applied mathematics and computationszhang/research/p/2020-25.pdfy. yang, j. wang and s. zhang et al....

Applied Mathematics and Computation 387 (2020) 124489

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Applied Mathematics and Computation

journal homepage: www.elsevier.com/locate/amc

Convergence analysis of space-time Jacobi spectral collocation

method for solving time-fractional Schrödinger equations

Yin Yang a , ∗, Jindi Wang a , Shangyou Zhang b , Emran Tohidi c

a Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing &

Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan,

Hunan 411105, China b Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA c Department of Mathematics, Kosar University of Bojnord, Bojnord, P. O. Box 9415615458, Iran

a r t i c l e i n f o

Article history:

Available online 5 July 2019

MSC:

33C45

35Q40

41A55

41A25

65M70

Keywords:

Convergence analysis

Time-fractional Schr ̈o dinger equation

Jacobi spectral-collocation method

Gauss-type quadrature

a b s t r a c t

In this paper, the space-time Jacobi spectral collocation method (JSC Method) is used to

solve the time-fractional nonlinear Schr ̈o dinger equations subject to the appropriate initial

and boundary conditions. At first, the considered problem is transformed into the associ-

ated system of nonlinear Volterra integro partial differential equations (PDEs) with weakly

singular kernels by the definition and related properties of fractional derivative and in-

tegral operators. Therefore, by collocating the associated system of integro-PDEs in both

of the space and time variables together with approximating the existing integral in the

equation using the Jacobi-Gauss-Type quadrature formula, then the problem is reduced to

a set of nonlinear algebraic equations. We can consider solving the system by some robust

iterative solvers. In order to support the convergence of the proposed method, we provided

some numerical examples and calculated their L ∞ norm and weighted L 2 norm at the end of the article.

© 2019 Elsevier Inc. All rights reserved.

1. Introduction

The Schr ̈o dinger equation is a significant development in the theory of quantum mechanics [1] . This equation is a pow-

erful differential structure that is related to the quantum systems changes over the temporal when the effects of quantum

are considerable.

It should be noted that, the classic variant of this equation is stated in terms of the integer first order temporal and

second order spatial partial derivatives. Because of non-locality of fractional differential operators, the fractional variant of

Schr ̈o dinger equation can better describe the physical and chemical events in real world applications [2,3] . The notion of

fractional Schr ̈o dinger equation was first introduced by Laskin [4] , in which the Feyman path integral is extended. Because

of nonlinearity, complexity and non-locality of the fractional Schr ̈o dinger equations, the classical methods for solving PDEs

are not efficient. On the other hand, solution of such these equations has considerable importance for researchers to have a

deterministic behavior for simulating the events accurately. Therefore, numerical and analytical schemes should be explored

and extended to compute the solution of fractional Schr ̈o dinger equations successfully.

∗ Corresponding author. E-mail address: [email protected] (Y. Yang).

https://doi.org/10.1016/j.amc.2019.06.003

0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

https://doi.org/10.1016/j.amc.2019.06.003http://www.ScienceDirect.comhttp://www.elsevier.com/locate/amchttp://crossmark.crossref.org/dialog/?doi=10.1016/j.amc.2019.06.003&domain=pdfmailto:[email protected]://doi.org/10.1016/j.amc.2019.06.003

2 Y. Yang, J. Wang and S. Zhang et al. / Applied Mathematics and Computation 387 (2020) 124489

In recent years, in order to solve the fractional Schr ̈o dinger equation, Scientists proposed many analytical methods and

numerical methods. Among the analytical methods, one can point out to the homotopy analysis method (HAM) [5] . Ana-

lytical methods are very straightforward for solving any nonlinear PDE, which do not need to discretization or linearization

process. But, one of the disadvantages of these methods is being time consuming. Since, in these approaches the integration

and differentiation processes are performed symbolically. Meanwhile, in numerical methods, we have two robust tools such

as operational matrices of differentiation and Gaussian quadrature rules which accelerate (and reduce the computational

time of) the process of differentiation and integration, respectively. Moreover, in some of the numerical approaches such as

Krylov subspace methods [6] , the no smooth solutions may be computed without any regularization tool, since the solu-

tion of these schemes depend on (and keep the behavior of) the right hand side functions of operator equations. Therefore,

numerical methods are more attractive to be implemented by researchers. Among the numerical methods, one can point

out to low order numerical methods such as finite elements [7] , local discontinuous Galerkin (DG) [8,9] and reproducing

kernel [10] techniques which was proposed for solving time-fractional Schr ̈o dinger equations recently. It should be noted

that, since the fractional differential operators are global, it is better to use some global numerical approaches to solve the

considered equations. Taking into account that, if for solving a PDE, space variables are discretized by global methods such

as radial basis function collocation technique and time variable is localized by local schemes such as Galerkin finite element

methods (FEMs) to solve nonlinear time fractional parabolic problems [11] and finite difference methods (FDMs), this may

results to an unbalanced numerical scheme [12–14] which has spectral accuracy in space variable and low order algebraic

accuracy in time variable. Therefore, it is desirable to propose a balanced numerical scheme that has spectral accuracy in

both time and space directions.

Fractional differential equations (FDEs) are easy to be implemented with respect to the spectral Galerkin methods spe-

cially for solving nonlinear FDEs [15] . These methods are successfully applied for solving nonlinear fractional boundary

value problems (BVPs) [16,18] and integral equations (IEs) [19,20] with a rigorous convergence analysis and also regularized

for solving FDEs with nonsmooth solutions [21] . Also, in [22,23] the author solved fractional Schr ̈o dinger equations just in

numerical implementation point of view via using spectral collocation method also used in [17] . In [24,25] a linearized L 1-

Galerkin finite element method and linearized compact alternating direction implicit (ADI) schemes are proposed to solve

the multidimensional nonlinear time-fractional Schr ̈o dinger equation, respectively. To the authors’ knowledge, space-time

Jacobi spectral collocation methods (that supported by a rigorous convergence analysis) for solving nonlinear time-fractional

Schr ̈o dinger equations have had few results. This motivate us to propose a Jacobi spectral collocation scheme together with

a full convergence analysis for solving time-fractional Schr ̈o dinger equations which is a balanced approach and has spectral

accuracy in both time and space directions.

The time-fractional PDE we will considered as follows:

i ∂ μψ(x, y, t)

∂t μ= a 1 ∂

2 ψ(x, y, t)

∂x 2 + a 2 ∂

2 ψ(x, y, t)

∂y 2 + γ | ψ(x, y, t) | 2 ψ(x, y, t) + δR (x, y, t) ,

0 < μ < 1 , (x, y, t) ∈ �1 × �2 × �3 , (1.1) where �1 = [0 , L 1 ] , �2 = [0 , L 2 ] and �3 = [0 , T ] , with the initial time conditions,

ψ(x, y, 0) = ζ1 (x, y ) , (x, y ) ∈ �1 × �2 , (1.2) and two-dimensional boundary space conditions,

ψ(0 , y, t) = ζ2 (y, t) , ψ(L 1 , y, t) = ζ3 (y, t) , (y, t) ∈ �2 × �3 , ψ(x, 0 , t) = ζ4 (x, t) , ψ(x, L 2 , t) = ζ5 (x, t) , (x, t) ∈ �1 × �3 ,

while ζ 1 , ζ 2 , ζ 3 , ζ 4 , ζ 5 and R ( x , y , t ) are given functions. It should be noted that the definition of Caputo fractional derivative as follows,

∂ μ

∂τμ(g(τ )) =

⎧ ⎨ ⎩

∂ m g(τ )

∂τ m , μ = m ∈ N ,

1

(m − μ) ∫ τ

0

g (m ) (s )

(τ − s ) μ−m +1 ds, m − 1 < μ < m,

where ∂ μ

∂t μ(·) denotes the of order μ derivative.

The definition of Riemann-Liouville (R-L) fractional integral as follows, indicated by l μτ ,

l μτ ( g(τ ) ) =

1

(μ)

∫ τ0

(τ − s ) μ−1 g(s ) ds, s > 0 .

Moreover

l μτ

(∂ μ

∂τμg(τ )

)= g(τ ) −

m −1 ∑ i =0

g (i ) (0) τ i

i ! , m − 1 < μ < m.

In the next section, we proposed space-time Jacobi spectral collocation method to solve the Eq. (1.1) . Some useful lemmas

and preliminaries are provided in Section 3 , such as the error estimation of interpolation by Jacobi-Gauss points. Rigorous

Y. Yang, J. Wang and S. Zhang et al. / Applied Mathematics and Computation 387 (2020) 124489 3

convergence analysis associated to the used numerical method in weighted L 2 norms and L ∞ norms is provided in Section 4 .In Section 5 , algorithm implementation and numerical results are given. Finally, in the last section, some concluding remarks

and considered problems for the future research works are stated.

2. Jacobi spectral collocation methods

We first split all of the known and unknown functions into real and imaginary parts and transform the basic equation

into a system of coupled time-fractional PDEs in Caputo sense. In the next step, we impose the Riemann-Liouville fractional

integral operator on both sides of the equations to change this system into a nonlinear system of Volterra integro-PDEs with

weakly singular kernels that contains initial conditions. Finally, both of the space and time variables are collocated and the

existing integrals are approximated by the powerful Gaussian quadrature rules to change the considered problem to a set of

nonlinear algebraic equations. We consider solving the system by some robust iterative solvers.

Now, one can split all of the known and unknown functions into real and imaginary parts as follows

ψ = u + i v , R = f + ig, ζ1 = g 1 + ig 2 , ζ2 = g 3 + ig 4 , ζ3 = g 5 + ig 6 , ζ4 = g 7 + ig 8 , ζ5 = g 9 + ig 10 , (2.1)

where, u, v , f, g, g 1 , g 2 , g 3 , g 4 , g 5 , g 6 , g 7 , g 8 , g 9 and g 10 are the real functions. By considering assumptions of (2.1) , the basic Eq. (1.1) can be rewritten in the following form

i

(∂ μu

∂t μ− a 1 ∂

2 v ∂x 2

− a 2 ∂ 2 v

∂y 2 − γ (u 2 + v 2 ) v − δg

)−

(∂ μv ∂t μ

+ a 1 ∂ 2 u

∂x 2 + a 2 ∂

2 u

∂y 2 + γ (u 2 + v 2 ) u + δ f

)= 0 . (2.2)

Therefore, the aforementioned complex equation can be transformed into the following coupled real time-fractional

PDEs

∂ μu

∂t μ= a 1 ∂

2 v ∂x 2

+ a 2 ∂ 2 v

∂y 2 + γ (u 2 + v 2 ) v + δg,

−∂ μv

∂t μ= a 1 ∂

2 u

∂x 2 + a 2 ∂

2 u

∂y 2 + γ (u 2 + v 2 ) u + δ f, (2.3)

the initial boundary conditions became

u (x, y, 0) = g 1 (x, y ) , v (x, y, 0) = g 2 (x, y ) , (x, y ) ∈ �1 × �2 , u (0 , y, t) = g 3 (y, t) , v (0 , y, t) = g 4 (y, t) ,

u (L 1 , y, t) = g 5 (y, t) , v (L 1 , y, t) = g 6 (y, t) , (y, t) ∈ �2 × �3 , u (x, 0 , t) = g 7 (x, t) , v (x, 0 , t) = g 8 (x, t) ,

u (x, L 2 , t) = g 9 (x, t) , v (x, L 2 , t) = g 10 (y, t) , (x, t) ∈ �1 × �3 . (2.4)Taking into account that, Eqs. (2.3) and (2.4) are equal to Eqs. (1.1) and (1.2) . So, instead of solving (1.1) and (1.2) numer-

ically, we compute numerical solution of (2.3) and (2.4) using the Jacobi spectral collocation method. Before the space-time

collocation method, first using the R-L fractional integral of order μ, we will transform (2.3) into the associated system ofweakly singular Volterra integro-PDEs,

u (x, y, t) = 1 (μ)

∫ t 0

(t − τ ) μ−1 k 1 ( x, y, τ, u ( x, y, τ ) , v ( x, y, τ ) ) dτ + δ ˜ g (x, y, t) + g 1 (x, y ) ,

v (x, y, t) = 1 (μ)

∫ t 0

(t − τ ) μ−1 k 2 ( x, y, τ, u ( x, y, τ ) , v ( x, y, τ ) ) dτ + δ ˜ f (x, y, t) + g 2 (x, y ) , (2.5)

where

k 1 (x, y, τ, u (x, y, τ ) , v (x, y, τ )) = a 1 ∂ 2 v (x, y, τ )

∂x 2 + a 2 ∂

2 v (x, y, τ ) ∂y 2

+ γ (u 2 (x, y, τ ) + v 2 (x, y, τ )) v (x, y, τ ) ,

k 2 (x, y, τ, u (x, y, τ ) , v (x, y, τ )) = −a 1 ∂ 2 u (x, y, τ )

∂x 2 − a 2 ∂

2 u (x, y, τ )

∂y 2 − γ (u 2 (x, y, τ ) + v 2 (x, y, τ )) u (x, y, τ ) ,

˜ g (x, y, t) = 1 (μ)

∫ t 0

(t − τ ) μ−1 g(x, y, τ ) dτ,

˜ f (x, y, t) = − 1 (μ)

∫ t 0

(t − τ ) μ−1 f (x, y, τ ) dτ. (2.6)

Since (2.5) has weakly singular kernels around t = 0 + , the numerical process of (2.5) may be difficult. To apply theorthogonal Jacobi polynomials for solving equations of (2.5) , we should consider the variable transformations as follows,


x = L 1 2

(1 + x̄ ) , x̄ = 2 x L 1

− 1 , x̄ ∈ [ −1 , 1] ,

y = L 2 2

(1 + ȳ ) , ȳ = 2 y L 2

− 1 , ȳ ∈ [ −1 , 1] ,

t = T 2

(1 + ̄t ) , t̄ = 2 t T

− 1 , t ∈ [ −1 , 1] ,

τ = T 2

(1 + s ) , s = 2 τT

− 1 , s ∈ [ −1 , t] .

For the sake of simplicity, we still use x , y , t to indicate x̄ , ȳ , t̄ , therefore, equations of (2.5) should be rewritten with the

following form

ū (x, y, t) = ∫ t

−1 (t − s ) μ−1 k̄ 1 (x, y, s, ū (x, y, s ) , ̄v (x, y, s )) ds + δḡ (x, y, t) + ḡ 1 (x, y ) ,

v̄ (x, y, t) = ∫ t

−1 (t − s ) μ−1 k̄ 2 (x, y, s, ū (x, y, s ) , ̄v (x, y, s )) ds + δ f̄ (x, y, t) + ḡ 2 (x, y ) , (2.7)

with the boundary conditions

ū (−1 , y, t) = ḡ 3 (y, t) , ū (1 , y, t) = ḡ 5 (y, t) , ū (x, −1 , t) = ḡ 7 (x, t) , ū (x, 1 , t) = ḡ 9 (x, t) , v̄ (−1 , y, t) = ḡ 4 (y, t) , v̄ (1 , y, t) = ḡ 6 (y, t) , v̄ (x, −1 , t) = ḡ 8 (x, t) , v̄ (x, 1 , t) = ḡ 10 (y, t) .

where

ū (x, y, t) = u (

L 1 2

(1 + x ) , L 2 2

(1 + y ) , T 2

(1 + t) ),

v̄ (x, y, t) = v (

L 1 2

(1 + x ) , L 2 2

(1 + y ) , T 2

(1 + t) ),

ḡ (x, y, t) = ˜ g (

L 1 2

(1 + x ) , L 2 2

(1 + y ) , T 2

(1 + t) ),

f̄ (x, y, t) = ˜ f (

L 1 2

(1 + x ) , L 2 2

(1 + y ) , T 2

(1 + t) ),

ḡ i (x, y ) = g i (

L 1 2

(1 + x ) , L 2 2

(1 + y ) ), i = 1 , 2 ,

ḡ i (y, t) = g i (

L 2 2

(1 + y ) , T 2 2

(1 + t) ), i = 3 , 4 , 5 , 6 ,

ḡ i (x, t) = g i (

L 1 2

(1 + x ) , T 2 2

(1 + t) ), i = 7 , 8 , 9 , 10 ,

k̄ 1 (x, y, s, ū (x, y, s ) , ̄v (x, y, s )) = 1 (μ)

(T

2

)μ(a 1

(2

L 1

)2 ∂ 2 v̄ ∂x 2

+ a 2 (

2

L 2

)2 ∂ 2 v̄ ∂y 2

+ γ ( ̄u 2 + ̄v 2 ) ̄v )

,

k̄ 2 (x, y, s, ū (x, y, s ) , ̄v (x, y, s )) = 1 (μ)

(T

2

)μ(− a 1

(2

L 1

)2 ∂ 2 ū ∂x 2

− a 2 (

2

L 2

)2 ∂ 2 ū ∂y 2

− γ ( ̄u 2 + ̄v 2 ) ̄u )

.

Now, we can collocate the variable t of equations of (2.7) at the Jacobi Gauss nodes corresponding to θ = ϑ = −μ, in whichω θ,ϑ (t) = (1 − t) −μ(1 + t) −μ = (1 − t 2 ) −μ, for 0 ≤ l ≤ M ,

ū (x, y, t l ) = ∫ t l

−1 (t l − s ) μ−1 k̄ 1 (x, y, s, ū (x, y, s ) , ̄v (x, y, s )) ds + δḡ (x, y, t l ) + ḡ 1 (x, y ) ,

v̄ (x, y, t l ) = ∫ t l

−1 (t l − s ) μ−1 k̄ 2 (x, y, s, ū (x, y, s ) , ̄v (x, y, s )) ds + δ f̄ (x, y, t l ) + ḡ 2 (x, y ) . (2.8)

We should apply the following change of variables for implementing Jacobi Gauss quadrature rule

s (θ ) = s l (θ ) = 1 + t l θ + t l − 1 , 0 ≤ l ≤ M, θ ∈ [ −1 , 1] . (2.9)

2 2


Therefore,

ū (x, y, t l ) = (

1 + t l 2

)μ ∫ 1 −1

(1 − θ ) μ−1 k̄ 1 (x, y, s (θ ) , ū (x, y, s (θ )) , ̄v (x, y, s (θ ))

)dθ + δḡ (x, y, t l ) + ḡ 1 (x, y ) ,

v̄ (x, y, t l ) = (

1 + t l 2

)μ ∫ 1 −1

(1 − θ ) μ−1 k̄ 2 (x, y, s (θ ) , ū (x, y, s (θ )) , ̄v (x, y, s (θ ))

)dθ + δ f̄ (z, y, t l ) + ḡ 2 (x, y ) . (2.10)

Applying Gaussian quadrature formula to change the integral parts of the above formulas to a summation form, ∫ 1 −1

(1 − θ ) μ−1 k̄ 1 (x, y, s (θ ) , ū (x, y, s (θ )) , ̄v (x, y, s (θ ))) dθ ≈L ∑

k =0 k̄ 1 (x, y, s (θk ) , ū (x, y, s (θk )) , ̄v (x, y, s (θk ))) ω

μ−1 , 0 k

,

∫ 1 −1

(1 − θ ) μ−1 k̄ 2 (x, y, s (θ ) , ū (x, y, s (θ )) , ̄v (x, y, s (θ ))) dθ ≈L ∑

k =0 k̄ 2 (x, y, s (θk ) , ū (x, y, s (θk )) , ̄v (x, y, s (θk ))) ω

μ−1 , 0 k

,

(2.11)

where { θk } L k =0 are Jacobi-Gauss-Lobatto nodes and { ω μ−1 , 0 k } L k =0 is the weight function in [ −1 , 1] and L ≥ M , ω μ−1 , 0 (t) =(1 − t) μ−1 .

Let ū l (x, y ) and v̄ l (x, y ) denote to the ū (x, y, t l ) and v̄ (x, y, t l ) , respectively. One can approximate ū (x, y, t) and v̄ (x, y, t) bytheir Lagrange interpolation polynomials in the following form

ū (x, y, t) ≈ ū M (x, y, t) = M ∑

l=0 ū l (x, y ) F l (t) ,

v̄ (x, y, t) ≈ v̄ M (x, y, t) = M ∑

l=0 v̄ l (x, y ) F l (t) ,

where F l ( t ) is the l th Lagrange polynomial associated to the t l for 0 ≤ l ≤ M . By using the aforementioned approximations together with the implemented Gauss quadrature rule, (2.10) can be re-

duced as follows (0 ≤ l ≤ M ),

ū l (x, y ) = (

1 + t l 2

)μ L ∑ k =0

k̄ 1 (x, y, s (θk ) , ū M (x, y, s (θk )) , ̄v M (x, y, s (θk ))) ω

μ−1 , 0 k

+ δḡ (x, y, t l ) + ḡ 1 (x, y ) ,

v̄ l (x, y ) = (

1 + t l 2

)μ L ∑ k =0

k̄ 2 (x, y, s (θk ) , ū M (x, y, s (θk )) , ̄v M (x, y, s (θk ))) ω

μ−1 , 0 k

+ δ f̄ (x, y, t l ) + ḡ 2 (x, y ) . (2.12)

For applying JSC method to the space variable, we use the Legendre-Gauss-Lobatto points { x i } N 1 i =0 , { y j } N 2 j=0 on the interval[-1,1] corresponding to ω 0 , 0 (x, y ) = (1 − x ) 0 (1 + x ) 0 (1 − y ) 0 (1 + y ) 0 = 1 . Then, (2.12) has the following form ( 1 ≤ i ≤ N 1 −1 , 1 ≤ j ≤ N 2 − 1 ),

ū j (x i , y j ) = (

1 + t l 2

)μ L ∑ k =0

k̄ 1 (x i , y j , s (θk ) , ū M (x i , y j , s (θk )) , ̄v M (x i , y j , s (θk ))) ω

μ−1 , 0 k

+ δḡ (x i , y j , t l ) + ḡ 1 (x i , y j ) ,

v̄ j (x i , y j ) = (

1 + t l 2

)μ L ∑ k =0

k̄ 2 (x i , y j , s (θk ) , ū M (x i , y j , s (θk )) , ̄v M (x i , y j , s (θk ))) ω

μ−1 , 0 k

+ δ f̄ (x i , y j , t j ) + ḡ 2 (x i , y j ) . (2.13)

Again, let ū l i j

and v̄ l i j

denote to the ū (x i , y j , t l ) and v̄ (x i , y j , t l ) , respectively. One can consider the Lagrange interpolationpolynomial for both of the functions ū and v̄ and write

ū (x, y, t) ≈ ū M N 1 N 2 (x, y, t) = M ∑

l=0

N 1 ∑ i =0

N 2 ∑ j=0

ū l i j H i (x ) H j (y ) F l (t) = M ∑

l=0

N 1 −1 ∑ i =1

N 2 −1 ∑ j=1

ū l i j H i (x ) H j (y ) F l (t) + M ∑

l=0

N 2 ∑ j=0

(ḡ 3 H 0 (x )

+ ̄g 4 H N 1 (x ) )H j (y ) F l (t) +

M ∑ l=0

N 1 ∑ i =0

(ḡ 7 H 0 (y ) + ḡ 8 H N 2 (y )

)H i (x ) F l (t) ,

v̄ (x, y, t) ≈ v̄ M N 1 N 2 (x, y, t) = M ∑

l=0

N 1 ∑ i =0

N 2 ∑ j=0

v̄ l i j H i (x ) H j (y ) F l (t) = M ∑

l=0

N 1 −1 ∑ i =1

N 2 −1 ∑ j=1

v̄ l i j H i (x ) H j (y ) F l (t) + M ∑

l=0

N 2 ∑ j=0

(ḡ 5 H 0 (x )

+ ̄g 6 H N 1 (x ) )H j (y ) F l (t) +

M ∑ l=0

N 1 ∑ i =0

(ḡ 9 H 0 (y ) + ḡ 10 H N 2 (y )

)H i (x ) F l (t) , (2.14)


where H i ( x ) is the i th Lagrange polynomial corresponding to x i for 0 ≤ i ≤ N 1 , where H j ( y ) is the j th Lagrange polynomialcorresponding to y i for 0 ≤ j ≤ N 2 .

Therefore, the full discrete system of algebraic equations arised from the space-time Jacobi spectral collocation method

to solve (1.1) and (1.2) can be stated as ( 1 ≤ i ≤ N 1 − 1 , 1 ≤ j ≤ N 2 − 1 , 0 ≤ l ≤ M)

ū l i j = (

1 + t l 2

)μ L ∑ k =0

k̄ 1 (x i , y j , s ( θk ) , ū

M N 1 N 2

(x i , y j , s ( θk )

), ̄v M N 1 N 2

(x i , y j , s ( θk )

))ω μ−1 , 0

k + δḡ (x i , y j , t l ) + ḡ 1 (x i , y j )

= (

1 + t l 2

)μ L ∑ k =0

k̄ 1 (x i , y j , s (θk ) ,

N 1 ∑ n 1 =0

N 2 ∑ n 2 =0

M ∑ m =0

ū m n 1 n 2 H n 1 (x i ) H n 2 (y j ) F m (s (θk )) ,

N 1 ∑ n 1 =0

N 2 ∑ n 2 =0

M ∑ m =0

v̄ m n 1 n 2 H n 1 (x i ) H n 2 (y j ) F m (s (θk )) )ω μ−1 , 0

k + δḡ (x i , y j , t l ) + ḡ 1 (x i , y j ) ,

v̄ l i j = (

1 + t l 2

)μ L ∑ k =0

k̄ 1 (x i , y j , s (θk ) , ū M N 1 N 2

(x i , y j , s (θk )) , ̄v M N 1 N 2 (x i , y j , s (θk ))) ω μ−1 , 0 k

+ δ f̄ (x i , y j , t l ) + ḡ 2 (x i , y j )

= (

1 + t l 2

)μ L ∑ k =0

k̄ 1 (x i , y j , s (θk ) ,

N 1 ∑ n 1 =0

N 2 ∑ n 2 =0

M ∑ m =0

ū m n 1 n 2 H n 1 (x i ) H n 2 (y j ) F m (s (θk )) ,

N 1 ∑ n 1 =0

N 2 ∑ n 2 =0

M ∑ m =0

v̄ m n 1 n 2 H n 1 (x i ) H n 2 (y j ) F m (s (θk )) )ω μ−1 , 0

k + δ f̄ (x i , y j , t l ) + ḡ 2 (x i , y j ) , (2.15)

We can solve the question about the above-mentioned system of nonlinear algebraic equations can be solved using

Newton Raphson iterative method. The practical implementation can be done by a well-known command “fsolve” in matlab

or maple softwares.

3. Some preliminaries and useful Lemmas

In the next section, we will introduction the convergence analysis associated to the proposed space-time JSC method to

solve (1.1) and (1.2) . Therefore, we need to some definitions and lemmas for stablishing the proof of the main theorem. These

lemmas include error of the Gauss quadrature rules, estimation of the interpolation errors, Lebesgue constant corresponding

to the Legendre series, and finally the Gronwall inequality.

Definition 3.1 [26] . Let I is a bounded interval in R and L p ( I ) is a measurable function space, where 1 ≤ p < ∞ , that is, for∀ u ∈ I → R , if ∫ b a | u (x ) | p dx < ∞ , we can define its norm as

‖ u ‖ L p (I) = (∫ b

a

| u (x ) | p dx ) 1 p

. (3.1)

Obviously, it is a Banach space.

Definition 3.2 [26] . Let I is a bounded interval in R , one can define H m ( I ), if it satisfied that for u ∈ L 2 ( I ), we can always findthe function v ∈ L 2 (I) so that d k u

dx k = v . In other words,

H m (I) = { u ∈ L 2 (I) : d k u

dx k ∈ L 2 (I) , for 0 ≤ k ≤ m } , (3.2)

define the inner product in H m ( I ) as,

(u, v ) m = m ∑

k =0

∫ b a

d k u (x )

dx k d k v (x )

dx k dx, (3.3)

obviously, H m ( I ) is a Hilbert space. Norm of function can be given,

‖ v ‖ H m (I) = ( m ∑

k =0 ‖ d

k v dx k

‖ 2 L 2 (I) ) 1 2

. (3.4)

Lemma 3.1 [27] . (Integration error) Assume that a set of quadrature nodes of three types of Gaussian formulas include Gauss

or Gauss-Radau or Gauss-Lobatto, for the function u , φ, where u ∈ H m (I) , φ ∈ P N (the set of all algebraic polynomials of degree≤ n), and for some m ≥ 1, with I := (−1 , 1) , we can find a constant C independent of N such that ∣∣ ∫ 1

−1 u (x ) φ(x ) dx − (u, φ) N

∣∣ ≤ CN −m | u | H m,N (I) ‖ φ‖ L 2 (I) , (3.5)


where

| u | H m,N (I) = (

m ∑ j= min (m,N+1)

‖ u ( j) ‖ 2 L 2 (I) ) 1

2

,

(u, φ) N = N ∑

j=0 ω j u (x j ) φ(x j ) . (3.6)

Lemma 3.2 [27] . For u ∈ H m,N ω α,β

(I) and by I α,βN

u denote its interpolation polynomial, where { x i } N i =0 is a set of Jacobi-Gauss points,namely,

I α,βN

u = N ∑

i =0 u (x i ) F i (x ) , (3.7)

then, the following estimates hold

‖ u − I α,βN

u ‖ L 2 ω α,β

(I) ≤ CN −m | u | H m,N ω α,β

(I) ,

‖ u − I α,βN

u ‖ L ∞ (I) ≤{

CN 1 2 −m | u | H m,N

ω c (I) , −1 ≤ α, β < − 1 2 ,

CN 1+ �−m logN| u | H m,N ω c

(I) , otherwise, (3.8)

where F i (x ) , i = 0 , 1 , . . . , N, are the lagrange interpolation basis functions associated with the Jacobi collocation points { x i } N i =0 ,where � = max (α, β) and ω c = ω − 1 2 , − 1 2 denotes to the Chebyshev weight function.

For u ∈ H m , N ( I ) and I N u denote its interpolation polynomial, where { x i } N i =0 is a set of Legendre-Gauss points. Namely,

I N u = N ∑

i =0 u (x i ) F i (x ) . (3.9)

Then, the following estimates hold [20]

‖ u − I N u ‖ L 2 (I) ≤ CN −m | u | H m,N (I) , ‖ u − I N u ‖ L ∞ (I) ≤ CN 3 4 −m | u | H m,N (I) . (3.10)

Lemma 3.3 [26] . For every function u , bounded, we can find a constant C , independent of u , satisfited

sup N

∥∥∥∥∥N ∑

i =0 u (x i ) F i (x )

∥∥∥∥∥L 2 ω α,β

(I)

≤ max x ∈ [ −1 , 1]

| u (x ) | , (3.11)

where F i ( x ) is the Lagrange interpolation basis function, and { x i } N i =0 is the Jacobi collocation points. From Mastroianni and Occorsto [28] , we get the following results related to the Lebesgue constant, for the Lagrange interpola-

tion polynomial associated with the zero of the Jacobian polynomial,

Lemma 3.4 [29] . Suppose { F j (x ) } N j=0 is the N Lagrangian polynomial taken at the Gaussian node of the Jacobi polynomial, then,

‖ I α,βN

‖ L ∞ (I) ≤ max x ∈ [ −1 , 1]

N ∑ j=0

| F j (x ) | (3.12)

= {

O (logN) , −1 < α, β ≤ 1 2 ,

O (N �+ 1 2 ) , � = max (α, β) , otherwise. (3.13)

The following lemma is a generalization of the singular kernel of the Gronwall lemma, which is very important in partial

differential equations, and can be found in the references, for example, in [30] , to a large extent affected the main results

of our research.

Lemma 3.5 [31] . Suppose u and v are non-negative local integrable functions defined on the interval [ −1 , 1] , L ≥ 0, 0 < μ< 1 . Inaddition,

u (x ) ≤ v (x ) + L ∫ x

−1 (x − τ ) −μu (τ ) dτ,

we can find a constant C = C(μ) satisfited,

u (x ) ≤ v (x ) + CL ∫ x

−1 (x − τ ) −μv (τ ) dτ, for − 1 ≤ x < 1 .


If E ( x ) satisfies

E(x ) ≤ L ∫ x

−1 E(s ) ds + J(x ) , −1 < x < 1 ,

where E ( x ) is a nonnegative integrable function, J ( x ) is an integrable function, then

‖ E‖ L ∞ (−1 , 1) ≤ C‖ J‖ L ∞ (−1 , 1) , ‖ E‖ L p

ω α,β(−1 , 1) ≤ C‖ J‖ L p

ω α,β(−1 , 1) . (3.14)

Lemma 3.6 [32] . ∀ ν ∈ C r,κ [ −1 , 1] , you can always find the polynomial function T N ν ∈ P N , making it for non-negative integersr and κ ∈ (0, 1), satisfying a constant C r , κ > 0,

‖ ν − T N ν‖ L ∞ (I) ≤ C r,κN −(r+ κ) ‖ ν‖ r,κ , (3.15) the standard norm in which we define C r,κ [ −1 , 1] is ‖ · ‖ r , κ , and T N is a linear operator from C r,κ [ −1 , 1] to P N . Lemma 3.7 [32] . Let κ ∈ (0, 1) and defined that

(M ν)(x ) = ∫ x

−1 (x − τ ) −μK(x, τ ) ν(τ ) dτ.

Then, for ∀ ν ∈ C[ −1 , 1] , you can always find a positive constant C such that |M ν(x ′ ) − M ν(x ′′ ) |

| x ′ − x ′′ | ≤ C max x ∈ [ −1 , 1] | ν(x ) | , assume that 0 < κ < 1 − μ, for any x ′ , x ′′ ∈ [ −1 , 1] and x ′ � = x ′′ . This established that

‖M ν‖ 0 ,κ ≤ C max x ∈ [ −1 , 1]

| ν(x ) | , 0 < κ < 1 − μ.

4. Convergence analysis of Jacobi spectral collocation method

The main purpose of this section is to make a specific convergence analysis for the numerical scheme presented in the

previous article. According to the convergence result, we show that the Jacobi collocation method we use is the approximate

solution obtained in (2.15) exponentially approximates the exact solution. First, we will derive the error estimate for the

function in L ∞ -norm. Here, we assume that the kernel function K ( x , s , U ( s )) has the two following properties which are required for the proof

of the convergence analysis:

(1) the Lipschitz property, in other words

| K(x, s, ˆ U (s )) − K(x, s, U(s )) | ≤ L K | ̂ U − U(s ) | , ∀ ̂ U , U ∈ [ −1 , 1] (4.1) where U(s ) = [ u (s ) , v (s )] T and ˆ U (s ) = [ ̂ u (s ) , ̂ v (s )] T ;

(2) K(x, s, 0) = 0 2 ×1 . Since the one-dimensional and two-dimensional convergence methods and results of the space are the same, here we

make the proof process concise, and the following is only for the rigor of the one-dimensional Schr ̈o dinger equation con-

vergence analysis.

Theorem 1. Suppose U ( x , t ) is an exact solution of the Eq. (2.7) and sets U M (x, t) = ∑ M j=0 U j (x ) F j (t) is the discrete solution ofthe Eq. (2.12) in the time direction, �t = max { αt , βt } . Then when M is big enough,

‖ U(x, t) − U M (x, t) ‖ L ∞ (I) ≤{

C M −m logMK ∗ + C M 1 2 −m U ∗, −1 < α, β ≤ − 1 2 ,

C M �+ 1 2 −m K ∗ + C M 1+ �−m logMU ∗, − 1

2 ≤ � < μ − 1

2 .

‖ U(x, t) − U M (x, t) ‖ L 2 (I) ≤

⎧ ⎪ ⎨ ⎪ ⎩

CM −m (K ∗ + U ∗) + CM −m −κ logMK ∗+ CM 1 2 −m −κU ∗, −1 < α, β ≤ − 1

2 ,

CM −m (K ∗ + U ∗) + CM �+ 1 2 −m −κK ∗+ CM 1+ �−m −κ logMU ∗, − 1

2 ≤ � < μ − 1

2 .

(4.2)

Where

K ∗ = max x ∈ I

| K(x, s (θ ) , U M (z, s (θ ))) | H m,M ω c

(I) , U ∗ = | U(x, t) | H m,M

ω c (I) .

Proof. According to Lamma 1

(K (x, s ( θ ) , U M ( x, s ( θ ) )

))N,s

:= N ∑

j=0 K (x, s ( θk ) , U

M ( x, s ( θk ) ) )ω μ−1 , 0

k , (4.3)


then, the numerical schemes (2.12) can be written as

U j (x ) −(

1 + t j 2

)μ−1 (K(x, s (θ ) , U M (z, s (θ )))) N,s = δG (x, t j ) + F (x ) , (4.4)

which gives

U j (x ) −(

1 + t j 2

)μ−1 ∫ 1 −1

(1 − θ ) μ−1 K(x, s (θ ) , U M (z, s (θ ))) dθ = δG (x, t) + F (x ) + J 1 (x ) , (4.5)

where

J 1 (x ) = (

1 + t j 2

)μ−1 (K(x, s (θ ) , U M (x, s (θ )))) N,s −

(1 + t j

2

)μ−1 ∫ 1 −1

( 1 − θ ) μ−1 K(x, s (θ ) , U M (x, s (θ ))) dθ, (4.6)

then, we have

| J 1 (x ) | ≤ CN −m (

1 + t j 2

)μ−1 | K(x, s (θ ) , U M (x, s (θ ))) | H m,M

ω c (I) ≤ CN −m | K(x, s (θ ) , U M (x, s (θ ))) | H m,M

ω c (I) . (4.7)

On the other hand, (4.5) can be rewritten as follows:

U j (x ) −∫ t j

−1 (t j − s ) μ−1 K(x, s, U M (x i , s )) ds = δG (x, t j ) + F (z) , 0 ≤ i ≤ M. (4.8)

Multipling F j ( t ) on the both sides of (4.8) and summing up from 0 to M yield,

U M (x, t) − I M ( ∫ t

−1 (t − s ) μ−1 K(x, s, U M (x, s )) ds = δI M G (x, t) + I M F (x ) + I M (J 1 ) , (4.9)

which can be restated in the following form:

δI M G (x, t) + F (x ) + I M (J 1 ) = U M (x, t) − I M ∫ t

−1 (t − s ) μ−1 K(x, t, U(x, t)) ds

−I M ( ∫ 1

−1 (t − s ) μ−1 [ K(x, s, U M (x, s )) − K(x, s, U(x, s ))] ds, (4.10)

where the interpolation operator I M is defined by (3.7) . It follows from (4.10) and (2.7) , that

δI M G (x, t) + F (x ) + I M (J 1 ) = U M (x, t) − I M (U(x, t) − δG (x, t) − F (x )) (4.11)

−I M (∫ t

−1 (t − s ) μ−1 [ K(x, s, U M (x, s )) − K(x, s, U(x, s ))] ds

).

Let e (x, t) = U M (x, t) − U(x, t) , x ∈ [ −1 , 1] , denote the error function. Then, we have

I M (J 1 ) = e (x, t) + (U − I M U)(x, t) − I M (∫ t


), (4.12)

consequently,

e (x, t) = ∫ t

−1 (t − s ) μ−1 [ K(x, s, U M (x, s )) − K(x, s, U(x, s ))] ds ) + I M (J 1 ) + J 2 (x, t) + J 3 (x, t) , (4.13)

where

J 2 (x, t) = I M U(x, t) − U(x, t) ,

J 3 (x, t) = −∫ t


+ I M (∫ t

−1 (t − s ) μ−1

[K (x, s, U M (x, s )

)− K ( x, s, U(x, s ) )

]ds

).

According to the Lipschitz property of the kernel K, we have

| e (x, t) | L 2 (I) ≤∣∣∣∣(∫ t

−1 (t − s ) μ−1 [ K

(x, s, U M (x, s )

)− K ( x, s, U(x, s ) ) ] ds

)∣∣∣∣ + | I M (J 1 ) + J 2 (x, t) + J 3 (x, t) | ≤ L k

∫ t −1

(t − s ) μ−1 | U M (x, s ) − U(x, s ) | ds + | I M (J 1 ) + J 2 (x, t) + J 3 (x, t) |

= L k ∫ t

−1 (t − s ) μ−1 | e (x, s ) | ds + | I M (J 1 ) + J 2 (x, t) + J 3 (x, t) | . (4.14)


Using the Gronwall inequality yield,

‖ e (x ) ‖ L 2 (I) ≤ C (‖ I M (J 1 ) ‖ L 2 (I) + ‖ J 2 (x ) ‖ L 2 (I) + ‖ J 3 (x ) ‖ L 2 (I) ), (4.15)

‖ e (x ) ‖ L ∞ (I) ≤ C (‖ I M (J 1 ) ‖ L ∞ (I) + ‖ J 2 (x ) ‖ L ∞ (I) + ‖ J 3 (x ) ‖ L ∞ (I) ).

From (4.7) and lemma 5, we have

‖ I M (J 1 ) ‖ L 2 (I) ≤ max t∈ I

| J 1 | ≤ CM −m K ∗,

‖ I M (J 1 ) ‖ L ∞ (I) ≤ CM −m max x ∈ I

| K(x, s (θ ) , U M (x, s (θ ))) | H m,M ω c

(I) × max x ∈ I

M ∑ j=0

| F j (t) |

≤ CM −m K ∗ ×{

O (logM) , −1 < α, β ≤ − 1 2 ,

O (M �+ 1 2 ) , otherwise,

≤{

CM −m logMK ∗, −1 < α, β ≤ − 1 2 ,

CM �+ 1 2 −m K ∗, otherwise.

(4.16)

Using L 2 − error and L ∞ − error bounds for the interpolation polynomials gives ‖ J 2 ‖ L 2 (I) = ‖ I M U(x, t) − U(x, t) ‖ L 2 (I)

≤ CM −m | U(x, t) | H m,M ω c

(I) ,

‖ J 2 ‖ L ∞ (I) = ‖ I M U(x, t) − U(x, t) ‖ L ∞ (I)

≤{

CM 1 2 −m | U(x, t) | H m,M

ω c (I) , −1 < α, β ≤ − 1 2 ,

CM 1+ �−m logM| U(x, t) | H m,M ω c

(I) , otherwise. (4.17)

By letting m = 1 , 2 , . . . , n, we have

‖ J 3 ‖ L 2 (I) = ‖ (I M − I) ∫ t j

−1 (t j − s ) μ−1 [ K(x, s, U M (x, s )) − K(x, s, U(x, s ))] ds ‖ L 2 (I)

≤ ‖ (I M − I) ∫ t j

−1 (t j − s ) μ−1 (U M (x, s ) − U(x, s )) ds ‖ L 2 (I)

≤ ‖ (I M − I) ∫ t j

−1 (t j − s ) μ−1 e (x, s ) ds ‖ L 2 (I)

= ‖ (I M − I) M e ‖ L 2 (I) = ‖ (I M − I)(M e − T M e ) ‖ L 2 (I) ≤ ‖ I M (M e − T M e ) ‖ L 2 (I) + ‖ (M e − T M e ) ‖ L 2 (I) ≤ C‖M e − T M e ‖ L ∞ (I) ≤ CM −κ‖M e ‖ 0 ,κ≤ CM −κ‖ e ‖ L ∞ (I) ,

‖ J 3 ‖ L ∞ (I) = ‖ (I M − I) M e ‖ L ∞ (I) = ‖ (I M − I)(M e − T M e ‖ L ∞ (I) ≤ (1 + ‖ I M ‖ L ∞ (I) ) CM −κ‖M e ‖ 0 ,κ , κ ∈ (0 , μ) ≤

{CM −κ logM‖ e ‖ L ∞ (I) , −1 < α, β ≤ − 1 2 , CM

1 2 + �−κ‖ e ‖ L ∞ (I) , − 1 2 ≤ � < μ − 1 2 .

(4.18)

Considering all above estimates together, yield

‖ e | L 2 (I) ≤ C M 1 2 −m K ∗ + C M −m | U(x, t) | H m,M ω c

(I) + C M −κ‖ e ‖ L ∞ (I) ,

‖ e | L ∞ (I) ≤

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

C M −m logMK ∗ + C M 1 2 −m | U(x, t) | H m,M ω c

(I) + C M −κ logM‖ e ‖ L ∞ (I) , −1 < α, β ≤ − 1

2 ,

C M �+ 1 2 −m K ∗ + C M 1+ �−m logM| U(x, t) | H m,M

ω c (I) C M

1 2 + �−κ‖ e ‖ L ∞ (I) ,

− 1 2

≤ � < μ − 1 2 .

(4.19)

�

Theorem 2. Suppose U M (x, t) = ∑ M j=0 U j (x ) F j (t) is the discrete solution in the time direction of the Eq. (2.12) , and U M N (x, t) =∑ N i =0

∑ M j=0 u

j i H i (x ) F j (t) is the Eq. (2.15) at the completely discrete solution of time space, then for large enough N , we can get the


following error estimate:

‖ U M − U M N ‖ L ∞ ≤{

ClogMN 3 4 −m U ∗,

CM 1 2 + �N

3 4 −m U ∗,

‖ U M − U M N ‖ L 2 ≤ CM −m U ∗. (4.20)Proof. By subtracting (2.15) from (2.13) , using the Lipschitz condition, we obtain

U j (x i ) − U j i = 1

(μ)

(T

2

)α ∫ t j −1

(t j − s ) μ−1 (

K (x i , s, U

M ( x i , s ) )

− K (x i , s, U

M N ( x i , s )

))ds

≤ L k ∫ t j

−1 (t j − s ) μ−1

∣∣U M ( x i , s ) − U M N ( x i , s ) ∣∣ds ≤ L k

∫ t j −1

(t j − s ) μ−1 ∣∣∣∣∣

M ∑ m =0

( U m (x i ) −

N ∑ n =0

U m n H n (x i )

) F m (s )

∣∣∣∣∣ds = L k

∣∣∣∣∣M ∑

m =0

( U m (x i ) −

N ∑ n =0

U m n H n (x i )

) ∣∣∣∣∣∫ t j

−1 (t j − s ) μ−1 F m (s ) ds

= L k M ∑

m =0 C j m

( U m (x i ) −

N ∑ n =0

U m n H n (x i )

) , (4.21)

where

C j m = ∫ t j

−1 (t j − s ) μ−1 F m (s ) ds.

Multiplying by H i ( z ) on the both sides of (4.21) and summing from 0 to N for i , then

N ∑ i =0

U j (x i ) H i (x ) −N ∑

i =0 U j

i H i (x ) ≤ L k

N ∑ i =0

[ M ∑

m =0 C j m

( U m (x i ) −

N ∑ n =0

U m n H n (x i )

) ] H i (x ) , (4.22)

so,

I N U j (x ) −

N ∑ i =0

U j i H i (x ) ≤ L k

M ∑ m =0

C j m

[ I N U

m (x ) − I N (

N ∑ n =0

U m n H n (x )

) ] , (4.23)

where,

I N U m (x ) =

N ∑ i =0

U m (x i ) H i (x ) ,

I N

( N ∑

n =0 U m n H n (x )

) =

N ∑ i =0

N ∑ n =0

U m n H n (x i ) H i (x ) . (4.24)

Let e (x, t j ) = U j (x ) −∑ N

i =0 U j

i H i (x ) , we can rewrite (4.23) as,

max t∈ [ −1 , 1]

| e (x, t) | ≤ C| J 4 | , (4.25)where,

J 4 = I N U j (x ) − U j (x ) , (4.26)According to lemma 2, yield,

‖ J 4 ‖ L 2 = ‖ I N U j (x ) − U j (x ) ‖ L 2 ≤ CN −m | u | H m,N (I) , (4.27)‖ J 4 ‖ L ∞ ≤ CN 3 4 −m | u | H m,N (I) .

On the other hand,

U M − U M N = M ∑

j=0

( U j (x ) −

N ∑ i =0

U j i H i (x )

) F j (t)

= M ∑

j=0 e (x, t j ) F j (t) = I M e (x, t) . (4.28)


Fig. 1. The numerical solution and exact solution of (5.1) , μ = 3 / 4 .

Using lemma 4, we have

‖ U M − U M N ‖ L ∞ ≤{

ClogM max t∈ [ −1 , 1]

| e (x, t) | , −1 ≤ α, β < − 1 2 ,

CM 1+ � max t∈ [ −1 , 1]

| e (x, t) | , otherwise. (4.29)

Using lemma 3, we have

‖ U M − U M N ‖ L 2 ≤ ClogM max t∈ [ −1 , 1]

| e (x, t) | . (4.30) Thus, combine (4.25), (4.27), (4.29) with (4.30) , we can compute the results showed in (4.20) . �

Theorem 3. Suppose U ( x , t ) is an exact solution of the Eq. (2.7) and sets U M N (x, t) = ∑ N

i =0 ∑ M

j=0 u j i H i (x ) F j (t) is the entire discrete

solution of (2.15) and satisfies the initial conditions and boundary conditions. If U(x, t) ∈ H n,N ω α,β

, then for big enough M and N ,

there are the following error estimates:

| U(x, t) − U M N (x, t) | L ∞ (I) ≤

⎧ ⎪ ⎨ ⎪ ⎩

C M −m logMK ∗ + C M 1 2 −m U ∗ + C logMN 3 4 −m U ∗, −1 < α, β ≤ − 1

2 ,

C M �+ 1 2 −m K ∗ + C M 1+ �−m logMU ∗ + C M 1 2 + �N 3 4 −m U ∗,

− 1 2

≤ � < μ − 1 2 .

‖ U(x, t) − U M N (x, t) ‖ L 2 (I) ≤

⎧ ⎪ ⎨ ⎪ ⎩

CM −m (K ∗ + U ∗) + CM −m −κ logMK ∗+ C M 1 2 −m −κU ∗ + C N −m U ∗, −1 < α, β ≤ − 1

2 ,

CM −m (K ∗ + U ∗) + CM �+ 1 2 −m −κK ∗+ C M 1+ �−m −κ logMU ∗ + C N −m U ∗, − 1 ≤ � < μ − 1 .

(4.31)

2 2


Fig. 2. The numerical solution and exact solution of (5.2) , μ = 7 / 8 .

Proof. Using the triangle inequality

| U(x, t) − U M N (x, t) | ≤ | U(x, t) − U M (x, t) | + | U M (x, t) − U M N (x, t) | , (4.32)and combine Theorems 1 and 2 , leads to estimation (4.31) . �

5. Algorithm implementation and numerical results

5.1. Example 1. 1D linear equation.

The domain is (x, t) ∈ (−1 , 1) × (0 , 2) ,

i ∂ 3 / 4

∂t 3 / 4 ψ (x, t) + ∂

2

∂x 2 ψ (x, t) = f 1 (x, t) , (x, t) ∈ (−1 , 1) × (0 , 2) , ψ(x, 0) = 0 , x ∈ (−1 , 1) ,

ψ(−1 , t) = ψ(1 , t) = −t 2 , t ∈ (0 , 2) , (5.1)where

f 1 = 16 √

2

5 π(

3

4

)t 5 / 4 (i cos πx − sin πx ) + t 2 (−π2 cos πx − i π2 sin πx ) .

The exact solution is

ψ(x, t) = t 2 ( cos πx + i sin πx ) . In the space direction x , we use P M+2 Lagrange-Gauss-Lobatto orthogonal polynomials, where the node x 0 = −1 and

x M+1 = 1 . In the time direction t , we use P N Jacobi orthogonal polynomials with the index (μ − 1 , 0) = (−1 / 4 , 0) , i.e., (1 +s ) −1 / 4 (1 − s ) 0 , where s = t − 1 . The number of collocation points is MN , instead of (M + 2) N, as the boundary values are


Fig. 3. The solution and the error for (5.3) , μ = 7 / 8 .

Table 1

Example 1. The L ∞ error and the L 2 error at t = 2 , for (5.1) .

M , N ‖ ψ − ψ h ‖ L ∞ ‖ ψ − ψ h ‖ L 2 4, 4 2.2628E-001 6.3147E-002

6, 6 9.1637E-003 1.2706E-003

8, 8 2.5137E-004 2.1386E-005

10, 10 5.4994E-006 1.8660E-006

12, 12 8.5810E-007 8.3305E-007
given. The numerical solution and the exact solution at time t = 2 are plotted in Fig. 1 , for M = 4 and N = 4 . We list thenumerical L ∞ error and the L 2 error at time t = 2 in Table 1 , from [26] , The method converges exponentially.
5.2. Example 2. 1D nonlinear equation.

Consider 1D nonlinear equation

i ∂ 7 / 8

∂t 7 / 8 ψ + ∂

2

∂x 2 ψ + | ψ | 2 ψ = f 2 , (x, t) ∈ (−1 , 1) × (0 , 2) ,

ψ(x, 0) = 0 , x ∈ (−1 , 1) , ψ(−1 , t) = ψ(1 , t) = 0 , t ∈ (0 , 2) , (5.2)

where f 2 is defined such that the exact solution is

ψ(x, t) = t 15 / 8 (

1

8 sin 2 πx + i

12 sin πx

).


Fig. 4. The solution and the error for (5.3) , μ = 7 / 8 .

Table 2


M , N ‖ ψ − ψ h ‖ L ∞ ‖ ψ − ψ h ‖ L 2 4, 4 8.1372E-001 5.6689E-001

6, 6 7.3603E-002 4.1019E-002

8, 8 6.6883E-003 2.7107E-003

10, 10 4.7876E-004 1.4701E-004

12, 12 2.7144E-005 1.2426E-005

14, 14 5.0659E-006 5.1181E-006

In the space direction x , we use P M+2 Lagrange-Gauss-Lobatto orthogonal polynomials. In the time direction t , we use P NJacobi orthogonal polynomials with the weight index (μ − 1 , 0) = (−1 / 8 , 0) , i.e., (1 + s ) −1 / 8 (1 − s ) 0 , s = t − 1 . The numberof collocation points is MN . The numerical solution and the exact solution at time t = 2 are plotted in Fig. 2 , for M = 6and N = 6 . We list the numerical L ∞ error and the L 2 error at time t = 2 in Table 2 . On the first few levels, the numericalsolution converges exponentially.

5.3. Example 3. 2D nonlinear equation.

We solve the following time-fractional Schrödinger equation.

i ∂ 7 / 8 ψ

∂t 7 / 8 + ∂

2 ψ

∂x 2 + ∂

2 ψ

∂y 2 + | ψ | 2 ψ = f 3 , (x, y, t) ∈ (−1 , 1) 2 × (0 , 2) , ψ(x, y, 0) = 0 , (x, y ) ∈ (−1 , 1) 2 ,

ψ(±1 , ±1 , t) = 0 , t ∈ (0 , 2) , (5.3)


Table 3


M , N ‖ ψ − ψ h ‖ L ∞ ‖ ψ − ψ h ‖ L 2 4, 4 6.2351E-001 2.8044E-001

6, 6 7.1289E-002 2.6120E-002

8, 8 7.6262E-003 1.8877E-003

10, 10 1.7519E-003 7.8029E-004

where

f 3 = (

7

8

)t

(− 35

256 sin (πx ) sin (2 πy ) + i 105

512 sin (2 πx ) sin (πy )

)+ t 15 / 8

(− 5

8 sin (2 πx ) sin (πy ) − i 5

12 sin (πx ) sin (2 πy )

)+ t 45 / 8

(1

64 sin

2 (2 πx ) sin 2 (πy ) + 1 144

sin 2 (πx ) sin 2 (2 πy )

)·(

1

8 sin (2 πx ) sin (πy ) + i 1


).

The exact solution is

ψ(x, y, t) = t 15 / 8 (

1

8 sin (2 πx ) sin (πy ) + i 1


).

In the space direction, we use 2D tenor P M+2 Lagrange-Gauss-Lobatto orthogonal polynomials. In the time direction t , weuse P N Jacobi orthogonal polynomials with the weight index (μ − 1 , 0) = (−1 / 8 , 0) , i.e., (1 + s ) −1 / 8 (1 − s ) 0 , s = t − 1 . Thenumber of collocation points is M 2 N . The exact solution and the error of the numerical solution at time t = 2 are plotted inFigs. 3 and 4 , for M = 10 and N = 10 . We list the numerical L ∞ error and the L 2 error at time t = 2 in Table 3 . On the firstfew levels, the numerical solution converges exponentially.

6. Conclusions

In this paper, space-time JSC method has been implemented to solve 1D and 2D time-fractional Schr ̈o dinger equations

with the appropriate initial and boundary conditions. In this regard, we first transform the basic equation into the associ-

ated system of nonlinear weakly singular integro-PDEs and then apply the collocation scheme together with approximating

the existing integrals with high accurate Gaussian quadrature rules, which reduce the basic equation into the corresponding

system of nonlinear algebraic equations. This system of algebraic equations can be solved by robust iterative solvers such as

Newton Raphson iterative method. The presented method has two basic advantages with respect to recent numerical meth-

ods. The proposed method is a balanced technique that has spectral accuracy in both of the spatial and temporal variables.

Moreover, a rigorous convergence analysis is provided to support the proposed numerical idea theoretically. From the results

of numerical experiments, one can conclude that, even by using small number of collocation points, high accurate solutions

are computed by the proposed spectral method. At the current time, this approach is considered to be implemented and

extended to solve other time-fractional PDEs. But, some modifications are need to be applied.

Acknowledgments

The author would like to thank the referees for the helpful suggestions. The work was supported by NSFC Project

( 11671342 , 91430213 , 11671157 , 11771369 ), Project of Scientific Research Fund of Hunan Provincial Science and Technology

Department ( 2018JJ2374 , 2018WK4006 ) and Key Project of Hunan Provincial Department of Education ( 17A210 ).

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Convergence analysis of space-time Jacobi spectral collocation method for solving time-fractional Schrödinger equations1 Introduction2 Jacobi spectral collocation methods3 Some preliminaries and useful Lemmas4 Convergence analysis of Jacobi spectral collocation method5 Algorithm implementation and numerical results5.1 Example 1. 1D linear equation.5.2 Example 2. 1D nonlinear equation.5.3 Example 3. 2D nonlinear equation.

6 ConclusionsAcknowledgmentsReferences

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