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Page 1: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche
Page 2: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Fudziah Ismail, Sufia Zulfa and Norazak Senu

Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor,

Malaysia

Extra Derivative Multistep Collocation Methods with Trigonometrically-fitting for Oscillatory Problems

Page 3: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Contents

โ€ข Introduction

โ€ข Derivation of the method

โ€ข Trigonometrically fitting the method

โ€ข Numerical Results

โ€ข Discussion and Conclusion

Page 4: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Objectives 1. To derive extra derivative multistep methods of order 3 and 4,

using collocation technique. 2. Trigonometrically-fitting the methods. 3. Validating the methods using a set of special second order differential equations which have oscillatory solutions. 4. Comparing the numerical results with other well known existing methods in the scientific literature.

Page 5: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

1. Introduction

The special second order ordinary differential equations (ODEs) can be

represented by

๐‘ฆ" = ๐‘“(๐‘ฅ, ๐‘ฆ), ๐‘ฆ ๐‘ฅ0 = ๐‘ฅ0, ๐‘ฆโ€ฒ ๐‘ฅ0 = ๐‘ฆ0โ€ฒ (1)

in which the first derivative does not appear explicitly.

This type of problems often appears in the field of science, mathematics and

engineering such as quantum mechanics, spatial semi-discretizations of wave

equations, populations modeling and celestial mechanics.

Page 6: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Currently, equation (1) can be solved directly without converting it first to a

system of first order ODEs. Such methods are direct multistep method, Runge-

Kutta Nystrำงm (RKN) method, and hybrid method.

Work RKN can be seen can be seen in Dormand et al. [1] and Sommeijer [2].

Work on hybrid method can be seen in Franco [3] and Coleman [4],where they

have developed hybrid algorithm and constructed the order condition of the

hybrid methods for directly solving equation (1).

Research on directly solving equation (1) using methods which involved, the

interpolation and collocation techniques has been done by various researchers

such as the work of authors in [5]-[6]. As well as the work of Awoyemi [7] and

Jator [8].

Page 7: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

The special second order ODEs also often exhibit oscillatory solutions which cannot

be solved efficiently using conventional methods.

To obtain a more efficient process for solving oscillatory problems, numerical

methods are constructed by taking into account the nature of the problem.

This results in methods in which the coefficients depend on the problem.

Numerical methods can adapted to the special structure of the problem, such

techniques for this purpose are trigonometrically-fitted and phase-fitted

techniques.

Page 8: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

A good theoretical foundation of exponentially fitted technique was given by

Lyche [9]. Since then a lot of exponentially fitted linear multistep have been

constructed mostly for special second order ODEs such as (1).

Vanden Berghe et. al [10]. Introduced an explicit exponentially fitted explicit

Runge-Kutta which integrate exactly first order systems whose solution can be

expressed as linear combination of functions of the form ๐‘’๐œ†๐‘ก, ๐‘’โˆ’๐œ†๐‘ก or

cos ๐œ”๐‘ก , sin(๐œ”๐‘ก) .

This idea is extended to Runge-Kutta method by Simos [11] and Franco [12].

Page 9: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Fang and Wu [13] , developed a Trigonometrically fitted explicit Numerov-type

method for second-order initial value problems with oscillating solutions.

In this research, we are going to construct extra derivative multistep methods of

order 3 and 4 using collocation technique and Chebyshev polynomial will be used

as the basis function

In order to improve the efficiency of the methods, we trigonometrically fitted the

methods so that the coefficients will depend on the fitted frequency and step size

of the problems. And used the methods for solving oscillatory Differential

Equations

Page 10: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

2. Derivation of Linear Multistep Methods (LMM) Using Collocation Technique

The general ๐‘˜-step LMM for solving special second order ODEs is given as

๐›ผ๐‘—๐‘ฆ๐‘›+๐‘—๐‘˜๐‘—=0 = โ„Ž2 ๐›ฝ๐‘—๐‘“๐‘›+๐‘—

๐‘˜๐‘—=0 ,

Here, we will use Chebyshev Polynomials as basis function. The following are the

first five terms of the sequence from Chebyshev Polynomials :

ฮค0 ๐‘ฅ = 1, ฮค1 ๐‘ฅ = ๐‘ฅ, ฮค2 ๐‘ฅ = 2๐‘ฅ2 โˆ’ 1, ฮค3 ๐‘ฅ = 4๐‘ฅ3 โˆ’ 3๐‘ฅ, ฮค4 ๐‘ฅ = 8๐‘ฅ4 โˆ’

8๐‘ฅ2 + 1 (2)

Here, we are going to develop linear multistep method with extra derivatives of the

form of

๐›ผ๐‘—๐‘ฆ๐‘›+๐‘—๐‘˜๐‘—=0 = โ„Ž2 ๐œ‡๐‘—๐‘“๐‘›+๐‘—

๐‘˜๐‘—=0 + โ„Ž3 ๐œ‚๐‘—๐‘”๐‘›+๐‘—

๐‘˜๐‘—=0 , (3)

Page 11: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

where ๐›ผ๐‘—, ๐œ‡๐‘— and ๐œ‚๐‘— are constant values, ๐‘“๐‘›+๐‘— = ๐‘ฆโ€ฒโ€ฒ๐‘›+๐‘— and ๐‘”๐‘›+๐‘— = ๐‘ฆโ€ฒโ€ฒโ€ฒ๐‘›+๐‘—.

We proceed to approximate the exact solution ๐‘ฆ(๐‘ฅ) by the interpolating

function of the form

๐‘ฆ ๐‘ฅ = ๐‘Ž๐‘—๐‘‡ ๐‘› (๐‘ฅ โˆ’ ๐‘ฅ๐‘˜)

๐‘›

๐‘—=0

, (4)

which is the polynomial of degree n and satisfied the equations

๐‘ฆโ€ฒโ€ฒ ๐‘ฅ = ๐‘“ ๐‘ฅ, ๐‘ฆ ๐‘ฅ , ๐‘ฅ๐‘˜ โ‰ค ๐‘ฅ โ‰ค ๐‘ฅ๐‘˜+๐‘ , ๐‘ฆ ๐‘ฅ๐‘˜ = ๐‘ฆ๐‘˜ . (5)

Page 12: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

2.1. Derivation of LMMC(3)

For ๐‘› = 4, equation (4) can be written as

๐‘ฆ ๐‘ฅ = ๐‘Ž0 + ๐‘Ž1 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž2 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 1 + ๐‘Ž3 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜

3 โˆ’ 3 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜

+๐‘Ž4 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜4 โˆ’ 8 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜

2 + 1 . (6)

Differentiating equation (6) three times and we get the first, second, and third

derivatives of the equations as follows:

๐‘ฆโ€ฒ ๐‘ฅ = ๐‘Ž1 + ๐‘Ž22 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž33 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 1

+๐‘Ž44 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜3 โˆ’ 4 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ , (7)

๐‘ฆโ€ฒโ€ฒ ๐‘ฅ = 2๐‘Ž2 + ๐‘Ž36 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž44 3 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 4 , (8)

๐‘ฆโ€ฒโ€ฒโ€ฒ ๐‘ฅ = 6๐‘Ž3 + ๐‘Ž424 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ , (9)

Page 13: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Next, equations (6) and (8) are collocated at ๐‘ฅ = ๐‘ฅ๐‘˜+1, ๐‘ฅ๐‘˜+2 and interpolated equation

(9) at ๐‘ฅ = ๐‘ฅ๐‘˜+1 which yields

๐‘ฆ ๐‘ฅ๐‘˜+1 = ๐‘Ž0 + ๐‘Ž1 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž2 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 1

+ ๐‘Ž3 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜3 โˆ’ 3 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜

+๐‘Ž4 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜4 โˆ’ 8 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜

2 + 1 = ๐‘ฆ๐‘˜+1, (10)

๐‘ฆ ๐‘ฅ๐‘˜+2 = ๐‘Ž0 + ๐‘Ž1 ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž2 ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 1

+ ๐‘Ž3 ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜3 โˆ’ 3 ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜

+๐‘Ž4 ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜4 โˆ’ 8 ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜

2 + 1 = ๐‘ฆ๐‘˜+2, (11)

Page 14: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

๐‘ฆโ€ฒโ€ฒ ๐‘ฅ๐‘˜+1 = 2๐‘Ž2 + ๐‘Ž36 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž44 3 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 4 = ๐‘“๐‘˜+1, (12)

๐‘ฆโ€ฒโ€ฒ ๐‘ฅ๐‘˜+2 = 2๐‘Ž2 + ๐‘Ž36 ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž44 3 ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 4 = ๐‘“๐‘˜+2, (13)

๐‘ฆโ€ฒโ€ฒโ€ฒ ๐‘ฅ๐‘˜+1 = 6๐‘Ž3 + ๐‘Ž424 ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜ = ๐‘”๐‘˜+1. (14)

By substituting โ„Ž = ๐‘ฅ๐‘˜+1 โˆ’ ๐‘ฅ๐‘˜ and 2โ„Ž = ๐‘ฅ๐‘˜+2 โˆ’ ๐‘ฅ๐‘˜ into equations (10)-(14), we obtain

the following:

๐‘ฆ๐‘˜+1 = ๐‘Ž0 + โ„Ž ๐‘Ž1 + ๐‘Ž2 โ„Ž2 โˆ’ 1 + ๐‘Ž3 โ„Ž

3 โˆ’ 3โ„Ž + ๐‘Ž4 โ„Ž4 โˆ’ 8โ„Ž2 + 1 , (15)

๐‘ฆ๐‘˜+2 = ๐‘Ž0 + 2โ„Ž ๐‘Ž1 + ๐‘Ž2 4โ„Ž2 โˆ’ 1 + ๐‘Ž3 8โ„Ž

3 โˆ’ 6โ„Ž + ๐‘Ž4 16โ„Ž4 โˆ’ 32โ„Ž2 + 1 , (16)

๐‘“๐‘˜+1 = 2๐‘Ž2 + ๐‘Ž3 6โ„Ž + ๐‘Ž4 12โ„Ž2 โˆ’ 16 , (17)

๐‘“๐‘˜+2 = 2๐‘Ž2 + ๐‘Ž3 12โ„Ž + ๐‘Ž4 48โ„Ž2 โˆ’ 16 , (18)

๐‘”๐‘˜+1 = 6๐‘Ž3 + ๐‘Ž4 24โ„Ž . (19)

Page 15: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Rearranging equations (15)-(19) into matrix form as follows:

1 โ„Ž โ„Ž2 โˆ’ 1 โ„Ž3 โˆ’ 3โ„Ž โ„Ž4 โˆ’ 8โ„Ž2 + 11 2โ„Ž 4โ„Ž2 โˆ’ 1 8โ„Ž3 โˆ’ 6โ„Ž 16โ„Ž4 โˆ’ 16โ„Ž2 + 10 0 2โ„Ž 6โ„Ž2 12โ„Ž2 โˆ’ 160 0 2โ„Ž 12โ„Ž2 48โ„Ž2 โˆ’ 160 0 0 6โ„Ž 24โ„Ž2

๐‘Ž0๐‘Ž1๐‘Ž2๐‘Ž3๐‘Ž4

=

๐‘ฆ๐‘˜+1๐‘ฆ๐‘˜+2โ„Ž๐‘“๐‘˜+1โ„Ž๐‘“๐‘˜+2โ„Ž๐‘”๐‘˜+1

which can be simplified as ๐‘‹๐ด = ๐‘Œ (20)

Page 16: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

we have ๐ด = ๐‘‹โˆ’1๐‘Œ (21) where

๐‘‹โˆ’1 =

2 โˆ’110โ„Ž4 โˆ’ 7

12โ„Ž32โ„Ž4 + 6โ„Ž2 + 7

12โ„Ž3โˆ’2โ„Ž4 + 12โ„Ž2 + 7

12โ„Ž2

โˆ’1

โ„Ž

1

โ„Žโˆ’13โ„Ž2 โˆ’ 12

12โ„Ž2โˆ’5โ„Ž2 + 12

12โ„Ž23 โ„Ž2 + 2

12โ„Ž2

0 0 โˆ’2

3โ„Ž33โ„Ž2 + 4

6โ„Ž3โˆ’3โ„Ž2 + 2

3โ„Ž2

0 01

3โ„Ž2โˆ’

1

3โ„Ž21

2โ„Ž

0 0 โˆ’1

12โ„Ž31

12โ„Ž3โˆ’

1

12โ„Ž2

Page 17: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Solving (21), the coefficients of ๐‘Ž0,๐‘Ž1, ๐‘Ž2, ๐‘Ž3, and ๐‘Ž4 are obtained in terms of ๐‘ฆ๐‘˜+1, ๐‘ฆ๐‘˜+2 , ๐‘“๐‘˜+1, ๐‘“๐‘˜+2 and ๐‘”๐‘˜+1.

๐‘Ž0 = 2๐‘ฆ๐‘˜+1 โˆ’ ๐‘ฆ๐‘˜+2 +1

12

10โ„Ž4 โˆ’ 7

โ„Ž2๐‘“๐‘˜+1 +

1

12

2โ„Ž4 + 6โ„Ž2 + 7

โ„Ž2๐‘“๐‘˜+2

โˆ’1

12

2โ„Ž4 + 12โ„Ž2 + 7

โ„Ž2๐‘”๐‘˜+1,

๐‘Ž1 = โˆ’๐‘ฆ๐‘˜+1โ„Ž

+๐‘ฆ๐‘˜+2โ„Ž

โˆ’1

12

13โ„Ž2 โˆ’ 12

โ„Ž๐‘“๐‘˜+1 โˆ’

1

12

5โ„Ž2 + 12

โ„Ž๐‘“๐‘˜+2

+3

4โ„Ž2 + 2 ๐‘”๐‘˜+1,

๐‘Ž2 = โˆ’2

3

๐‘“๐‘˜+1โ„Ž2

+1

6

3โ„Ž2 + 4

โ„Ž2๐‘“๐‘˜+2 โˆ’

1

3

3โ„Ž2 + 2

โ„Ž๐‘”๐‘˜+1,

๐‘Ž3 =1

3

๐‘“๐‘˜+1โ„Ž

โˆ’1

3

๐‘“๐‘˜+2โ„Ž

+1

2๐‘”๐‘˜+1,

๐‘Ž4 = โˆ’1

12

๐‘“๐‘˜+1โ„Ž2

+1

12

๐‘“๐‘˜+2โ„Ž2

โˆ’1

12

๐‘”๐‘˜+1โ„Ž

.

Substituting the coefficients into equation (6) and letting ๐‘ฅ = ๐‘ฅ๐‘˜+3, we obtain the following equation:

Page 18: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

๐‘ฆ ๐‘ฅ๐‘˜+3 = 2๐‘ฆ๐‘˜+1 โˆ’ ๐‘ฆ๐‘˜+2 +1

12

10โ„Ž4 โˆ’ 7

โ„Ž2๐‘“๐‘˜+1 +

1

12

2โ„Ž4 + 6โ„Ž2 + 7

โ„Ž2๐‘“๐‘˜+2

โˆ’1

12

2โ„Ž4 + 12โ„Ž2 + 7

โ„Ž2๐‘”๐‘˜+1

+ โˆ’๐‘ฆ๐‘˜+1โ„Ž

+๐‘ฆ๐‘˜+2โ„Ž

โˆ’1

12

13โ„Ž2 โˆ’ 12

โ„Ž๐‘“๐‘˜+1 โˆ’

1

12

5โ„Ž2 + 12

โ„Ž๐‘“๐‘˜+2

+3

4โ„Ž2 + 2 ๐‘”๐‘˜+1 ๐‘ฅ๐‘˜+3 โˆ’ ๐‘ฅ๐‘˜

+ โˆ’2

3

๐‘“๐‘˜+1โ„Ž2

+1

6

3โ„Ž2 + 4

โ„Ž2๐‘“๐‘˜+2 โˆ’

1

3

3โ„Ž2 + 2

โ„Ž๐‘”๐‘˜+1 ๐‘ฅ๐‘˜+3 โˆ’ ๐‘ฅ๐‘˜

2 โˆ’ 1

+1

3

๐‘“๐‘˜+1โ„Ž

โˆ’1

3

๐‘“๐‘˜+2โ„Ž

+1

2๐‘”๐‘˜+1 ๐‘ฅ๐‘˜+3 โˆ’ ๐‘ฅ๐‘˜

3 โˆ’ 3 ๐‘ฅ๐‘˜+3 โˆ’ ๐‘ฅ๐‘˜

+1

3

๐‘“๐‘˜+1โ„Ž

โˆ’1

3

๐‘“๐‘˜+2โ„Ž

+1

2๐‘”๐‘˜+1 ๐‘ฅ๐‘˜+3 โˆ’ ๐‘ฅ๐‘˜

4 โˆ’ 8 ๐‘ฅ๐‘˜+3 โˆ’ ๐‘ฅ๐‘˜2 + 1 = ๐‘ฆ๐‘˜+3.

Page 19: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Letting 3โ„Ž = ๐‘ฅ๐‘˜+3 โˆ’ ๐‘ฅ๐‘˜ , we obtain the discrete form of LMMC as

๐‘ฆ๐‘˜+3 = 2๐‘ฆ๐‘˜+2 โˆ’ ๐‘ฆ๐‘˜+1 +โ„Ž2

6โˆ’๐‘“๐‘˜+1 + 7๐‘“๐‘˜+2 โˆ’ โ„Ž๐‘”๐‘˜+1 , (22)

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2.2 Order and Consistency of LMMC method

Definition 1: The linear difference operator L is defined by

๐ฟ ๐‘ฆ ๐‘ฅ ; โ„Ž = ๐›ผ๐‘—๐‘ฆ ๐‘ฅ + ๐‘—โ„Ž โˆ’ โ„Ž2๐œ‡๐‘—๐‘“ ๐‘ฅ + ๐‘—โ„Ž โˆ’ โ„Ž3๐œ‚๐‘—๐‘” ๐‘ฅ + ๐‘—โ„Ž

๐‘˜

๐‘—=0

,

where ๐‘ฆ(๐‘ฅ) is an arbitrary function that is sufficiently differentiable on [๐‘Ž, ๐‘]. By

expanding the test function and its first derivative as Taylor series about ๐‘ฅ and

collecting the terms to obtain

๐ฟ ๐‘ฆ ๐‘ฅ ; โ„Ž = ๐‘0๐‘ฆ ๐‘ฅ + ๐‘1โ„Ž๐‘ฆโ€ฒ ๐‘ฅ +โ‹ฏ+ ๐‘๐‘žโ„Ž

(๐‘ž)๐‘ฆ(๐‘ž) ๐‘ฅ + โ‹ฏ ,

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where the coefficients of ๐‘๐‘ž are constants independent of ๐‘ฆ(๐‘ฅ). In particular

๐‘0 = ๐›ผ๐‘—๐‘˜๐‘—=0 , ๐‘1 = ๐‘—๐›ผ๐‘—

๐‘˜๐‘—=0 ,

๐‘2 = ๐‘—(2)

2!๐›ผ๐‘— โˆ’ ๐œ‡๐‘—

๐‘˜๐‘—=0 ,

๐‘3 = ๐‘—(3)

3!๐›ผ๐‘— โˆ’ ๐‘—๐œ‡๐‘— โˆ’ ๐œ‚๐‘—

๐‘˜

๐‘—=0

,

โ‹ฎ

๐‘๐‘ž = ๐‘—(๐‘ž)

๐‘ž!๐›ผ๐‘— โˆ’

๐‘—(๐‘žโˆ’2)

(๐‘ž โˆ’ 2)!๐œ‡๐‘— โˆ’

๐‘—(๐‘žโˆ’3)

(๐‘ž โˆ’ 3)!๐œ‚๐‘—

๐‘˜

๐‘—=0

.

(23)

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Definition 2:

The associated linear multistep method (22) is said to be of the order ๐œŒ if

๐‘0 = ๐‘1 = โ‹ฏ = ๐‘๐œŒ+1 = 0 and ๐‘๐œŒ+2 โ‰  0.

Definition 3 [Consistency of the method]

The method is said to be consistence if it has order at least one.

By substituting the coefficients into equations (23), we obtain

๐‘0 = ๐‘1 = ๐‘2 = ๐‘3 = ๐‘4 = 0 and ๐‘5 =1

8.

The new method is consistent and has order ๐‘ = 3.

And denoted as linear multistep method with extra derivative using collocation

technique of order three (LMMC(3)).

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2.2. Derivation of LMMC(4) , order and consistency of the method

In this section, we derive the LMMC of order four. For ๐‘› = 5, we obtain equation (4) as

๐‘ฆ ๐‘ฅ = ๐‘Ž0 + ๐‘Ž1 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž2 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 1 + ๐‘Ž3 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜

3 โˆ’ 3 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜

+๐‘Ž4 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜4 โˆ’ 8 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜

2 + 1

+๐‘Ž5 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜5 โˆ’ 20 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜

3 + 5 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ . (24)

Differentiating equation (24) three times gives

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Differentiating equation (24) three times gives

๐‘ฆโ€ฒ ๐‘ฅ = ๐‘Ž1 + ๐‘Ž22 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž33 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 1

+๐‘Ž44 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜3 โˆ’ 4 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž55 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜

4 โˆ’ 12 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜2 + 1 , (25)

๐‘ฆโ€ฒโ€ฒ ๐‘ฅ = 2๐‘Ž2 + ๐‘Ž36 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž44 3 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 4

+๐‘Ž520 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜3 โˆ’ 6 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ , (26)

๐‘ฆโ€ฒโ€ฒโ€ฒ ๐‘ฅ = 6๐‘Ž3 + ๐‘Ž424 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜ + ๐‘Ž560 ๐‘ฅ โˆ’ ๐‘ฅ๐‘˜2 โˆ’ 2 , (27)

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Equations (24) and (27) are collocated at ๐‘ฅ = ๐‘ฅ๐‘˜+1, ๐‘ฅ๐‘˜+2 , and equation (26) at

๐‘ฅ = ๐‘ฅ๐‘˜+2, ๐‘ฅ๐‘˜+3 which yields

๐‘ฆ๐‘˜+1 = ๐‘Ž0 + โ„Ž ๐‘Ž1 + ๐‘Ž2 โ„Ž2 โˆ’ 1 + ๐‘Ž3 โ„Ž

3 โˆ’ 3โ„Ž + ๐‘Ž4 โ„Ž4 โˆ’ 8โ„Ž2 + 1

+๐‘Ž5 โ„Ž5 โˆ’ 20โ„Ž2 + 5โ„Ž , (28)

๐‘ฆ๐‘˜+2 = ๐‘Ž0 + 2โ„Ž ๐‘Ž1 + ๐‘Ž2 4โ„Ž2 โˆ’ 1 + ๐‘Ž3 8โ„Ž

3 โˆ’ 6โ„Ž + ๐‘Ž4 16โ„Ž4 โˆ’ 32โ„Ž2 + 1

+๐‘Ž5 32โ„Ž5 โˆ’ 160โ„Ž3 + 10โ„Ž , (29)

๐‘“๐‘˜+2 = 2๐‘Ž2 + ๐‘Ž3 6โ„Ž + ๐‘Ž4 12โ„Ž2 โˆ’ 16 + ๐‘Ž5 20โ„Ž

3 โˆ’ 120โ„Ž , (30)

๐‘“๐‘˜+3 = 2๐‘Ž2 + ๐‘Ž3 18โ„Ž + ๐‘Ž4 48โ„Ž2 โˆ’ 16 + ๐‘Ž5 540โ„Ž

3 โˆ’ 360โ„Ž , (31)

๐‘”๐‘˜+1 = 6๐‘Ž3 + ๐‘Ž4 24โ„Ž + ๐‘Ž5 60โ„Žโ„Ž2 โˆ’ 120 , (32)

๐‘”๐‘˜+2 = 6๐‘Ž3 + ๐‘Ž4 48โ„Ž + ๐‘Ž5 240โ„Žโ„Ž2 โˆ’ 120 . (33)

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Equations (29)-(33) are written in matrix form as follows:

๐‘‹๐ด = ๐‘Œ

Where

๐‘‹ =

1 โ„Ž โ„Ž2 โˆ’ 1 โ„Ž3 โˆ’ 3โ„Ž โ„Ž4 โˆ’ 8โ„Ž2 + 1 โ„Ž5 โˆ’ 20โ„Ž3 + 5โ„Ž1 2โ„Ž 4โ„Ž2 โˆ’ 1 8โ„Ž3 โˆ’ 6โ„Ž 16โ„Ž4 โˆ’ 32โ„Ž2 + 1 32โ„Ž5 โˆ’ 160โ„Ž3 + 10โ„Ž0 0 2โ„Ž 12โ„Ž2 48โ„Ž3 โˆ’ 16โ„Ž 160โ„Ž4 โˆ’ 240โ„Ž2

0 0 2โ„Ž 18โ„Ž2 108โ„Ž3 โˆ’ 16โ„Ž 540โ„Ž4 โˆ’ 360โ„Ž2

0 0 0 6โ„Ž 24โ„Ž2 60โ„Ž3 โˆ’ 120โ„Ž0 0 0 6โ„Ž 48โ„Ž2 240โ„Ž3 โˆ’ 120โ„Ž

,

๐ด = ๐‘Ž0 ๐‘Ž1 ๐‘Ž2 ๐‘Ž3 ๐‘Ž4 ๐‘Ž5 ๐‘‡ , and ๐‘Œ = ๐‘ฆ๐‘˜+1 ๐‘ฆ๐‘˜+2 โ„Ž๐‘“๐‘˜+2 โ„Ž๐‘“๐‘˜+3 โ„Ž๐‘”๐‘˜+1 โ„Ž๐‘”๐‘˜+2

๐‘‡.

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๐‘Ž0 = 2๐‘ฆ๐‘˜+1 โˆ’ ๐‘ฆ๐‘˜+2 +3 6โ„Ž4 + 6โ„Ž2 + 7

20โ„Ž2๐‘“๐‘˜+2 +

2โ„Ž4 โˆ’ 8โ„Ž2 โˆ’ 21

20โ„Ž2๐‘“๐‘˜+3

โˆ’32โ„Ž4 + 72โ„Ž2 + 49

60โ„Ž๐‘”๐‘˜+1 โˆ’

17โ„Ž4 โˆ’ 18โ„Ž2 โˆ’ 56

30โ„Ž๐‘”๐‘˜+2,

๐‘Ž1 = โˆ’1

โ„Ž๐‘ฆ๐‘˜+1 +

1

โ„Ž๐‘ฆ๐‘˜+2 โˆ’

153โ„Ž4 + 120โ„Ž2 + 110

100โ„Ž3๐‘“๐‘˜+2

+3โ„Ž4 + 120โ„Ž2 + 100

100โ„Ž3๐‘“๐‘˜+3 +

392โ„Ž4 + 480โ„Ž2 + 165

300โ„Ž2๐‘”๐‘˜+1

+149โ„Ž4 โˆ’ 690โ„Ž2 โˆ’ 495

300โ„Ž2๐‘”๐‘˜+2,

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๐‘Ž2 =3 3โ„Ž2 + 4

10โ„Ž2๐‘“๐‘˜+2 โˆ’

2 โ„Ž2 + 3

5โ„Ž2๐‘“๐‘˜+3 โˆ’

2 9โ„Ž2 + 7

15โ„Ž2๐‘”๐‘˜+1

+9โ„Ž2 + 32

15โ„Ž2๐‘”๐‘˜+2

๐‘Ž3 = โˆ’2 โ„Ž2 + 1

5โ„Ž2๐‘“๐‘˜+2 +

2 โ„Ž2 + 1

5โ„Ž2๐‘“๐‘˜+3 +

8โ„Ž2 + 3

15โ„Ž2๐‘”๐‘˜+1

โˆ’23โ„Ž2 + 18

30โ„Ž2๐‘”๐‘˜+2,

๐‘Ž4 =3

20โ„Ž2๐‘“๐‘˜+2 โˆ’

3

20โ„Ž2๐‘“๐‘˜+3 โˆ’

7

60โ„Ž๐‘”๐‘˜+1 +

4

15โ„Ž๐‘”๐‘˜+2,

๐‘Ž5 = โˆ’1

50โ„Ž3๐‘“๐‘˜+2 +

1

50โ„Ž3๐‘“๐‘˜+3 +

1

100โ„Ž2๐‘”๐‘˜+1 โˆ’

3

100โ„Ž2๐‘”๐‘˜+2.

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We substitute๐‘Ž0,๐‘Ž1, ๐‘Ž2, ๐‘Ž3, ๐‘Ž4 and ๐‘Ž5 into equation (24) and by letting ๐‘ฅ = ๐‘ฅ๐‘˜+3, and

3โ„Ž = ๐‘ฅ๐‘˜+3 โˆ’ ๐‘ฅ๐‘˜ we obtain the discrete form of LMMC as

๐‘ฆ๐‘˜+3 = 2๐‘ฆ๐‘˜+2 โˆ’ ๐‘ฆ๐‘˜+1 +โ„Ž2

109๐‘“๐‘˜+2 + ๐‘“๐‘˜+3 โˆ’

โ„Ž3

30๐‘”๐‘˜+1 + 2๐‘”๐‘˜+2 , (34)

By substituting back the coefficients into equations (23), we obtain

๐‘0 = ๐‘1 = ๐‘2 = ๐‘3 = ๐‘4 = ๐‘5 = 0 and ๐‘6 = โˆ’1

144.

From Definition 2, the new method is consistent and the order is ๐‘ = 4.

And denoted as linear multistep method with extra derivative using collocation

technique (LMMC(4)).

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3. Trigonometrically-Fitting The Methods

In this section, we adapt the trigonometrically-fitting technique to

LMMC(3). By letting some of the coefficients to be unknown values of ๐œ…๐‘–,

for ๐‘– = 1,2,3, (22) is written as follows:

๐‘ฆ๐‘›+1 = 2๐‘ฆ๐‘› โˆ’ ๐‘ฆ๐‘›โˆ’1 + โ„Ž2 ๐œ…1๐‘“๐‘›โˆ’1 + ๐œ…2๐‘“๐‘› + โ„Ž3 ๐œ…3๐‘”๐‘›โˆ’1 . (35)

Assuming that ๐‘ฆ ๐‘ฅ is a linear combination of the functions

{sin(๐‘ฃ๐‘ฅ), cos(๐‘ฃ๐‘ฅ)} for ๐‘ฃ โˆˆ โ„›. We obtain the following equations:

cos ๐ป = 2 โˆ’ cos ๐ป โˆ’ ๐ป2 ๐œ…1 cos ๐ป + ๐œ…2 + ๐œ…3๐ป sin ๐ป , (36)

๐œ…1 sin ๐ป = ๐œ…3๐ป cos ๐ป ,

where ๐ป = ๐‘ฃโ„Ž, โ„Ž is the step size and๐‘ฃ is fitted frequency.

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Solving equation (36) and letting, ๐œ…1 = โˆ’1/6, the value of the remaining coefficients

is obtain in terms of ๐ป.

๐œ…2 =7

6+

3

80๐ป4 +

851

60480๐ป6 +

20777

3628800๐ป8 +

7939

3421440๐ป10 + ๐‘‚ ๐ป12 ,

๐œ…3 = โˆ’1

6โˆ’

1

18๐ป2 โˆ’

1

45๐ป4 โˆ’

17

1890๐ป6 โˆ’

31

8505๐ป8 โˆ’

691

467775๐ป10 + ๐‘‚ ๐ป12 .

This new method is denoted as Trigonometrically-fitted linear multistep method

with extra derivative using collocation technique of order three (TF-LMMC(3)).

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Then we apply the trigonometrically-fitting technique to LMMC(4). By letting

some of the coefficients to be unknown values of ๐œ…๐‘–, for ๐‘– = 1,2,3,4, rewrite the

formula in (34) as

๐‘ฆ๐‘›+1 = 2๐‘ฆ๐‘› โˆ’ ๐‘ฆ๐‘›โˆ’1 + โ„Ž2 ๐œ…1๐‘“๐‘› + ๐œ…2๐‘“๐‘›+1 + โ„Ž3 ๐œ…3๐‘”๐‘›โˆ’1 + ๐œ…4๐‘”๐‘› . (37)

Assuming that ๐‘ฆ ๐‘ฅ is the linear combination of the function {sin(๐‘ฃโ„Ž), cos(๐‘ฃโ„Ž)}

for ๐‘ฃ โˆˆ โ„›. Therefore, the following equations are obtained.

cos ๐ป = 2 โˆ’ cos ๐ป โˆ’ ๐ป2 ๐œ…1 + ๐œ…2 cos ๐ป + ๐œ…3๐ป sin ๐ป ,

(38)

๐œ…2 sin ๐ป = โˆ’๐ป ๐œ…3 cos ๐ป +๐œ…4 ,

where ๐ป = ๐‘ฃโ„Ž, โ„Ž is the step size and๐‘ฃ is fitted frequency.

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Solving equations in (38) simultaneously by letting ๐œ…1 = 9/10 and ๐œ…3 = โˆ’1/30

the value of the remaining coefficients is obtained in terms of ๐ป as follows:

๐œ…2 =1

10โˆ’

1

144๐ป4 โˆ’

313

100800๐ป6 โˆ’

923

725760๐ป8 โˆ’

6437

12474000๐ป10 + ๐‘‚ ๐ป12 ,

๐œ…4 = โˆ’1

15+

3

400๐ป4 +

83

432000๐ป6 +

983

1209600๐ป8 + ๐‘‚ ๐ป10 .

This new method is denoted as Trigonometrically-fitted linear multistep method

with extra derivative using collocation technique of order four (TF-LMMC(4)).

The other coefficients of the method remain the same. This method is

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4. Numerical Results and Discussion

In this section, the new methods LMMC(3), LMMC(4), TF-LMMC(3) and TF-

LMMC(4) are tested for problems .1-5 . Comparisons are made with other

existing methods.

The following are the notation used in figures 1-10:

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TF-LMMC(3) Trigonometrically-fitted linear multistep method with collocation

method of order three developed in this paper.

LMMC(3) A linear multistep method with collocation method of order three

developed in this paper.

TF-LMMC(4) Trigonometrically-fitted linear multistep method with collocation

method of order four developed in this paper.

LMMC(4) A linear multistep method with collocation method of order four

developed in this paper.

EHM3(4) Explicit three-stage fourth-order hybrid method derived by Franco

[12]

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RKN3(4) Explicit three-stage fourth-order RKN method by Hairer et al.[14].

PFRKN4(4) Explicit four-stage fourth-order Phase-fitted RKN method by

Papadopoulos et al.[15].

DIRKN(HS) Diagonally implicit three-stage fourth-order RKN method derived in

Sommeijer [16]

DIRKN3(4) Diagonally implicit three-stage fourth-order RKN method derived in

Senu et al. [17]

SIHM3(4) Semi-implicit three-stage fourth-order hybrid method developed in

Ahmad et al. [18]

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Problem 1(An almost Periodic Orbit problem studied by Stiefel and Bettis [19]) ๐‘ฆ1" ๐‘ฅ + ๐‘ฆ1 ๐‘ฅ = 0.001cos ๐‘ฅ , ๐‘ฆ1 0 = 1,๐‘ฆ1โ€ฒ 0 = 0, ๐‘ฆ2" ๐‘ฅ + ๐‘ฆ2 ๐‘ฅ = 0.001sin ๐‘ฅ , ๐‘ฆ2 0 = 0,๐‘ฆ2โ€ฒ 0 = 0.9995, Exact solution is ๐‘ฆ1 = cos ๐‘ฅ + 0.0005๐‘ฅsin ๐‘ฅ , and ๐‘ฆ2 = sin ๐‘ฅ โˆ’0.0005๐‘ฅcos ๐‘ฅ . The fitted frequency is ๐‘ฃ = 1.

Figure 1: The efficiency curve for TF-LMMC(3) for

Problem 1 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.9

2๐‘– for ๐‘– = 1,โ€ฆ , 5.

The following are efficiency curves TF-LMMC(3) Method.

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Problem 2 (Inhomogeneous system

by Lambert and Watson [20])

๐‘‘2๐‘ฆ1 ๐‘ฅ

๐‘‘๐‘ก2= โˆ’๐‘ฃ2๐‘ฆ1 ๐‘ฅ + ๐‘ฃ2๐‘“ ๐‘ฅ + ๐‘“" ๐‘ฅ ,

๐‘ฆ1 0 = ๐‘Ž + ๐‘“ 0 , ๐‘ฆ1โ€ฒ 0 = ๐‘“โ€ฒ 0 ,

๐‘‘2๐‘ฆ2 ๐‘ฅ

๐‘‘๐‘ก2= โˆ’๐‘ฃ2๐‘ฆ2 ๐‘ฅ + ๐‘ฃ2๐‘“ ๐‘ฅ + ๐‘“" ๐‘ฅ

๐‘ฆ2 0 = ๐‘“ 0 , ๐‘ฆ2โ€ฒ 0 = ๐‘ฃ๐‘Ž + ๐‘“โ€ฒ 0

Exact solution is ๐‘ฆ1 ๐‘ฅ = ๐‘Žcos ๐‘ฃ๐‘ฅ +

๐‘“ ๐‘ฅ ,๐‘ฆ2 ๐‘ฅ = ๐‘Žsin ๐‘ฃ๐‘ฅ + ๐‘“ ๐‘ฅ , ๐‘“ ๐‘ฅ is chosen

to be ๐‘’โˆ’0.05๐‘ฅ and parameters ๐‘ฃ and ๐‘Ž

are 20 and 0.1 respectively.

Figure 2: The efficiency curve for TF-LMMC(3) for

Problem 2 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.125

2๐‘– for

๐‘– = 2,โ€ฆ , 6.

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Problem 3 (Inhomogeneous system studied by Franco [12])

๐‘ฆ" ๐‘ฅ =1012

โˆ’992

โˆ’992

1012

๐‘ฆ ๐‘ฅ = ๐›ฟ932cos(2๐‘ฅ) โˆ’99

2sin 2๐‘ฅ

932sin(2๐‘ฅ) โˆ’

992cos(2๐‘ฅ)

๐‘ฆ 0 =โˆ’1 + ๐›ฟ

1, ๐‘ฆโ€ฒ 0 =

โˆ’1010 + 2๐›ฟ

for ๐›ฟ = 10โˆ’3. Exact solution ๐‘ฆ ๐‘ก =โˆ’cos 10๐‘ฅ โˆ’ sin 10๐‘ฅ + ๐›ฟcos(2๐‘ฅ)

cos 10๐‘ฅ + sin 10๐‘ฅ + ๐›ฟsin(2๐‘ฅ).

The Eigen-value of the problem are ๐‘ฃ = 10 and ๐‘ฃ = 1. The fitted frequency is chosen to be ๐‘ฃ = 10 because it is dominant than ๐‘ฃ = 1.

Figure 3: The efficiency curve for TF-LMMC(3) for

Problem 3 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.125

2๐‘– for

๐‘– = 1,โ€ฆ , 5.

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Problem 4 (Homogenous given

in Attili et.al [2])

๐‘ฆ" ๐‘ฅ = โˆ’64๐‘ฆ ๐‘ฅ ,๐‘ฆ 0 = 14 ,

๐‘ฆโ€ฒ 0 = โˆ’ 12 .

Exact solution is ๐‘ฆ = 1716 sin 8๐‘ฅ + ๐œƒ ,

๐œƒ = ๐œ‹ โˆ’ tanโˆ’1 4 .

The fitted frequency is ๐‘ฃ = 8.

Figure 4: The efficiency curve for TF-LMMC(3) for

Problem 5 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.1

2๐‘– for

๐‘– = 3,โ€ฆ , 7.

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Problem 5 (Inhomogeneous equation

studied by Papadopoulos et.al [15])

๐‘ฆโ€ฒโ€ฒ(๐‘ฅ) = โˆ’๐œˆ2๐‘ฆ(๐‘ฅ) + (๐œˆ2 โˆ’ 1)sin(๐‘ฅ),

๐‘ฆ(0) = 1, ๐‘ฆโ€™(0) = ๐œˆ + 1.

Exact solution is y(๐‘ฅ) = cos(๐œˆ๐‘ฅ) +

sin(๐œˆ๐‘ฅ) + sin(๐‘ฅ).

The fitted frequency is ๐œˆ = 10.

Figure 6: The efficiency curve for TF-LMMC(3) for Problem

6 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.125

2๐‘– for ๐‘– = 3,โ€ฆ , 7.

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The efficiency curves are shown in Figures 1-5, where problems 1-5 are

tested for a very large interval ๐‘‡๐‘’๐‘›๐‘‘ = 10000.

It is observed that TF-LMMC(3) lies below all of the other methods efficiency

graphs. It prove that TF-LMMC(3) is superior compared to the other existing

methods in the literature.

LMMC(3) is as competitive as the other methods for solving oscillatory

problems, though LMMC(3) is method of order three which is one order less

compared to the other methods in comparison.

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Although explicit methods such as EHM3(4) and RKN3(4) needs less time to

do the computation, the methods are less efficient compared to the other

implicit and fitted methods.

Implicit methods: DIRKN(HS), DIRKN3(4), SIHM3(4) and phase-fitted

method: PFRKN4(4), need more time to do the computation, thus less

efficient compared to TF-LMMC(3). For all of the problems tested,

Page 44: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Problem 1(An almost Periodic Orbit problem studied by Stiefel and Bettis [19]) ๐‘ฆ1" ๐‘ฅ + ๐‘ฆ1 ๐‘ฅ = 0.001cos ๐‘ฅ , ๐‘ฆ1 0 = 1,๐‘ฆ1โ€ฒ 0 = 0, ๐‘ฆ2" ๐‘ฅ + ๐‘ฆ2 ๐‘ฅ = 0.001sin ๐‘ฅ , ๐‘ฆ2 0 = 0,๐‘ฆ2โ€ฒ 0 = 0.9995, Exact solution is ๐‘ฆ1 = cos ๐‘ฅ + 0.0005๐‘ฅsin ๐‘ฅ , and ๐‘ฆ2 = sin ๐‘ฅ โˆ’0.0005๐‘ฅcos ๐‘ฅ . The fitted frequency is ๐‘ฃ = 1.

Figure 6: The efficiency curve for TF-LMMC(4) for Problem 1

with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.9

2๐‘– for ๐‘– = 1,โ€ฆ , 5.

The following are efficiency curves TF-LMMC(4) Method.

Page 45: and Norazak Senu - Universiti Putra Malaysiaeinspem.upm.edu.my/sawona2018/Presentation For SAWONA...A good theoretical foundation of exponentially fitted technique was given by Lyche

Problem 2 (Inhomogeneous system

by Lambert and Watson [20])

๐‘‘2๐‘ฆ1 ๐‘ฅ

๐‘‘๐‘ก2= โˆ’๐‘ฃ2๐‘ฆ1 ๐‘ฅ + ๐‘ฃ2๐‘“ ๐‘ฅ + ๐‘“" ๐‘ฅ ,

๐‘ฆ1 0 = ๐‘Ž + ๐‘“ 0 , ๐‘ฆ1โ€ฒ 0 = ๐‘“โ€ฒ 0 ,

๐‘‘2๐‘ฆ2 ๐‘ฅ

๐‘‘๐‘ก2= โˆ’๐‘ฃ2๐‘ฆ2 ๐‘ฅ + ๐‘ฃ2๐‘“ ๐‘ฅ + ๐‘“" ๐‘ฅ

๐‘ฆ2 0 = ๐‘“ 0 , ๐‘ฆ2โ€ฒ 0 = ๐‘ฃ๐‘Ž + ๐‘“โ€ฒ 0

Exact solution is ๐‘ฆ1 ๐‘ฅ = ๐‘Žcos ๐‘ฃ๐‘ฅ +

๐‘“ ๐‘ฅ ,๐‘ฆ2 ๐‘ฅ = ๐‘Žsin ๐‘ฃ๐‘ฅ + ๐‘“ ๐‘ฅ , ๐‘“ ๐‘ฅ is chosen

to be ๐‘’โˆ’0.05๐‘ฅ and parameters ๐‘ฃ and ๐‘Ž

are 20 and 0.1 respectively.

Figure8: The efficiency curve for TF-LMMC(4) for

Problem 2 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.125

2๐‘– for

๐‘– = 2,โ€ฆ , 6.

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Problem 3 (Inhomogeneous system studied by Franco [12])

๐‘ฆ" ๐‘ฅ =1012

โˆ’992

โˆ’992

1012

๐‘ฆ ๐‘ฅ = ๐›ฟ932cos(2๐‘ฅ) โˆ’99

2sin 2๐‘ฅ

932sin(2๐‘ฅ) โˆ’

992cos(2๐‘ฅ)

๐‘ฆ 0 =โˆ’1 + ๐›ฟ

1, ๐‘ฆโ€ฒ 0 =

โˆ’1010 + 2๐›ฟ

for ๐›ฟ = 10โˆ’3. Exact solution ๐‘ฆ ๐‘ก =โˆ’cos 10๐‘ฅ โˆ’ sin 10๐‘ฅ + ๐›ฟcos(2๐‘ฅ)

cos 10๐‘ฅ + sin 10๐‘ฅ + ๐›ฟsin(2๐‘ฅ).

The Eigen-value of the problem are ๐‘ฃ = 10 and ๐‘ฃ = 1. The fitted frequency is chosen to be ๐‘ฃ = 10 because it is dominant than ๐‘ฃ = 1.

Figure 9: The efficiency curve for TF-LMMC(4) for

Problem 3 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.125

2๐‘– for

๐‘– = 1,โ€ฆ , 5.

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Problem 4 (Homogenous given

in Attili et.al [2])

๐‘ฆ" ๐‘ฅ = โˆ’64๐‘ฆ ๐‘ฅ ,๐‘ฆ 0 = 14 ,

๐‘ฆโ€ฒ 0 = โˆ’ 12 .

Exact solution is ๐‘ฆ = 1716 sin 8๐‘ฅ + ๐œƒ ,

๐œƒ = ๐œ‹ โˆ’ tanโˆ’1 4 .

The fitted frequency is ๐‘ฃ = 8.

Figure 9: The efficiency curve for TF-LMMC(4) for

Problem 4 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.1

2๐‘– for

๐‘– = 3,โ€ฆ , 7.

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Problem 5 (Inhomogeneous equation

studied by Papadopoulos et.al [15])

๐‘ฆโ€ฒโ€ฒ(๐‘ฅ) = โˆ’๐œˆ2๐‘ฆ(๐‘ฅ) +(๐œˆ2 โˆ’

1)sin(๐‘ฅ),

๐‘ฆ(0) = 1, ๐‘ฆโ€™(0) = ๐œˆ + 1.

Exact solution is y(๐‘ฅ) = cos(๐œˆ๐‘ฅ) +

sin(๐œˆ๐‘ฅ) + sin(๐‘ฅ).

The fitted frequency is ๐œˆ = 10.

Figure 10: The efficiency curve for TF-LMMC(4) for

Problem 5 with ๐‘‡๐‘’๐‘›๐‘‘ = 104 and โ„Ž =0.125

2๐‘– for ๐‘– = 3, โ€ฆ , 7.

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From the efficiency curves in Figures 6-10, we observed that methods with fitting

properties have smaller error.

DIRKN methods have more functions evaluation than LMMs and SIHMs, hence more

computational time is required to implement DIRKN methods.

It is shown that TF-LMMC(4) have better performance compared to the original

method LMMC(4), and superior compared to all the existing methods in comparisons.

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Conclusion

โ€ข In this research, we developed linear multistep methods with extra

derivatives using collocation technique of order three (LMMC(3)) and four

(LMMC(4))

โ€ข Trigonometrically-Fitted the Linear Multistep Method With Collocation

technique of Order Three (TF-LMMC(3)) and four (TF-LMMC(4)) respectively.

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โ€ข Numerical results for LMMC(3) which has order three is as comparable as other

existing methods which are of order four

โ€ข Numerical results for LMMC(4) which is order four is slightly better than other

existing methods in comparisons.

โ€ข Hence having extra derivtives in the multistep method do improved the accuracy

of the methods.

โ€ข TF-LMMC(3) and TF-LMMC(4) are clearly superior in solving special second order

ODEs with oscillatory solutions since it involves lesser computational time and

better accuracy

โ€ข We can conclude that Trigonometrically-fitting the methods improved the effiency

of the methods for integrating oscillatary problems.

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References [1] J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of Rungeโ€“ Kuttaโ€“Nystrom formulae, IMA Journal of Numerical Analysis, 7 (1987) 235โ€“250. [2] B.P. Sommeijer, A note on a diagonally implicit Rungeโ€“Kuttaโ€“Nystrom method, Journal of Computational and Applied Mathematics 19 (1987) 395โ€“399 [3]J.M. Franco, An explicit hybrid method of Numerov type for second-order periodic initial-value problems, J. Comput. Appl. Math. 59 (1995) 79โ€“90. [4] J. P. Coleman, Order conditions for class of two-step methods for ๐‘ฆโ€ = ๐‘“(๐‘ฅ, ๐‘ฆ). IMA Journal of Numerical Analysis, vol. 23,pp. 197-220, 2003.

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[5] Y. J. L. Guo, A third order equation arising in the falling film, Taiwanese Journal

of Mathematics, vo1. 11, no. 3, pp. 637-643, 2007.

[6]R.B. Adeniyi, and E.O. Adeyefa, Chebyshev collocation approach for a continuous

formulation of implicit hybrid methods for ivps in second order odes, IOSR Journal

of Mathematics, vol. 6, no. 9, pp. 9-12, 2013.

[7]D. O. Awoyemi, A new sixth-order algorithm for general second order ordinary

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[8] S. N. Jator, Numerical integrators for fourth order initial and boundary

problems, International Journal of pure and applied mathematics, vol. 47, no. 4,

pp. 563-576, 2008.

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[9] T. Lyche, Chebyshevian multistep methods for ordinary differential equations. Numersche Mathematick 19(1): 65-75, 1972. [10] G. Vanden Berghe, H. De Meyer, M. Van Daele, T. Van Hecke, Exponentially fitted explicit Runge-Kutta methods, Comput. Phys. Commun. 123 (1999) 7-15. [10] G. Vanden Berghe, H. De Meyer, M. Van Daele, T. Van Hecke, Exponentially fitted explicit Runge-Kutta methods, J. Comput. Appl. Math. 125 (2000) 107-115. [11] T. E. Simos, Exponentially fitted explicit Runge-Kutta Nystrom method for the solution of initial-value problems with oscillating solutions. Appl. Math. Lett. 15 (2002) 217-225.

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[12] J. M. Franco, Exponentially fitted explicit Runge-Kutta Nystrom methods, Appl. Numer. Math. 50 (2004) 427-443. [13] Y. Fang and X. Wu, A Trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions, Applied Numerical Mathematics ,vol. 58, pp. 341-451, 2008. [14] E. Hairer, S.P. Nรธrsett, G. Wanner, Solving Ordinary Differential Equations 1, Berlin : Springer-Verlag, 2010. [15] D.F. Papadopoulos, Z.A. Anastassi and T.E. Simos, A phase-fitted Runge-Kutta Nystrรถm method for the numerical solution of initial value problems with oscillating solutions, Journal of Computer Physics Communications, vol. 180, pp. 1839โ€“1846, 2009. [16] B.P. Sommeijer, A note on a diagonally implicit Rungeโ€“Kuttaโ€“Nystrom method, Journal of Computational and Applied Mathematics 19 (1987) 395โ€“399

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[17] N. Senu, M. Suleiman, F. Ismail, and M. Othman, A Singly Diagonally Implicit Runge-Kutta Nystrรถm Method for Solving Oscillatory Problems, Proceeding of the International Multi Conference of Engineers and Computer Scienticts. ISBN: 978-988-19251-2-1. 2011. [18] Z. Ahmad, F. Ismail, N. Senu, and M. Suleiman, Zero dissipative phase-fitted hybrid methods for solving oscillatory second order ordinary differential equations, Applied Mathematics and Computation, vol. 219, no. 19, pp. 10096โ€“10104, 2013. [19] E. Stiefel and D.G. Bettis, Stabilization of Cowellโ€™s methods, Numer. Math, vol. 13, pp. 154-175, 1969. [20]J. D. Lambert and I. A. Watson, Symmetric multistep methods for periodic initial-value problems, J. Inst. Maths Applics , vol. 18, pp. 189-202, 1976. [16] B.P. Sommeijer, A note on a diagonally implicit Rungeโ€“Kuttaโ€“Nystrom method, Journal of Computational and Applied Mathematics 19 (1987) 395โ€“399

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THANK YOU

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THANK YOU