 1, 2, . . . , K − 1, (5.2.4)

where wn+1(x) = w(x, tn+1) and α0 = Γ(2− α)∆tα,

with the boundary conditions (I, II, III) and the following initial condition

w0(x) = g0(x), x ∈ [0, L]. (5.2.5)


The presented scheme is a three time level scheme. To apply the presented

scheme, it is first needed to find the values of w at the nodal points at the

zeroth (w0) and first (w1) time levels.

For n = 1, the scheme (5.2.4) can be written as

w2(x) + α0µ∂4w2

∂x4= (1− b1)w

1(x) + b1w0(x) + α0h

2(x).

For n = 0, the scheme takes the following form

w1(x) + α0µ∂4w1

∂x4= w0(x) + α0h

1(x), (5.2.6)

where w0(x) = w(x, 0) = g0(x) is the value of w at the zeroth time level

(the initial condition).

Eqs.(5.2.4) and (5.2.6), along with boundary conditions (I, II, III) and

initial condition (5.2.5) form a complete set of the semi-dicscrete problem

of Eq.(5.1.1).

Following [61], the error term rn+1 is defined as

rn+1 := α0

(∂αw(x, tn+1)

∂tα− Lα

t w(x, tn+1)

). (5.2.7)

Using Eq.(5.2.2) and Eq.(5.2.3) the error term rn+1 becomes

∣∣rn+1∣∣ = Γ(2− α)∆tα

∣∣rn+1∆t

∣∣ ≤ Cw∆t2. (5.2.8)

Some functional spaces endowed with standard norms and inner products,

to be used hereafter, are defined as under

H2(Ω) =

v ∈ L2(Ω),

dv

dx,d2v

dx2∈ L2(Ω)

,


H20 (Ω) =

v ∈ H2(Ω), v|∂Ω = 0,

dv

dx|∂Ω = 0

,

Hm(Ω) =

v ∈ L2(Ω),

dkv

dxkfor all positive integer k ≤ m

,

where L2 (Ω) is the space of measurable functions whose square is Lebesgue

integrable in Ω. The inner products of L2 (Ω) and H2(Ω) are defined, re-

spectively, by

(w, v) =

∫

Ω

wvdx, (w, v)2 = (w, v) +

(dw

dx,dv

dx

)+

(d2w

dx2,d2v

dx2

),

and the corresponding norms by

‖v‖0 = (v, v)12 , ‖v‖2 = (v, v)

122 .

The norm ‖.‖ of the space Hm (Ω) is defined as

‖v‖m =

(m∑

k=0

∥∥∥∥dkv

dxk

∥∥∥∥2

0

) 12

.

Instead of using the above standard H2-norm, it is preferred to define ‖.‖2

by

‖v‖2 =

(‖v‖2

0 + α0µ

∥∥∥∥d2v

dx2

∥∥∥∥2

0

)1/2

, (5.2.9)

where α0 = Γ(2− α)∆tα.

In order to analize the stability and convergence, the following weak formu-

lation of Eq.(5.2.4) and Eq.(5.2.6) is required, i.e. finding wn+1 ∈ H20 (Ω),


such that for all v ∈ H20 (Ω),

((wn+1, v

)+ α0µ

(∂4wn+1

∂x4, v

))

=

((1− b1) (wn, v) +

n−1∑j=1

(bj − bj+1)(wn−j, v

)+ bn

(w0, v

)+ α0

(hn+1, v

))

,

(5.2.10)

and

(w1, v

)+ α0µ

(∂4w1

∂x4, v

)=

(w0, v

)+ α0

(h1, v

). (5.2.11)

The stability analysis for the semi-discrete problem is given in the fol-

lowing theorem.

Theorem 1

The semi-discrete problem is unconditionally stable in the sense that for all

∆t > 0, it holds

∥∥wn+1∥∥

2≤

(∥∥w0

∥∥0+ α0

n+1∑j=1

∥∥hj∥∥

0

), n = 0, 1, 2, . . . , K − 1, (5.2.12)

where ‖.‖2 is defined in (5.2.9).

Proof

The theorem is proved using mathematical induction. When n = 0, let

v = w1 in Eq.(5.2.11), it can be written as

(w1, w1

)+ α0µ

(∂4w1

∂x4, w1

)=

(w0, w1

)+ α0

(h1, w1

).

Using integration by parts two times, the above equation reduces to

(w1, w1

)+ α0µ

(∂2w1

∂x2,∂2w1

∂x2

)=

(w0, w1

)+ α0

(h1, w1

), (5.2.13)


where all the boundary contributions disappeared due to boundary condi-

tions on v.

Using the inequality ‖v‖0 ≤ ‖v‖2 and Schwarz inequality, Eq.(5.2.13) takes

the following form

∥∥w1∥∥2

2≤

∥∥w0∥∥

0

∥∥w1∥∥

0+ α0

∥∥h1∥∥

0

∥∥w1∥∥

0

≤ ∥∥w0∥∥

0

∥∥w1∥∥

2+ α0

∥∥h1∥∥

0

∥∥w1∥∥

2

∥∥w1∥∥

2≤ (∥∥w0

∥∥0+ α0

∥∥h1∥∥

0

).

Suppose that the result holds for v = wj i.e.

∥∥wj∥∥

2≤

(∥∥w0

∥∥0+ α0

j∑i=1

∥∥hi∥∥

0

), j = 2, 3, . . . , n. (5.2.14)

Taking v = wn+1 in Eq.(5.2.10), it can be written as

(wn+1, wn+1

)+ α0µ

(∂4wn+1

∂x4, wn+1

)

= (1− b1)(wn, wn+1

)+

n−1∑j=1

(bj − bj+1)(wn−j, wn+1

)+ bn

(w0, wn+1

)

α0

(hn+1, wn+1

). (5.2.15)

Using integration by parts two times, the above equation takes the following

form

(wn+1, wn+1

)+ α0µ

(∂2wn+1

∂x2,∂2wn+1

∂x2

)

= (1− b1)(wn, wn+1

)+

n−1∑j=1


)

+bn

(w0, wn+1

)+ α0

(hn+1, wn+1

),


where all the boundary contributions disappeared due to boundary condi-

tions on v.

Using the inequality ‖v‖0 ≤ ‖v‖2 and Schwarz inequality, the above equa-

tion becomes

∥∥wn+1∥∥2

2≤ (1− b1) ‖wn‖0

∥∥wn+1∥∥

0+

n−1∑j=1

(bj − bj+1)∥∥wn−j

∥∥0

∥∥wn+1∥∥

0

+bn

∥∥w0∥∥

0

∥∥wn+1∥∥

0+ α0

∥∥hn+1∥∥

0

∥∥wn+1∥∥

0,

or

∥∥wn+1∥∥2

2≤ (1− b1) ‖wn‖0

∥∥wn+1∥∥

2+

n−1∑j=1


∥∥0

∥∥wn+1∥∥

2

+bn

∥∥w0∥∥

0

∥∥wn+1∥∥

2+ α0

∥∥hn+1∥∥

0

∥∥wn+1∥∥

2,

or

∥∥wn+1∥∥

2≤ (1− b1) ‖wn‖0 +

n−1∑j=1


∥∥0+ bn

∥∥w0∥∥

0+ α0

∥∥hn+1∥∥

0.

Using Eq.(5.2.14), the above equation becomes

∥∥wn+1∥∥

2≤

[∥∥w0

∥∥0+ α0

n∑j=1

∥∥hj∥∥

0

]((1− b1) +

n−1∑j=1

(bj − bj+1) + bn

)

+α0

∥∥hn+1∥∥

0. (5.2.16)

Using properties of bj, it can be rewritten as

∥∥wn+1∥∥

2≤

(∥∥w0

∥∥0+ α0

n+1∑j=1

∥∥hj∥∥

0

).

The error analysis for the solution of the semi-discrete problem is

discussed in the following theorem.


Theorem 2

Let w be the exact solution of (5.1.1) and wnKn=0 be the time-discrete

solution of Eqs.(5.2.10) and (5.2.11) with initial condition (5.2.5), then it

holds

‖w(tn)− wn‖2 ≤ Cw,αTα∆t2−α, n = 1, 2, . . . , K. (5.2.17)

The proof of Theorem 2 needs the following lemma.

Lemma 1 Under the assumption of Theorem 2, it holds

‖w(tn)− wn‖2 ≤ Cwb−1n−1∆t2, n = 1, 2, . . . , K. (5.2.18)

Proof

Let en = w(x, tn) − wn(x), for n = 1, by combining Eqs (5.1.1), (5.2.11)

and (5.2.9), the error equation can takes the following form

(e1, v

)+ α0µ

(∂2e1

∂x2,∂2v

∂x2

)=

(e0, v

)+

(r1, v

), ∀v ∈ H2

0 (Ω).

Let v = e1, noting e0 = 0 yields

∥∥e1∥∥

2≤

∥∥r1∥∥

0.

This together with (5.2.8), gives

∥∥w(t1)− w1∥∥

2≤ Cwb−1

0 ∆t2. (5.2.19)

Hence (5.2.18) is proved for the case n = 1.

For inductive part, suppose (5.2.18) holds for n = 1, 2, 3, . . . , s, i.e.

‖w(tn)− wn‖2 ≤ Cwb−1n−1∆t2. (5.2.20)


To prove the lemma for n = s + 1 the Eqs.(5.1.1), (5.2.10) and (5.2.9) are

used and the error equation can be written, for all v ∈ H20 (Ω), as

(en+1, v

)+ α0µ

(∂2en+1

∂x2,∂2v

∂x2

)

= (1− b1) (en, v) +n−1∑j=1

(bj − bj+1)(en−j, v

)+ bn

(e0, v

)+

(rn+1, v

).

The above equation, for v = en+1, can be written as

∥∥en+1∥∥2

2≤ (1− b1) ‖en‖0

∥∥en+1∥∥

0+

n−1∑j=1

(bj − bj+1)∥∥en−j

∥∥0

∥∥en+1∥∥

0

+bn

∥∥e0∥∥

0

∥∥en+1∥∥

0+

∥∥rn+1∥∥

0

∥∥en+1∥∥

0.

Using the induction assumption and the fact thatb−1j

b−1j+1

< 1 for all non

negative integer j, it can be written as

∥∥en+1∥∥

2≤

[(1− b1) +

n−1∑j=1

(bj − bj+1) + bn

]Cwb−1

n ∆t2.

Using the properties of bj, the above equation becomes

∥∥en+1∥∥

2≤ Cwb−1

n ∆t2.

Proof

By the definition of bn, it can be shown that

limn→∞

b−1n−1

nα= lim

n→∞n−α

n1−α − (n− 1)1−α

= limn→∞

n−1

1− (1− 1n)1−α

=1

(1− α)

The function Φ(x) is introduced as Φ(x) := x−α

x1−α−(x−1)1−α .

Since Φ′(x) ≥ 0, ∀x > 1 therefore Φ(x) is increasing on x for all x > 1.


This means that n−αb−1n−1 increasingly tends to 1

(1−α)as 1 < n → ∞. It is

to be noted that n−αb−1n−1 = 1 for n = 1, hence it can takes the following

form

n−αb−1n−1 ≤

1

(1− α), n = 1, 2, . . . , K.

Consequently, for all n such that n∆t ≤ T ,

‖w(tn)− wn‖2 ≤ Cwb−1n−1∆t2

= Cwn−αb−1n−1n

α∆t2−α+α

≤ Cw1

1− α(n∆t)α∆t2−α

≤ Cw,αT α∆t2−α.

5.3 Discretization in space

Consider a uniform mesh with the grid points (xi, tn) to discretize the

region [0, L] × [0, T ], where xi = ih, i = 0, 1, 2, . . . , M , and tn = n∆t,

n = 0, 1, 2, . . . , K, K∆t = T . The quantities h and ∆t are the grid sizes in

the space and time directions, respectively.

Using

W n+1(x) =M+2∑i=−2

piPi(x), (5.3.1)

where pi are unknown parameters to be determined from the boundary con-

ditions and collocation form of the fractional PDE as defined in subsection

1.7.11.




as nodes. So, for i = 0, 1, 2, . . . , M the following relation can be obtained

as

(pn+1

i−2 + 26pn+1i−1 + 66pn+1

i + 26pn+1i+1 + pn+1

i+2

)

+α0µ120

h4

(pn+1

i−2 − 4pn+1i−1 + 6pn+1

i − 4pn+1i+1 + pn+1

i+2

)

= (1− b1)(pn

i−2 + 26pni−1 + 66pn

i + 26pni+1 + pn

i+2

)

+n−1∑j=1

(bj − bj+1)(pn−j

i−2 + 26pn−ji−1 + 66pn−j

i + 26pn−ji+1 + pn−j

i+2

)

+bn

(p0

i−2 + 26p0i−1 + 66p0

i + 26p0i+1 + p0

i+2

)+ α0h

n+1i ,

n = 1, 2, . . . , K − 1. (5.3.2)


linear equations in (M + 5) unknowns pn+1−2 , pn+1

−1 , pn+10 , . . . , pn+1

M+1, pn+1M+2.

((1 + α0µ

120

h4

)pn+1

i−2 +

(26− 4α0µ

120

h4

)pn+1

i−1 +

(66 + 6α0µ

120

h4

)pn+1

i

+

(26− 4α0µ

120

h4

)pn+1

i+1 +

(1 + α0µ

120

h4

)pn+1

i+2

)

= Hi, n = 1, 2, . . . , K − 1, i = 0, 1, 2, . . . , M, (5.3.3)

where

Hi = (1− b1)(pn

i−2 + 26pni−1 + 66pn

i + 26pni+1 + pn

i+2

)

+n−1∑j=1


i−2 + 26pn−ji−1 + 66pn−j

i + 26pn−ji+1 + pn−j

i+2

)

+bn

(p0

i−2 + 26p0i−1 + 66p0

i + 26p0i+1 + p0

i+2

)+ α0h

n+1i .


The unique solution of the system (5.3.3) is obtained by eliminating the pa-

rameters p−2, p−1, pM+1 and pM+2 using the boundary conditions (I,II,III).

After eliminating p−2, p−1, pM+1, pM+2 the system is reduced to a penta-

diagonal system of (M + 1) linear equations in (M + 1) unknowns.


21, . . . , p

2M ]

T, it is necessary to find the value

of C1 = [p10, p

11, . . . , p

1M ]


using collocation method with quintic B-spline basis functions, as

(1 + α0µ

120

h4

)p1

i−2 +

(26− 4α0µ

120

h4

)p1

i−1 +

(66 + 6α0µ

120

h4

)p1

i

+

(26− 4α0µ

120

h4

)p1

i+1 +

(1 + α0µ

120

h4

)p1

i+2

=(p0

i−2 + 26p0i−1 + 66p0

i + 26p0i+1 + p0

i+2

)+ α0h

1i , i = 0, 1, 2, . . . ,M.

(5.3.4)



1−1, p

10, . . . , p

1M , p1

M+1, p1M+2. The unique solution of this

system is obtained by eliminating the parameters p−2, p−1, pM+1 and pM+2

using the boundary conditions (I, II, III).

The time evolution of the numerical solution W n+1 is determined by the

time evolution of the vector Cn+1. This is found by repeatedly solving the

recurrence relationship, once the initial vector C0 = [p00, p

01, . . . , p

0M ]

T, has


In the following section, the time-fractional semilinear fourth-order PDE

has been discussed.


5.4 Time-fractional semilinear fourth-order

partial differential equation

The time-fractional semilinear fourth-order PDE with a fractional derivative

of order α, (0 < α < 1) is considered as

∂αw

∂tα+ µ

∂4w

∂x4+ p (x, t, w, wx, wxx, wxxx) = f(x, t), x ∈ Ω = [0, L], 0 < t ≤ T,


w(x, 0) = g1(x), 0 ≤ x ≤ L.

Using section 5.2, the numerical scheme for time-fractional semilinear fourth-

order PDE can be obtained as

(wn+1(x) + α0 µ

∂4wn+1

∂x4+ p

(x, tn+1, w

n+1, wn+1x , wn+1

xx , wn+1xxx

))

=

((1− b1)w

n(x) +n−1∑j=1

(bj − bj+1)wn−j(x) + bnw0(x) + α0f

n+1(x)

),

n = 1, 2, . . . , K − 1, (5.4.1)


with boundary conditions (I, II, III) and initial condition

w0(x) = g1(x), x ∈ [0, L].



as nodes as discussed in section 5.3.


To test the accuracy of the presented method, five examples are considered



The accuracy of the presented method has been discussed by calculating

maximum norm errors and L2 norm errors between numerical and exact

solutions.

Example 5.1

The following time-fractional fourth-order PDE is considered as

∂0.75w

∂t0.75+ 0.01

∂4w

∂x4= h(x, t), x ∈ [0, 4π], 0 < t ≤ T,


w(x, 0) = 1− cos 2x, x ∈ [0, 4π],

and clamped boundary condition

w(0, t) = w(4π, t) = 0,

wx(0, t) = wx(4π, t) = 0, 0 ≤ t ≤ T.


w(x, t) = 2 (t + 1) sin2 x.

The maximum norm error and L2 norm error for M = 50 and M = 100

with ∆t=0.00001 are tabulated in Table 5.1. In Table 5.1, ∆t is taken very


Table 5.1: The errors ‖eK‖∞ and ‖eK‖2 for different K taken ∆t=0.00001.M K ‖eK‖∞ ‖eK‖2

50 100 2.0964× 10−5 1.7489× 10−6

500 6.9909× 10−5 5.9425× 10−6

1000 1.17692× 10−4 1.0453× 10−5

1500 1.5975× 10−4 1.4443× 10−5

100 100 1.4771× 10−5 1.0535× 10−6

500 1.7313× 10−5 1.0858× 10−6

1000 2.9146× 10−5 1.5952× 10−6

1500 3.9562× 10−5 2.1897× 10−6

small in order to avoid contamination of temporal error and attribute most

of the errors to the spatial discretization. The accuracy of the presented

method is also tested by varying the time level K, which shows that the

presented method is efficient. The Table 5.2 indicates that the presented

method approximates the exact solution very accurately.

The maximum norm error and L2 norm error for different time steps ∆t

with M = 100 and the corresponding convergence rates at T = 0.1 are

tabulated in Table 5.2. From Table 5.2, it is clear that the temporal rates

of convergence of the numerical solutions obtained by the presented method

is in good agreement with the theoretical estimation.

The exact solution and the numerical solution are plotted using M = 40,

K = 1000 and ∆t = 0.00001 as shown in Fig. 5.1. In Fig. 5.2, the solid line

represents the exact solution and the dotted line represents the numerical

solution at K = 1000 time level.

The temporal convergence rate of L2 norm error as a function of ∆t for

α = 0.75 is shown in Fig. 5.3.


Table 5.2: The errors ‖eK‖∞ and ‖eK‖2 of different time steps with M = 100∆t ‖eK‖∞ Rate ‖eK‖2 Rate

0.001 1.8231× 10−3 1.1586× 10−4

0.0005 8.2748× 10−4 1.1396 5.5035× 10−5 1.07390.00025 3.2966× 10−4 1.3277 2.5335× 10−5 1.11920.000125 1.3960× 10−4 1.2397 1.1105× 10−5 1.1968

Exact solution

1.0

1.5

2.0t

020

40x

0.0

0.5

1.0

1.5

2.0

u

,

Numerical solution

1.0

1.5

2.0t

020

40x

0.0

0.5

1.0

1.5

2.0

u

Figure 5.1: The results at M=40, K=1000 and ∆t = 0.00001 for Example5.1

10 20 30 40 50x

0.5

1.0

1.5

2.0

u

Figure 5.2: The exact and numerical solution at K=1000. Dotted line:numerical solution, Solid line: exact solution


Figure 5.3: Errors as a function of the time ∆t for α = 0.75

Example 5.2

Following is the time-fractional fourth-order PDE taking α = 0.5

∂0.50w

∂t0.50+ 0.05

∂4w

∂x4= h(x, t), x ∈ [0, 1], 0 < t ≤ T,

For α = 1, the above equation becomes

∂w

∂t+ 0.05

∂4w

∂x4= h(x, t), x ∈ [0, 1], 0 < t ≤ T, (5.5.1)


w(x, 0) = sin πx, x ∈ [0, 1],

and simply-supported boundary condition

w(0, t) = w(1, t) = 0,

wxx(0, t) = wxx(1, t) = 0, 0 ≤ t ≤ T.


w(x, t) = (t + 1) sin πx.


The maximum norm error and L2 norm error for M = 40 and M = 80 with

∆t=0.00001 are tabulated in Table 5.3 for α = 0.5 and α = 1. In Table 5.3,

∆t is taken very small in order to avoid contamination of temporal error

and attribute most of the errors to the spatial discretization. The accuracy

of the presented method is also tested by varying the time level K, which

shows that the presented method is efficient. The Table 5.3 indicates that

the presented method approximates the exact solution accurately.

The maximum norm error and L2 norm error for different time steps ∆t with

M = 100 and the corresponding convergence rates at T = 0.1 are tabulated

in Table 5.4 for α = 0.5 and α = 1. From Table 5.4, it is clear that the

temporal rates of convergence of the numerical solutions obtained by the

presented method for time-fractional fourth-order PDE is of O(∆t2−α) and

for fourth-order PDE is of O(∆t).


K = 500 and ∆t =0.00001 as shown in Fig. 5.4. In Fig. 5.5, the solid line






Table 5.3: The errors ‖eK‖∞ and ‖eK‖2 for different M taken ∆t=0.00001.α = 0.5 M K ‖eK‖∞ ‖eK‖2

40 100 1.2964× 10−4 1.4494× 10−5

500 2.3241× 10−4 2.5985× 10−5

1000 2.8285× 10−4 3.1624× 10−5

1500 3.1247× 10−4 3.4935× 10−5

80 100 2.4905× 10−5 1.9689× 10−6

500 5.0599× 10−5 4.0002× 10−6

1000 6.3211× 10−5 4.9972× 10−6

1500 7.0617× 10−5 5.5828× 10−6

α = 1 M K ‖eK‖∞ ‖eK‖2

40 100 2.4509× 10−6 2.7402× 10−7

500 1.2161× 10−5 1.3596× 10−6

1000 2.4091× 10−5 2.6934× 10−6

1500 3.5796× 10−5 4.0021× 10−6

80 100 5.7615× 10−7 4.5548× 10−8

500 2.8591× 10−6 2.2603× 10−7

1000 5.6648× 10−6 4.4785× 10−7

1500 8.4186× 10−6 6.6555× 10−7

Exact solution

1.0 1.5 2.0

t

1020 30 40

x

0.0

0.5

1.0

u

,

Numerical solution

1.0 1.5 2.0

t

1020 30 40

x

0.0

0.5

1.0

u


0.2 0.4 0.6 0.8 1.0x

0.05

0.10

0.15

0.20

0.25

0.30

u



Table 5.4: The errors ‖eK‖∞ and ‖eK‖2 of different time steps with M = 100α = 0.5 ∆t ‖eK‖∞ Rate ‖eK‖2 Rate

0.001 9.0468× 10−4 7.1521× 10−5

0.0005 4.0476× 10−4 1.1603 3.1999× 10−5 1.16030.00025 1.5480× 10−4 1.3867 1.2238× 10−5 1.38670.000125 5.4821× 10−5 1.4976 4.3575× 10−6 1.4898

α = 1 ∆t ‖eK‖∞ Rate ‖eK‖2 Rate0.001 3.5145× 10−4 2.4851× 10−5

0.0005 1.5919× 10−4 1.1425 1.1257× 10−5 1.14250.00025 7.9929× 10−5 0.9939 5.6498× 10−6 0.99450.000125 3.9963× 10−5 1.0001 2.8239× 10−6 1.0005


Example 5.3

The following time-fractional fourth-order PDE is considered as

∂0.90w

∂t0.90+ 0.01

∂4w

∂x4= h(x, t), x ∈ [0, 1], 0 < t ≤ T,


w(x, 0) =

(π5 sin πx +

1

π5cos πx− 1

π5cos 3πx

), x ∈ [0, 1],


and transversely supported boundary condition

w(0, t) = w(1, t) = 0,

wxx(0, t)− 8

π9wx(0, t) = wxx(1, t)− 8

π9wx(1, t) = 0, 0 ≤ t ≤ T.


w(x, t) = (t + 1)

(π5 sin πx +

1

π5cos πx− 1

π5cos 3πx

).

The maximum norm error and L2 norm error M = 20 and M = 40 with

∆t=0.00001 are tabulated in Table 5.5. In Table 5.5, ∆t is taken very


of the errors to the spatial discretization. The accuracy of the developed


presented method is efficient.

The maximum norm error and L2 norm error for different time steps ∆t


tabulated in Table 5.6. From Table 5.6, it is clear that the temporal rate

of convergence of the numerical solutions obtained by the presented method



K = 1000 and ∆t = 0.00001 as shown in Fig. 5.7. In Fig. 5.8, the solid line





Table 5.5: The errors ‖eK‖∞ and ‖eK‖2 for different M taken ∆t=0.00001.M K ‖eK‖∞ ‖eK‖2

20 100 1.7601× 10−4 2.7996× 10−5

500 3.0843× 10−4 4.8934× 10−5

1000 3.7339× 10−4 5.9205× 10−5

1500 4.1152× 10−4 6.5235× 10−5

40 100 4.8103× 10−5 5.4639× 10−6

500 8.6046× 10−5 9.7178× 10−6

1000 1.0469× 10−4 1.1807× 10−5

1500 1.1564× 10−4 1.3034× 10−5



0.001 2.9978× 10−4 2.1119× 10−5

0.0005 1.4687× 10−4 1.0294 1.0303× 10−5 1.03550.00025 7.0485× 10−5 1.0592 4.8988× 10−6 1.07260.000125 3.2454× 10−5 1.1189 2.2043× 10−6 1.1521


Exact solution

1.0

1.5

2.0

t

0

20

40

60

80

x

0.0

0.5

1.0

u

,

Numerical solution

1.0

1.5

2.0

t

0

20

40

60

80

x

0.0

0.5

1.0

u


0.2 0.4 0.6 0.8 1.0x

0.2

0.4

0.6

0.8

1.0

u




Example 5.4

The following time-fractional semilinear fourth-order PDE is considered as

∂0.5w

∂t0.5+ 0.1

∂4w

∂x4+ w(1 + w) = f(x, t), x ∈ [0, 1], 0 < t ≤ T,


w(x, 0) = sin πx, x ∈ [0, 1],


w(0, t) = w(1, t) = 0, t ≥ 0

wxx(0, t) = wxx(1, t) = 0, t ≥ 0.


w(x, t) = (t + 1) sin πx.

The Maximum norm error and L2 norm error for M = 40 and M = 80

with ∆t = 0.00001 are tabulated in Table 5.7. In Table 5.7, ∆t is taken

small enough to avoid contamination of temporal error and attribute most



developed method is efficient. The Table 5.7 indicates that the presented

method approximates the exact solution very accurately.

The Maximum norm error and L2 norm error for different time steps ∆t

with M = 80 at T = 0.1 are tabulated in Table 5.8. From Table 5.8,


Table 5.7: The errors ‖eK‖∞ and ‖eK‖2 for different M taken ∆t = 0.00001M K ‖eK‖∞ ‖eK‖2

40 100 1.2248× 10−4 1.3699× 10−5

500 2.1295× 10−4 2.3817× 10−5

1000 2.5636× 10−4 2.8672× 10−5

1500 2.8210× 10−4 3.1551× 10−5

80 100 2.4135× 10−5 1.9087× 10−6

500 5.030× 10−5 3.9775× 10−6

1000 6.4622× 10−5 5.1098× 10−6

1500 7.3991× 10−5 5.8504× 10−6


0.001 5.6964× 10−4 4.5024× 10−5

0.0005 2.1929× 10−4 1.3772 1.9514× 10−5 1.20620.00025 8.2242× 10−5 1.4149 7.2560× 10−6 1.42730.000125 2.9006× 10−5 1.5035 2.5683× 10−6 1.4984

it is clear that the temporal rate of convergence of the numerical solutions

obtained by the presented method is in good agreement with the theoretical

estimation.


Example 5.5

The following time-fractional semilinear fourth-order PDE is considered as

∂0.99w

∂t0.99+ 10

∂4w

∂x4+ wwx = f(x, t), x ∈ [0, 1], 0 < t ≤ T,


w(x, 0) = sin πx, x ∈ [0, 1],


w(0, t) = w(1, t) = 0, t ≥ 0

wxx(0, t) = wxx(1, t) = 0, t ≥ 0.


w(x, t) = (t + 1) sin πx.

The Maximum norm error and L2 norm error for M = 50 and M = 100 with

∆t = 0.00001 are tabulated in Table 5.9. In Table 5.9, ∆t is taken very




presented method is efficient. The Table 5.9 indicates that the presented

method approximates the exact solution accurately.


with M = 60 at T = 0.1 are tabulated in Table 5.10. From Table 5.10, it


Table 5.9: The errors ‖eK‖∞ and ‖eK‖2 for different M taken ∆t = 0.00001M K ‖eK‖∞ ‖eK‖2

50 100 4.9710× 10−4 3.7272× 10−5

500 4.5161× 10−4 3.7668× 10−5

1000 4.5545× 10−4 3.7977× 10−5

1500 4.5897× 10−4 3.8253× 10−5

100 100 2.5416× 10−4 1.5143× 10−5

500 2.5594× 10−4 1.5263× 10−5

1000 2.5845× 10−4 1.5413× 10−5

1500 2.6085× 10−4 1.5561× 10−5

is clear that the temporal rates of convergence of the numerical solutions

obtained by the presented method is in good agreement with the theoretical

estimation.


0.001 8.6545× 10−4 7.1902× 10−5

0.0005 4.3327× 10−4 0.99818 3.5980× 10−5 0.99880.00025 2.4427× 10−5 0.82679 1.7904× 10−5 1.00690.000125 1.2229× 10−5 0.99817 8.8627× 10−6 1.0145

5.6 Conclusion

In this chapter, a numerical method has been presented for the numerical

solution of time-fractional fourth-order PDEs. The backward Euler formula

has been used for temporal discretization and quintic B-spline collocation

method has been used for spatial discretization. It has been shown that the

temporal discretization is unconditionally stable and the numerical solu-

tions converge to exact solutions with order O(∆t2−α). The parameters M ,

∆t and K are varied in order to test the accuracy of the presented method.


The presented method is also valid and efficient for different values of α,

(0 < α < 1) and provide efficient numerical results for the three boundary

conditions (I, II, III).

Chapter 6

Numerical Solution ofTime-FractionalConvection-Diffusion Equation

The time-fractional convection-diffusion equations found in many applica-

tions such as in the transport of air and ground water pollutants, oil reser-

voir flow, in the modeling of semiconductors, etc.

In this chapter, a numerical method is developed to solve the time-fractional

convection-diffusion equation using the cubic B-spline collocation technique.

The stability and convergence analysis related to the temporal discretiza-

tion are discussed and proven, theoretically.

The contents of this chapter have been submitted for publication in the

form of a research paper [50].

131

Ch 6: Solution of Time-fractional Convection-Diffusion Equation 132

6.1 Introduction

In the current decade, much attention has been made in the field of frac-

tional calculus. Many physical phenomena in different field such as en-

gineering and other sciences can be formulated very efficiently by models

using mathematical tools from fractional calculus.

In recent years, the use of fractional partial differential equation (FPDEs)

has become increasingly popular in mathematical models. FPDEs can be

classified into two principal types: space-fractional differential equation and

time-fractional differential equation.

The time-fractional convection-diffusion equation is obtained from the stan-

dard convection-diffusion equation by replacing the first-order time deriva-

tive with a fractional derivative (of order α, with 0 < α ≤ 1).

The equation takes the following form

∂αw

∂tα= m

∂2w

∂x2− d

∂w

∂x+ h(x, t), x ∈ Ω = [0, L], 0 < t ≤ T, (6.1.1)

subject to the initial condition

w(x, 0) = g0(x), x ∈ Ω, (6.1.2)


w(0, t) = w(L, t) = 0, t ≥ 0, (6.1.3)


where α (0 < α ≤ 1) is the order of the time-fractional derivative, m > 0

and d > 0 are considered to be positive constants quantifying the ad-

vection(convection) and diffusion processes, respectively. The source term

h(x, t), accounts for an insertion or extraction of mass from the system as

it evolves with time. Specifically, h(x, t) represents the time rate of change

of concentration due to external factors, such as a source or a sink.


in Eq.(6.1.1) is defined by the Caputo

fractional derivatives of order α, in [41], given by

∂αw(x, t)

∂tα=

1

Γ(1− α)

∫ t

0

∂w(x, s)

∂s

ds

(t− s)α, 0 < α < 1. (6.1.4)

For α = 1, Eq.(6.1.1) reduces to standard convection-diffusion equation

∂w

∂t= m

∂2w

∂x2− d

∂w

∂x+ h(x, t), (6.1.5)

while the case for α = 0 corresponds to the classical Helmholtz elliptic

equation.

The convection-diffusion equation is a parabolic PDE,which describes a

physical phenomena, how particles, energy or other physical quantities are

transferred inside a physical system through two processes: convection and

diffusion. Convection is a process that describes the movement of molecules

within fluids, whereas, diffusion describes the spread of particles from region

of higher concentration to region of lower concentration through random

motion of particles.


6.2 The cubic B-spline

Let ∆∗ = 0 = x0 < x1 < x2 < · · · < xM = L be the partition of [0, L].

The spacial step length is denoted by h, h = xi − xi−1, i = 1, 2, . . . , M .

Let Bi be cubic B-spline basis functions with knots at the points xi, i =

0, 1, . . . , M . Thus, an approximation W n+1(x) to the exact solution wn+1(x)

at n + 1 time level, can be expressed in terms of the cubic B-spline basis

functions Pi(x) as

W n+1(x) =M+1∑i=−1

piPi(x), (6.2.1)

where pi are unknown time dependent quantities to be determined from

the boundary conditions and collocation form of the fractional differential

equation.

The cubic B-spline Pi(x), i = −1, 0, . . . ,M + 1 can be defined as under

Pi(x) =1

h3

(x− xi−2)3, x ∈ [xi−2, xi−1[,

h3 + 3h2(x− xi−1) + 3h(x− xi−1)2 − 3(x− xi−1)

3, x ∈ [xi−1, xi[,

h3 + 3h2(xi+1 − x) + 3h(xi+1 − x)2 − 3(xi+1 − x)3, x ∈ [xi, xi+1[,

(xi+2 − x)3, x ∈ [xi+1, xi+2[,

0, otherwise

The values of successive derivatives P(r)i (x), i = −1, . . . , M + 1; r = 0, 1, 2

at nodes, are listed in Table 6.1.


Table 6.1: Coefficients of cubic B-spline and its derivatives at knots xi.xi−2 xi−1 xi xi+1 xi+2 else

Pi(x) 0 1 4 1 0 0P ′

i (x) 0 3h 0 −3

h 0 0P ′′

i (x) 0 6h2

−12h2

6h2 0 0



has been discretized by the first-order

backward Euler formula. Let tn = n∆t, n = 0, 1, 2, . . . , K, where ∆t = TK

is the temporal step size, wn(x) is an approximation to the value of w(x, t)

at a time point t = tn, n = 0, 1, . . . , K − 1.


approximated as

∂αw(x, tn+1)

∂tα=

1

Γ(1− α)

∫ tn+1

0

∂w(x, s)

∂s

ds

(tn+1 − s)α

=1

Γ(1− α)

n∑j=0

∫ tj+1

tj

∂w(x, s)

∂s

ds

(tn+1 − s)α,

=1

Γ(1− α)

n∑j=0


∆t

∫ tj+1

tj

ds

(tn+1 − s)α

+rn+1∆t ,

=1

Γ(1− α)

n∑j=0


∆t

∫ tn+1−j

tn−j

dτ

τα+ rn+1

∆t ,

=1

Γ(1− α)

n∑j=0


∆t

∫ tj+1

tj

dτ

τα+ rn+1

∆t ,

=1

Γ(2− α)

n∑j=0


∆tα((j + 1)1−α − j1−α

)

+rn+1∆t ,

=1

Γ(2− α)

n∑j=0


∆tα+ rn+1

∆t , (6.3.1)




• bj > 0, j = 0, 1, 2, . . . , n,

• 1 = b0 > b1 > b2 > . . . > bn, bn → 0 as n →∞,

• ∑nj=0(bj − bj+1) + bn+1 = (1− b1) +

∑n−1j=1 (bj − bj+1) + bn = 1.

The semi-discrete differential operator Lαt is defined as

Lαt w(x, tn+1) :=

1

Γ(2− α)

n∑j=0


∆tα.


∂αw(x, tn+1)

∂tα= Lα

t w(x, tn+1) + rn+1∆t .



by rn+1∆t , and

rn+1∆t ≤ Cw∆t2−α,


Using Lαt w(x, tn+1) as an approximation of 1

Γ(1−α)

∫ tn+1

0∂w(x,s)

∂sds

(tn+1−s)α , the

finite difference scheme corresponding to (6.1.1) takes the following form

Lαt w(x, tn+1)−m

∂2w(x, tn+1)

∂x2+ d

∂w(x, tn+1)

∂x= h(x, tn+1),

Substituting the value of Lαt w(x, tn+1), it can be written as

(1

Γ(2− α)

n∑j=0


∆tα−m

∂2w(x, tn+1)

∂x2+ d

∂w(x, tn+1)

∂x

)

= h(x, tn+1).



(wn+1(x)− α0 m

∂2wn+1

∂x2+ α0 d

∂wn+1

∂x

)

=

((1− b1)w

n(x) +n−1∑j=1

(bj − bj+1)wn−j(x) + bnw0(x) + α0h

n+1(x)

), n ≥ 1,

(6.3.2)


along with boundary conditions

wn+1(0) = wn+1(L) = 0, n ≥ 0, (6.3.3)

and the following initial condition

w0(x) = g0(x), x ∈ [0, L]. (6.3.4)

To apply the presented scheme, it is first needed to find the values of w at

the nodal points at the zeroth (w0) and first (w1) time levels.

For n = 1, the scheme (6.3.2) takes the following form

w2(x)− α0 m∂2w2

∂x2+ α0 d

∂w2

∂x= (1− b1)w

1(x) + b1w0(x) + α0h

2(x).

For n = 0, the presented scheme, takes the following form

w1(x)− α0 m∂2w1

∂x2+ α0 d

∂w1

∂x= w0(x) + α0h

1(x). (6.3.5)



Eqs.(6.3.2) and (6.3.5), together with boundary conditions (6.3.3) and ini-

tial condition (6.3.4) form a complete set of the semi-discrete problem of


Eqs.(6.1.1)-(6.1.3).

Following Wulan Li and Xu Da [61], the error term rn+1 is defined as

rn+1 := α0

(∂αw(x, tn+1)

∂tα− Lα

t w(x, tn+1)

)(6.3.6)

Substituting the value of α0, the error term rn+1 becomes

∣∣rn+1∣∣ = Γ(2− α)∆tα

∣∣rn+1∆t

∣∣ ≤ Cw∆t2, (6.3.7)

where Cw is a positive constant.



H10 (Ω) =

v ∈ H1(Ω), v|∂Ω = 0

,

(w, v)1 = (w, v) +

(∂w

∂x,∂v

∂x

),

‖v‖0 = (v, v)1/2, ‖v‖1 = (v, v)1/21 .


by

‖v‖1 =

(‖v‖2

0 + α0

∥∥∥∥∂v

∂x

∥∥∥∥2

0

)1/2

, (6.3.8)

where α0 = Γ(2− α)∆tα.

In order to analize the stability and convergence, the following weak formu-




((wn+1, v

)− α0m

(∂2wn+1

∂x2, v

)+ α0d

(∂wn+1

∂x, v

))

=

((1− b1) (wn, v) +

n−1∑j=1


)+ bn

(w0, v

)

+α0

(hn+1, v

)), (6.3.9)

and

((w1, v

)− α0m

(∂2w1

∂x2, v

)+ α0d

(∂w1

∂x, v

))

=((

w0, v)

+ α0

(h1, v

)). (6.3.10)

6.4 Stability and convergence analysis

The stability analysis for the semi-discrete problem is given in the following

theorem

Theorem 1

The semi-discrete problems for n ≥ 1 and n = 0 as given in (6.3.9) and

(6.3.10), are unconditionally stable in the sense that for all ∆t > 0, it holds

∥∥wn+1∥∥

1≤

(∥∥w0

∥∥1+ α0

n+1∑j=1

∥∥hj∥∥

0

), n = 0, 1, 2, . . . , K − 1 (6.4.1)

where ‖.‖1 is defined in (6.3.8).

Proof

Mathematical induction is used to prove the theorem. When n = 0, let


(w1, w1

)− α0m

(∂2w1

∂x2, w1

)+ α0d

(∂w1

∂x, w1

)=

(w0, w1

)+ α0

(h1, w1

).


Using integration by parts and the boundary conditions (6.3.3), the above

equation becomes

(w1, w1

)+ α0m

(∂w1

∂x,∂w1

∂x

)=

(w0, w1

)+ α0

(h1, w1

). (6.4.2)

Using Eq.(6.3.8), it can be written as

∥∥w1∥∥2

1=

(w1, w1

)+ α0m

(∂w1

∂x,∂w1

∂x

). (6.4.3)

From Eq.(6.3.8), it can be written as

‖v‖0 ≤ ‖v‖1 ,

∥∥∥∥∂v

∂x

∥∥∥∥0

≤√

1

α0

‖v‖1 . (6.4.4)

Using Eqs.(6.4.2) and (6.4.3), Schwarz inequality and Eq.(6.4.4), it can be

written as

∥∥w1∥∥2

1≤

∥∥w0∥∥

0

∥∥w1∥∥

0+ α0

∥∥h1∥∥

0

∥∥w1∥∥

0

≤∥∥w0

∥∥1

∥∥w1∥∥

1+ α0

∥∥h1∥∥

0

∥∥w1∥∥

1

∥∥w1∥∥

1≤ (∥∥w0

∥∥1+ α0

∥∥h1∥∥

0

)

Suppose that the result hold for v = wj i.e.

∥∥wj∥∥

1≤

(∥∥w0

∥∥1+ α0

j∑i=1

∥∥hi∥∥

0

), j = 2, 3, . . . , n (6.4.5)


((wn+1, wn+1

)− α0m

(∂2wn+1

∂x2, wn+1

)+ α0d

(∂wn+1

∂x, wn+1

))

=

((1− b1)

(wn, wn+1

)+

n−1∑j=1


)+ bn

(w0, wn+1

)

+α0

(hn+1, wn+1

)).


Using integration by parts and boundary conditions, the above equation

can be written as

((wn+1, wn+1

)+ α0m

(∂wn+1

∂x,∂wn+1

∂x

))

=

((1− b1)

(wn, wn+1

)+

n−1∑j=1


)+ bn

(w0, wn+1

)

+α0

(hn+1, wn+1

))

Using Eq. (6.3.8), Schwarz inequality and Eq. (6.4.4), it can be written as

∥∥wn+1∥∥2

1≤ (1− b1) ‖wn‖0

∥∥wn+1∥∥

0+

n−1∑j=1


∥∥0

∥∥wn+1∥∥

0

+bn

∥∥w0∥∥

0

∥∥wn+1∥∥

0+ α0

∥∥hn+1∥∥

0

∥∥wn+1∥∥

0,

or

∥∥wn+1∥∥2

1≤ (1− b1) ‖wn‖1

∥∥wn+1∥∥

1+

n−1∑j=1


∥∥1

∥∥wn+1∥∥

1

+bn

∥∥w0∥∥

1

∥∥wn+1∥∥

1+ α0

∥∥hn+1∥∥

0

∥∥wn+1∥∥

1,

or

∥∥wn+1∥∥

1≤ (1− b1) ‖wn‖1 +

n−1∑j=1


∥∥1+ bn

∥∥w0∥∥

1

+α0

∥∥hn+1∥∥

0.

Using Eq. (6.4.5), the above equation becomes

∥∥wn+1∥∥

1≤

[∥∥w0

∥∥1+ α0

n∑j=1

∥∥hj∥∥

0

]((1− b1) +

n−1∑j=1

(bj − bj+1) + bn

)

+α0

∥∥hn+1∥∥

0.


The above can be rewritten as

∥∥wn+1∥∥

1≤

(∥∥w0

∥∥1+ α0

n+1∑j=1

∥∥hj∥∥

0

).

The error analysis of the semi-discrete problem is discussed in the follow-

ing theorem.

Theorem 2

Let w be the exact solution of Eqs.(6.1.1)-(6.1.3) and wnKn=0 be the time-

discrete solution of Eqs.(6.3.9) and (6.3.10) with initial condition (6.3.4),

then it holds

‖w(tn)− wn‖1 ≤ Cw,αTα∆t2−α, n = 1, 2, . . . , K (6.4.6)

The proof of the Theorem needs the following lemma.

Lemma 1 Under the assumption of Theorem 2, the following can be

written as

‖w(tn)− wn‖1 ≤ Cwb−1n−1∆t2, n = 1, 2, . . . , K. (6.4.7)

Proof


and (6.3.7), the error equation becomes

(e1, v

)+ α0m

(∂e1

∂x,∂v

∂x

)=

(e0, v

)+

(r1, v

), ∀v ∈ H1

0 (Ω).

Let v = e1, noting e0 = 0, it can be written as

∥∥e1∥∥

1≤

∥∥r1∥∥

0.


Using Eq.(6.3.7), it leads to

∥∥w(t1)− w1∥∥

1≤ Cw∆t2 = Cwb−1

0 ∆t2. (6.4.8)


Suppose (6.4.8) holds for n = 1, 2, 3, . . . , s, i.e.

‖w(tn)− wn‖1 ≤ Cwb−1n−1∆t2. (6.4.9)

To prove the lemma for n = s + 1 the Eqs. (6.1.1), (6.3.9) and (6.3.7) are

used and the error equation takes the following form, for all v ∈ H10 (Ω), as

((en+1, v

)+ α0m

(∂en+1

∂x,∂v

∂x

))

=

((1− b1) (en, v) +

n−1∑j=1


)+ bn

(e0, v

)+

(rn+1, v

))

.

The above equation, for v = en+1, becomes

∥∥en+1∥∥2

1≤ (1− b1) ‖en‖0

∥∥en+1∥∥

0+

n−1∑j=1


∥∥0

∥∥en+1∥∥

0

+bn

∥∥e0∥∥

0

∥∥en+1∥∥

0+

∥∥rn+1∥∥

0

∥∥en+1∥∥

0.


b−1j+1

< 1 for all non

negative integer j, it can takes the following form

∥∥en+1∥∥

1≤

[(1− b1) +

n−1∑j=1

(bj − bj+1) + bn

]Cwb−1

n ∆t2.


∥∥en+1∥∥

1≤ Cwb−1

n ∆t2.


Proof


limn→∞

b−1n−1

nα= lim

n→∞n−α

n1−α − (n− 1)1−α

= limn→∞

n−1

1− (1− 1n)1−α

=1

(1− α)


x1−α−(x−1)1−α .



(1−α)as 1 < n → ∞. It is

to be noted that n−αb−1n−1 = 1 for n = 1, hence it can be written as

n−αb−1n−1 ≤

1

(1− α), n = 1, 2, . . . , K.


‖w(tn)− wn‖1 ≤ Cwb−1n−1∆t2,


α∆t2−α+α,

≤ Cw1

1− α(n∆t)α∆t2−α,



Consider a uniform mesh with the grid points λin to discretize the region

[0, L] × [0, T ]. Each, λin is the grid point (xi, tn) where xi = ih, i =


0, 1, 2, . . . , M and tn = n∆t, n = 0, 1, 2, . . . , K. h and ∆t are the mesh

sizes in the space and time directions, respectively.




as

((pn+1

i−1 + 4pn+1i + pn+1

i+1

)− α06

h2

(pn+1

i−1 − 2pn+1i + pn+1

i+1

)+ α0

3

h

(−pn+1i−1 + pn+1

i+1

))

=

((1− b1)

(pn

i−1 + 4pni + pn

i+1

)+

n−1∑j=1


i−1 + 4pn−ji + pn−j

i+1

)

+bn

(p0

i−1 + 4p0i + p0

i+1

)+ α0h

n+1i

), n ≥ 1. (6.5.1)


linear equations in (M + 3) unknowns pn+1−1 , pn+1

0 , pn+11 , . . . , pn+1

M , pn+1M+1,

((1− α0

6

h2m− α0

3

hd

)pn+1

i−1 +

(4 + α0

12

h2

)pn+1

i +

(1− α0

6

h2m + α0

3

hd

)pn+1

i+1

)

= Fi, n ≥ 1, i = 0, 1, 2, . . . , M, (6.5.2)

where

Fi = (1− b1)(pn

i−1 + 4pni + pn

i+1

)+

n−1∑j=1



i+1

)

+bn

(p0

i−1 + 4p0i + p0

i+1

)+ α0h

n+1i .



w(0, t) = (p−1 + 4p0 + p1) = 0,

w(L, t) = (pM−1 + 4pM + pM+1) = 0.


p−1 = −4p0 − p1,

pM+1 = −4pM − pM−1.

After eliminating p−1 and pM+1, the system is reduced to a tri-diagonal

system of (M + 1) linear equations in (M + 1) unknowns. This system can

be rewritten in matrix form as

ACn+1 = F, n = 1, 2, 3, . . . , K, (6.5.3)

where

Cn+1 =[pn+1

0 , pn+11 , . . . , pn+1

M

]T, n = 1, 2, 3, . . . , K.


A =

α0

(36h2 m + 12

hd)

α06hd

x y z

x y z. . . . . . . . .

x y z

α06hd α0

(36h2 m + 12

hd)

,

where

x =

(1− α0

6

h2m− α0

3

hd

),

y =

(4 + α0

12

h2m

),

z =

(1− α0

6

h2m + α0

3

hd

).


21, . . . , p

2M ]

T, it is first needed to find the value

of C1 = [p10, p

11, . . . , p

1M ]



using cubic B-spline collocation method, as

((1− a0

3

h2

)p1

i−1 +

(4 + a0

6

h2

)p1

i +

(1− a0

3

h2

)p1

i+1

)

= Fi, i = 0, 1, 2, . . . , M, (6.5.4)

where

Fi =(p0

i−1 + 4p0i + p0

i+1

)+

1

2a0h

1i .



10, . . . , p

1M , p1

M+1. The unique solution of this system is

obtained by eliminating the parameters p−1 and pM+1 using the boundary

conditions.

The time evolution of the numerical solution W n+1 is determined by the


recurrence relationship, after the initial vector C0 = [p00, p

01, . . . , p

0M ]

T, has


To test the accuracy of the proposed method, two examples are considered



In order to test the accuracy of the presented method, the maximum norm

errors and L2 norm errors between numerical and exact solution are calcu-

lated.


Example 6.1

Following is the time-fractional convection-diffusion equation

∂0.5w

∂t0.5=

∂2w

∂x2− 0.1

∂w

∂x+ h(x, t), x ∈ [0, L], 0 < t ≤ T,


w(x, 0) = sin πx, x ∈ [0, 1],


w(0, t) = w(1, t) = 0, t ≥ 0.


w(x, t) = (t + 1) sin πx

The Maximum norm error and L2 norm error M = 100 and M = 200 with

∆t=0.00001 are tabulated in Table 6.2. In Table 6.2, ∆t is taken very




presented method is efficient.


with M = 100 and the corresponding rates of convergence at T = 0.1 are

tabulated in Table 6.3. From Table 6.3, it is clear that the temporal rate of

convergence of the numerical solutions obtained by the presented method



100 100 1.3410× 10−5 1.0603× 10−6

500 3.1053× 10−5 2.3020× 10−6

1000 3.9710× 10−5 2.9143× 10−6

1500 4.4784× 10−5 3.2737× 10−6

200 100 4.1912× 10−6 2.0283× 10−7

500 2.5886× 10−6 6.7328× 10−7

1000 2.6470× 10−6 1.5061× 10−7

1500 3.7971× 10−6 2.1056× 10−7


0.001 9.2156× 10−4 6.5057× 10−5

0.0005 4.2161× 10−4 1.1282 2.9707× 10−5 1.13090.00025 1.7168× 10−4 1.2962 1.2036× 10−5 1.30340.000125 6.0681× 10−5 1.5004 4.2184× 10−6 1.5126


Example 6.2

The following time-fractional convection-diffusion equation is considered as

∂0.85w

∂t0.85= 0.1

∂2w

∂x2− 0.1

∂w

∂x+ h(x, t), x ∈ [0, L], 0 < t ≤ T,


w(x, 0) = sin πx, x ∈ [0, 1],

and boundary conditions

w(0, t) = w(1, t) = 0, t ≥ 0.


w(x, t) =(t2 + 2t + 1

)sin πx.



100 100 6.8578× 10−5 1.6233× 10−6

500 8.4971× 10−5 1.8680× 10−6

1000 9.0433× 10−5 2.0131× 10−6

1500 9.3537× 10−5 2.1129× 10−6

200 100 2.1075× 10−5 1.010× 10−6

500 2.3506× 10−5 1.0164× 10−6

1000 2.4360× 10−5 1.0179× 10−6

1500 2.4894× 10−5 1.1179× 10−6

The Maximum norm error and L2 norm error for M = 100 and M =

200 with ∆t=0.00001 are tabulated in Table 6.4. In Table 6.4, ∆t is

taken very small in order to avoid contamination of temporal error and

attribute most of the errors to the spatial discretization. The accuracy of

the developed method is also tested by varying the time level K, which

shows that the developed method is efficient. It is clear from the Table 6.4

that the developed method approximates the exact solution accurately.



tabulated in Table 6.5. From Table 6.5, it is clear that the temporal rates

of convergence of the numerical solutions obtained by the developed method



0.001 2.2279× 10−3 1.5696× 10−4

0.0005 1.1052× 10−3 1.0114 7.760× 10−5 1.01630.00025 5.4513× 10−4 1.0196 3.8062× 10−5 1.02770.000125 2.6564× 10−4 1.0371 1.8432× 10−5 1.04610.0000625 1.1210× 10−4 1.1345 8.4059× 10−6 1.1327


6.7 Conclusion

In this chapter, an accurate method has been presented for the numerical

solution of time-fractional convection-diffusion equations. The backward

Euler formula has been used for temporal discretization and cubic B-spline

collocation technique has been used for spatial discretization. It has been

proved that the temporal discretization is unconditionally stable and the nu-

merical solutions converge to exact solutions with order O(∆t2−α) where ∆t

is the temporal step size. The cubic B-spline collocation method possesses

high level of accuracy in dealing with the given problems. The efficiency

of the presented method was measured by calculating the maximum norm

error and L2-norm error, tabulated in Tables 6.2-6.5. The accuracy of the

presented method is also tested by varying the values of the parameters M ,

∆t and K. The presented method is also valid and accurate for different

values of α, (0 < α < 1). It is observed from the numerical examples that

the numerical solutions efficiently approximate the exact solutions.

Chapter 7

B-Spline Solution ofTime-Fractional IntegroPartial Differential EquationWith a Weakly SingularKernel

The fractional integro PDEs are commonly used in the mathematical model-

ing of various physical phenomena. In this chapter, a collocation technique

is proposed for the numerical solution of fractional integro PDE with a

weakly singular kernel, appeared in viscoelastic forces. The method is based

on finite difference scheme in time direction and cubic B-spline collocation

method for spatial derivatives. The stability and convergence analysis of

temporal discretization are discussed and proven, theoretically.

The contents of this chapter have been submitted in the form of a research

paper [49].

152

Ch 7: Solution of Fractional-Integro PDE 153

7.1 Introduction

Fractional partial differential equation and fractional integro partial differ-

ential equation are used in mathematical modeling of various physical and

chemical processes.

Collocation method with cubic B-spline as basis functions has been pro-

posed for the numerical solution of the following fractional integro PDE

with a weakly singular kernel of the form

∂αw(x, t)

∂tα−

∫ t

0

(t− s)−1/2 wxx(x, s)ds = f(x, t), (7.1.1)

where x ∈ [0, 1], 0 < t ≤ T, 0 < α < 1,

subject to initial condition

w(x, 0) = g0(x), 0 ≤ x ≤ 1, (7.1.2)

along with boundary conditions

w(0, t) = 0, w(1, t) = 0, 0 < t ≤ T, (7.1.3)

where g0(x) and f(x, t) are known smooth functions.


in Eq.(7.1.1) is described by the Caputo

fractional derivative of order α, given by

∂αw(x, t)

∂tα=

1

Γ(1−α)

∫ t

0∂w(x,s)

∂sds

(t−s)α , 0 < α < 1,∂w(x,t)

∂t, α = 1.


For α = 1, Eq.(7.1.1) becomes a PIDE with a weakly singular kernel of the

following form

wt(x, t)−∫ t

0

(t− s)−1/2 wxx(x, s)ds = f(x, t), (7.1.4)

where x ∈ [0, 1], 0 < t ≤ T.

This type of equation (7.1.4) often arises from the applications such as

physical phenomena involving viscoelastic forces [32], heat conduction in

materials with memory [36] and [20], biological models and chemical kinet-

ics, compression of viscoelastic media, fluid dynamics and nuclear reactor

dynamics [46].



has been discretized by the backward

Euler formula. Let tn = n∆t, n = 0, 1, 2, . . . , K, where ∆t = TK

is the

temporal step size, wn(x) is an approximation to the value of w(x, t) at a

time point t = tn, n = 0, 1, . . . , K − 1.



approximated as

∂αw(x, tn+1)

∂tα=

1

Γ(1− α)

∫ tn+1

0

∂w(x, s)

∂s

ds

(tn+1 − s)α

=1

Γ(1− α)

n∑j=0

∫ tj+1

tj

∂w(x, s)

∂s

ds

(tn+1 − s)α,

=1

Γ(1− α)

n∑j=0


∆t

∫ tj+1

tj

ds

(tn+1 − s)α+ rn+1

∆t ,

=1

Γ(1− α)

n∑j=0


∆t

∫ tn+1−j

tn−j

dτ

τα

+rn+1∆t ,

=1

Γ(1− α)

n∑j=0


∆t

∫ tj+1

tj

dτ

τα+ rn+1

∆t ,

=1

Γ(2− α)

n∑j=0


∆tα((j + 1)1−α − j1−α

)

+rn+1∆t ,

=1

Γ(2− α)

n∑j=0


∆tα+ rn+1

∆t , (7.2.1)



• bj > 0, j = 0, 1, 2, . . . , n,

• 1 = b0 > b1 > b2 > . . . > bn, bn → 0 as n →∞,

• ∑nj=0(bj − bj+1) + bn+1 = (1− b1) +

∑n−1j=1 (bj − bj+1) + bn = 1.

The semi-discrete differential operator Lαt is defined as

Lαt w(x, tn+1) :=

1

Γ(2− α)

n∑j=0


∆tα.



∂αw(x, tn+1)

∂tα= Lα

t w(x, tn+1) + rn+1∆t . (7.2.2)



by rn+1∆t , and

rn+1∆t ≤ Cw∆t2−α, (7.2.3)


The integral term in Eq.(7.1.1) can be considered as under

∫ tn+1

0

(tn+1 − s)−1/2 wxx(x, s)ds =

∫ tn+1

0

s−1/2wxx(x, tn+1 − s)ds,

=n∑

j=0

∫ tj+1

tj

s−1/2wxx (x, tn+1 − s) ds,

≈n∑

j=0

wxx (x, tn−j+1)

∫ tj+1

tj

s−1/2ds,

= 2∆t1/2

n∑j=0

bjwxx (x, tn−j+1) . (7.2.4)

Using Eq.(7.2.1) and Eq.(7.2.4), the following time discretization of Eq.(7.1.1)

is obtained as

Lαt w(x, tn+1)− 2∆t1/2

n∑j=0

bjwn+1−jxx (x) = fn+1(x), n = 0, 1, 2, . . . , K − 1,

(7.2.5)

where wn+1(x) = w(x, tn+1) and f(x, tn+1) = fn+1(x).

Substituting the value of Lαt w(x, tn+1), it can be written as

(1

Γ(2− α)

n∑j=0


∆tα− 2∆t1/2

n∑j=0

bjwn+1−jxx (x)

)

= fn+1(x), n = 0, 1, 2, . . . , K − 1. (7.2.6)


Eq.(7.2.6) can be rewritten as

(wn+1(x)− β0

∂2wn+1

∂x2

)

=

((1− b1)w

n(x) +n−1∑j=1

(bj − bj+1)wn−j(x) + bnw

0(x)

+ β0

n∑j=1

bj∂2wn+1−j

∂x2+ α0f

n+1(x)

), n ≥ 1, (7.2.7)

where α0 = 2∆tαΓ(2 − α) and β0 = 2∆tα+1/2Γ(2 − α) along with the

boundary conditions

wn+1(0) = wn+1(1) = 0, n = 0, 1, 2, . . . , K − 1, (7.2.8)

and the initial condition

w0(x) = g0(x), x ∈ [0, 1]. (7.2.9)

For n = 0, the proposed scheme takes the following form

w1(x)− β0w1xx(x) = w0(x) + α0f

1(x), (7.2.10)



Eqs.(7.2.10) and (7.2.7), along with boundary conditions (7.2.8) and ini-

tial condition (7.2.9) form a complete set of the semi-discrete problem of

Eqs.(7.1.1)-(7.1.3).

Following [30], the error term rn+1 is defined by

rn+1 := α0

(∂αw(x, tn+1)

∂tα− Lα

t w(x, tn+1)

). (7.2.11)


Using Eq.(7.2.2) and Eq.(7.2.3) the error term rn+1 becomes

∣∣rn+1∣∣ = Γ(2− α)∆tα

∣∣rn+1∆t

∣∣ ≤ Cw∆t2. (7.2.12)



H10 (Ω) =

v ∈ H1(Ω), v|∂Ω = 0

,

(w, v)1 = (w, v) +

(∂w

∂x,∂v

∂x

),

‖v‖0 = (v, v)1/2, ‖v‖1 = (v, v)1/21 .


by

‖v‖1 =

(‖v‖2

0 + β0

∥∥∥∥∂v

∂x

∥∥∥∥2

0

)1/2

. (7.2.13)

In order to analyze the stability and convergence, the following weak formu-



((wn+1, v

)− β0

(∂2wn+1

∂x2, v

))

=

((1− b1) (wn, v) +

n−1∑j=1


)+ bn

(w0, v

)

+β0

n∑j=1

bj

(∂2wn+1−j

∂x2, v

)+ α0

(fn+1, v

))

, n ≥ 1,(7.2.14)

and

((w1, v

)− β0

(∂2w1

∂x2, v

))=

((w0, v

)+ α0

(f 1, v

))(7.2.15)


Stability and convergence analysis

The stability analysis is presented in the following theorem. For conve-

nience and without loss of generality, suppose f(x, t) = 0 in Eq.(7.2.14)

and Eq.(7.2.15).

Theorem 1

The semi-discrete problems for n ≥ 1 and n = 0 as given in (7.2.14) and

(7.2.15), are unconditionally stable in the sense that for all ∆t > 0, it holds

∥∥wn+1∥∥

1≤ D

∥∥w0∥∥

0, n = 0, 1, 2, . . . , K − 1 (7.2.16)

where ‖.‖1 is defined in (7.2.13) and D is a constant.

Proof

The theorem is proved using mathematical induction. When n = 0, let


(w1, w1

)− β0

(∂2w1

∂x2, w1

)=

(w0, w1

)

Using integration by parts and the boundary conditions (7.2.8), the above

equation becomes

(w1, w1

)+ β0

(∂w1

∂x,∂w1

∂x

)=

(w0, w1

)(7.2.17)

Using Eq.(7.2.13), it can be written as

∥∥w1∥∥2

1=

(w1, u1

)+ β0

(∂w1

∂x,∂w1

∂x

)(7.2.18)


From Eq.(7.2.13), it can be written as

‖v‖0 ≤ ‖v‖1 ,

∥∥∥∥∂v

∂x

∥∥∥∥0

≤√

1

β0

‖v‖1 (7.2.19)

Using Eqs.(7.2.17) and (7.2.18), Schwarz inequality and Eq.(7.2.19), it can

be written as

∥∥w1∥∥2

1≤

∥∥w0∥∥

0

∥∥w1∥∥

0

≤∥∥w0

∥∥0

∥∥w1∥∥

1

∥∥w1∥∥

1≤

∥∥w0∥∥

0

Suppose that the result hold for v = wj i.e.

∥∥wj∥∥

1≤

∥∥w0∥∥

0, j = 2, 3, . . . , n (7.2.20)


((wn+1, wn+1

)− β0

(∂2wn+1

∂x2, wn+1

))

=

((1− b1)

(wn, wn+1

)+

n−1∑j=1


)

+bn

(w0, wn+1

)+ β0

n∑j=1

bj

(∂2wn+1−j

∂x2, wn+1

))

Using integration by parts and boundary conditions, the above equation

becomes

((wn+1, wn+1

)+ β0

(∂wn+1

∂x,∂wn+1

∂x

))

=

((1− b1)

(wn, wn+1

)+

n−1∑j=1


)

+bn

(w0, wn+1

)− β0

n∑j=1

bj

(∂wn+1−j

∂x,∂wn+1

∂x

))


Using Eq. (7.2.13), Schwarz inequality and Eq. (7.2.19), it can be written

as

∥∥wn+1∥∥2

1≤ (1− b1) ‖wn‖0

∥∥wn+1∥∥

0+

n−1∑j=1


∥∥0

∥∥wn+1∥∥

0

+bn

∥∥w0∥∥

0

∥∥wn+1∥∥

0+

n∑j=1

bj

∥∥wn+1−j∥∥

0

∥∥wn+1∥∥

0,

or

∥∥wn+1∥∥2

1≤ (1− b1) ‖wn‖1

∥∥wn+1∥∥

1+

n−1∑j=1


∥∥1

∥∥wn+1∥∥

1

+bn

∥∥w0∥∥

1

∥∥wn+1∥∥

1+

n∑j=1

bj

∥∥wn+1−j∥∥

1

∥∥wn+1∥∥

1,

or

∥∥wn+1∥∥

1≤ (1− b1) ‖wn‖1 +

n−1∑j=1


∥∥1+ bn

∥∥w0∥∥

1

+n∑

j=1

bj

∥∥wn+1−j∥∥

1.

Using induction hypothesis and properties of bj, it can be rewritten as

∥∥wn+1∥∥

1≤ D

∥∥w0∥∥

0.

The error analysis is discussed in the following theorem.

Theorem 2

Let w be the exact solution of Eqs.(7.1.1)-(7.1.3) and wnKn=0 be the time-

discrete solution of Eqs.(7.2.14) and (7.2.15) with initial condition (7.2.9),

then it holds

‖w(tn)− wn‖1 ≤ Cw,αTα∆t2−α, n = 1, 2, . . . , K (7.2.21)


The proof of the Theorem needs the following lemma.

Lemma 1 Under the assumption of Theorem 2, the following can be

written as

‖w(tn)− wn‖1 ≤ Cwb−1n−1∆t2, n = 1, 2, . . . , K. (7.2.22)

Proof


and (7.2.12), the error equation can be written as

(e1, v

)+ β0

(∂e1

∂x,∂v

∂x

)=

(e0, v

)+

(r1, v

), ∀v ∈ H1

0 (Ω).

Let v = e1, noting e0 = 0, and Eq. (7.2.19), it can be written as

∥∥e1∥∥

1≤

∥∥r1∥∥

0.

Using Eq.(7.2.12), it leads to

∥∥w(t1)− w1∥∥

1≤ Cw∆t2 = Cwb−1

0 ∆t2. (7.2.23)


For inductive part, suppose (7.2.22) holds for n = 1, 2, 3, . . . , s, i.e.

‖w(tn)− wn‖1 ≤ Cwb−1n−1∆t2. (7.2.24)


To prove the lemma for n = s + 1 the Eqs. (7.1.1), (7.2.14) and (7.2.12)

are used and the error equation can be written, for all v ∈ H10 (Ω), as

((en+1, v

)+ β0

(∂en+1

∂x,∂v

∂x

))

=

((1− b1) (en, v) +

n−1∑j=1


)+ bn

(e0, v

)

−β0

n∑j=1

bj

(∂en+1−j

∂x,∂v

∂x

)+

(rn+1, v

))

.

The above equation, for v = en+1, can be written as

∥∥en+1∥∥2

1≤ (1− b1) ‖en‖0

∥∥en+1∥∥

0+

n−1∑j=1


∥∥0

∥∥en+1∥∥

0

+bn

∥∥e0∥∥

0

∥∥en+1∥∥

0+

n∑j=1

bj

∥∥en+1−j∥∥

0

∥∥en+1∥∥

0+

∥∥rn+1∥∥

0

∥∥en+1∥∥

0.


b−1j+1

< 1 for all non

negative integer j, it can be written as

∥∥en+1∥∥

1≤ Cwb−1

n ∆t2.

Proof


limn→∞

b−1n−1

nα= lim

n→∞n−α

n1−α − (n− 1)1−α

= limn→∞

n−1

1− (1− 1n)1−α

=1

(1− α)


x1−α−(x−1)1−α .




(1−α)as 1 < n → ∞. It is

to be noted that n−αb−1n−1 = 1 for n = 1, hence it can be written as

n−αb−1n−1 ≤

1

(1− α), n = 1, 2, . . . , K.


‖w(tn)− wn‖1 ≤ Cwb−1n−1∆t2,


α∆t2−α+α,

≤ Cw1

1− α(n∆t)α∆t2−α,



Using

W n+1(x) =N+1∑i=−1

piPi(x), (7.3.1)

where pi are unknown parameters to be determined from the boundary

conditions and collocation form of the fractional integro partial differential

equation as defined in subsection 1.7.10.

The region [0, 1] × [0, T ] has been discretized as grid points (xi, tj) where

xi = ih, i = 0, 1, 2, . . . , M and tn = n∆t, n = 0, 1, 2, . . . , K, K∆t = T .


respectively.

The space discretization of Eq. (7.2.7) is carried out using Eq. (7.3.1) and




as

((pn+1

i−1 + 4pn+1i + pn+1

i+1

)− β06

h2

(pn+1

i−1 − 2pn+1i + pn+1

i+1

))

=

((1− b1)

(pn

i−1 + 4pni + pn

i+1

)+

n−1∑j=1



i+1

)

+bn

(p0

i−1 + 4p0i + p0

i+1

)+ β0

6

h2

n∑j=1

bj

(pn+1−j

i−1 − 2pn+1−ji + pn+1−j

i+1

)

+α0fn+1i

), n ≥ 1. (7.3.2)

Simplifying, the above relation yields the following system of (M +1) linear

equations in (M + 3) unknowns pn+1−1 , pn+1

0 , pn+11 , . . . , pn+1

M , pn+1M+1,

(1− β0

6

h2

)pn+1

i−1 +

(4 + β0

12

h2

)pn+1

i +

(1− β0

6

h2

)pn+1

i+1 = Fi,

n ≥ 1, i = 0, 1, 2, . . . , M, (7.3.3)

where

Fi = (1− b1)(pn

i−1 + 4pni + pn

i+1

)+

n−1∑j=1



i+1

)

+bn

(p0

i−1 + 4p0i + p0

i+1

)+ β0

6

h2

n∑j=1

bj

(pn+1−j

i−1 − 2pn+1−ji + pn+1−j

i+1

)

+α0fn+1i .




w(0, t) = (p−1 + 4p0 + p1) = 0,

w(L, t) = (pM−1 + 4pM + pM+1) = 0.

p−1 = −4p0 − p1,

pM+1 = −4pM − pM−1.

On substituting the values of parameters p−1 and pM+1, the above system

reduces to a diagonal system of (M+1) linear equations in (M+1) unknowns


ACn+1 = F, n = 1, 2, 3, . . . , K, (7.3.4)

where

Cn+1 =[pn+1

0 , pn+11 , . . . , pn+1

M

]T, n = 1, 2, 3, . . . , K.


A =

β036h2

x y x

x y x. . . . . . . . .

x y x

β036h2

,

where

x =

(1− β0

6

h2

),

y =

(4 + β0

12

h2

).


To find the value of C2 = [p20, p2

1, . . . , p2M ]

T, it is first needed to find the

value of C1 = [p10, p

11, . . . , p

1M ]

T. The value of C1 is obtained, solving Eq.

(7.2.10) using cubic B-spline collocation method, as

(1− β0

6

h2

)p1

i−1 +

(4 + β0

12

h2

)p1

i +

(1− β0

6

h2

)p1

i+1 = Fi , i = 0, 1, 2, . . . , M,

(7.3.5)

where

Fi =(p0

i−1 + 4p0i + p0

i+1

)+ α0f

1i .



10, . . . , p

1M , p1

M+1. To obtain the unique solution of this

system, eliminate p−1 and pM+1 using boundary conditions.

The time evolution of the approximate solution W n+1 is determined by the


recurrence relationship, once the initial vector C0 = [p00, p

01, . . . , p

0M ]

T, has



The L∞ errors and L2 errors between numerical solution W and exact

solution w are given with the following definitions

L∞ := max0≤i≤M

|Wi − wi||wi| ,

L2 :=1

M

(M∑i=0

(Wi − wi)2

) 12

.


Four examples have been considered to illustrate the effectiveness of the

method developed.

Example 7.1

The following fractional integro partial differential equation is considered

as

∂0.6w(x, t)

∂t0.6−

∫ t

0

(t− s)−1/2 wxx(x, s)ds = f(x, t),

x ∈ [0, 1], t > 0,


w(x, 0) = 2 sin2 πx, 0 ≤ x ≤ 1,


w(0, t) = w(1, t) = 0, 0 < t ≤ T.


w(x, t) = 2(t2 + t + 1

)sin2 πx.

L∞ and L2 norm errors for different time steps ∆t with M = 100 and

the corresponding rates of convergence at T = 0.1 are presented in Table

7.1. From Table 7.1, it is clear that the temporal convergence rate of the

numerical results obtained by the presented method is in good agreement

with the theoretical estimation.

Example 7.2


Table 7.1: The errors L∞ and L2 of different time steps with M = 100∆t L∞ Rate L2 Rate0.001 1.3640e− 003 1.0490e− 0040.0005 5.3980e− 004 1.3373 4.6417e− 005 1.17620.00025 2.1603e− 004 1.3212 1.8338e− 005 1.33980.000125 8.6935e− 005 1.3132 7.4331e− 006 1.3028

The following partial integro-differential equation is considered as

∂0.5w(x, t)

∂t0.5−

∫ t

0

(t− s)−1/2 wxx(x, s)ds = f(x, t),

x ∈ [0, 1], t > 0,


w(x, 0) = sin πx, 0 ≤ x ≤ 1,


w(0, t) = w(1, t) = 0, 0 < t ≤ T.


w(x, t) = (t + 1) sin πx.

L∞ and L2 norm error for different time steps ∆t with M = 60 and the

corresponding rates of convergence at T = 0.1 are presented in Table 7.2.

From Table 7.2, it is clear that the temporal convergence rate of the nu-

merical results obtained by the presented method is in good agreement with

the theoretical estimation. It is observed from the Table 7.2 that the con-

vergence rate of numerical solutions is of order 2−α as ∆t approaches zero.


Table 7.2: The errors L∞ and L2 of different time steps with M = 60∆t L∞ Rate L2 Rate0.001 9.4447e− 004 8.6218e− 0050.0005 4.4876e− 004 1.0736 4.0966e− 005 1.07360.00025 2.0655e− 004 1.1195 1.8855e− 005 1.11950.000125 8.7159e− 005 1.2448 7.9565e− 006 1.24470.0000625 3.0715e− 005 1.5047 2.8110e− 006 1.5011

Example 7.3

For α = 1, the fractional integro partial differential equation becomes par-

tial integro-differential equation as considered in [12]

wt =

∫ t

0

(t− s)−1/2 wxx(x, s)ds, x ∈ [0, 1],


w(x, 0) = sin πx, 0 ≤ x ≤ 1,


w(0, t) = 0,

w(1, t) = 0, 0 < t ≤ T.


w(x, t) =∞∑

n=0

(−1)nΓ

(3

2n + 1

)−1 (π5/2t3/2

)nsin(πx).

The numerical solutions with h = 150

, ∆t = 0.0001 and at K = 10000 time

level is tabulated in Table 7.3. Table 7.3 shows that the presented method

gives better results as compared with [12].


Table 7.3: Results for u with h = 1/50 and T = 1.0

xExactSolution

PresentedMethod

Explicit[12]

Implicit[12]

Crandall’s[12]

Crank-Nicolson[12]

0.1 0.00796105 3.63787e− 005 7.5e− 003 7.1e− 003 6.2e− 004 5.1e− 0030.2 0.01514283 6.91964e− 005 7.5e− 003 7.2e− 003 6.1e− 004 5.2e− 0030.3 0.02084231 9.52406e− 005 7.6e− 003 7.4e− 003 6.5e− 004 5.3e− 0030.4 0.02450162 1.11962e− 004 7.4e− 003 7.5e− 003 6.6e− 004 5.2e− 0030.5 0.02576251 1.17724e− 004 7.5e− 003 7.5e− 003 6.6e− 004 5.4e− 0030.6 0.02450161 1.11962e− 004 7.4e− 003 7.3e− 003 6.6e− 004 5.3e− 0030.7 0.02084231 9.52406e− 005 7.3e− 003 7.2e− 003 6.4e− 004 5.5e− 0030.8 0.01514283 6.91964e− 005 7.7e− 003 7.4e− 003 6.5e− 004 5.3e− 0030.9 0.00796105 3.63787e− 005 7.8e− 003 7.6e− 003 6.4e− 004 5.2e− 003

Example 7.4

For α = 1, the fractional integro partial differential equation becomes par-

tial integro-differential equation as considered in [33]

wt −∫ t

0

(t− s)−1/2 wxx(x, s)ds = f(x, t), 0 < x < 1, 0 < t ≤ T,


w(x, 0) = sin πx, 0 ≤ x ≤ 1,


w(0, t) = 0,

w(1, t) = 0, 0 < t ≤ T.


w(x, t) = sin πx− 4t3/2

3√

πsin 2πx.

For numerical consideration set T = 1. The results of 50th, 150th, 250th,

350th, 450th time levels of the two different grid sizes, ∆t = 0.00001 and


∆t = 0.000001 are presented in Tables 7.4 and 7.5. The Tables 7.4 and

7.5 show that the proposed method gives better results as compared with

[33].

Table 7.4: The errors L∞ when ∆t = 0.00001M W. Long et al. [33] L∞ Presented method L∞10 4.9343e− 004 2.0454e− 004

2.5228e− 003 1.0423e− 0035.3616e− 003 2.2325e− 0038.7631e− 003 3.6892e− 0031.2588e− 002 5.3691e− 003

20 1.5825e− 003 2.0553e− 0047.8974e− 003 1.0476e− 0031.6182e− 002 2.2438e− 0032.5304e− 002 3.7080e− 0033.4578e− 002 5.3965e− 003

Table 7.5: The errors L∞ when ∆t = 0.000001M W. Long et al. [33] L∞ Proposed method L∞10 1.5630e− 005 6.4688e− 006

8.0470e− 005 3.2975e− 0051.7272e− 004 7.0662e− 0052.8572e− 004 1.1684e− 0044.1611e− 004 1.7016e− 004

20 5.0435e− 005 6.5001e− 0062.6001e− 004 3.3141e− 0055.5766e− 004 7.1021e− 0059.2134e− 004 1.1744e− 0041.3397e− 003 1.7104e− 004

7.5 Conclusion

The collocation method using cubic B-spline as basis functions has been

applied to develop numerical solution of fractional integro partial differen-

tial equation with a weakly singular kernel. The backward Euler formula


has been used for temporal discretization and collocation method has been

used for spatial discretiztion. It has been shown that the discretization

in time is unconditionally stable and the numerical solution converges to

exact solution with order O(∆t2−α) where ∆t is the time step size. It has

been observed from Examples 7.1 and 7.2 that the temporal convergence

rate of the numerical results obtained by the proposed method is in good

agreement with the theoretical estimation. For α = 1, the fractional integro

partial differential equation reduces to partial integro-differential equation.

In Example 7.3, the presented method is compared with [12] and the nu-

merical results are tabulated in Table 7.3. The Table 7.3 demonstrates

that the presented method efficiently approximates the exact solution than

explicit, implicit, Crandall’s and Crank-Nicolson finite difference methods

developed by M. Dehghan [12]. In Example 7.4, the presented method is

also compared with [33] and the approximate results are tabulated in Tables

7.4 and 7.5 which show that the proposed method efficiently approximate

the exact solution than the quasi wavelet based numerical method devel-

oped by W. T. Long et al. [33]. The numerical results tabulated in Tables

7.1-7.5, verify the superiority of the presented method.

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Appendix

List of Research Work

On the basis of this dissertation, the following research works have beenpublished, revised and submitted for possible publications and are attachedherewith.

Published

1. Shahid S. Siddiqi and Saima Arshed, Numerical solution of convection-diffusion integro-differential equations with a weakly singular kernel,Journal of basic and applied scientific research. 3(11) (2013) 106-120.

2. Shahid S. Siddiqi and Saima Arshed, Quintic B-Spline for the numer-ical solution of fourth-order parabolic partial differential equations,World applied sciences journal. 23 (12) (2013) 115-122.

3. Shahid S. Siddiqi and Saima Arshed, Cubic B-spline for the numericalsolution of parabolic integro-differential equation with a weakly sin-gular kernel, Research Journal of Applied Sciences, Engineering andTechnology. 7 (10) (2014) 2065-2073.

4. S. S. Siddiqi and S. Arshed, Quintic B-spline for the numerical solutionof the good Boussinesq equation, Journal of the egyptian mathematicalsociety. 22 (2014) 209-213.

5. Shahid S. Siddiqi and Saima Arshed, Numerical solution of time-fractional fourth-order partial differential equations, International jour-nal of computer mathematics. DOI:1080/00207160.2014.948430, (2014).

Revised

1. Shahid S. Siddiqi and Saima Arshed, Numerical solution of time-fractional convection-diffusion equation, Research Journal of AppliedSciences, Engineering and Technology.

182

References 183

Submitted

1. Shahid S. Siddiqi and Saima Arshed, B-spline solution of time-fractionalintegro partial differential equation with a weakly singular kernel, Jour-nal of the Korean Mathematical Society.

numerical solutions of partial differential...

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