Caputo’s finite difference solution of fractional two-point boundary value problems using SOR iteration

Rahman, Rostang; Ali, Nur Afza Mat; Sulaiman, Jumat; Muhiddin, Fatihah Anas

doi:10.1063/1.5054233

Cited by 6 publications

(3 citation statements)

References 13 publications

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“…In addition, the successive over-relaxation iterative (SOR) method is also used in this study to solve a linear system generated from the five-point rotated similarity finite difference approximation equation. Presently, this method has been dramatically considered and exploited in a variety of applications, see [37][38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

Ali¹,

Sulaiman²,

Saudi³

et al. 2023

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In this paper, we transformed a two-dimensional unsteady convection-diffusion equation into a two-dimensional steady convection-diffusion equation using the similarity transformation technique. This technique can be easily applied to linear or nonlinear problems and is capable of reducing the size of computational works since the main idea of this technique is to reduce at least one independent variable. The corresponding similarity equation is then solved numerically using an effective numerical technique, namely a new five-point rotated similarity finite difference scheme via half-sweep successive over-relaxation iteration. This work compared the performance of the proposed method with Gauss-Seidel and successive over-relaxation with the full-sweep concept. Numerical tests were carried out to obtain the performance of the proposed method using C simulation. The results revealed that the combination of the five-point rotated similarity finite difference scheme via half-sweep successive over-relaxation iteration is the most superior method in terms of the iteration number and computational time compared to all these methods. Additionally, in terms of accuracy, all three iterative methods are also comparable.

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Section: Introductionmentioning

confidence: 99%

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

Ali¹,

Sulaiman²,

Saudi³

et al. 2023

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show abstract

“…Since the generated linear system has a large and sparse matrix, the use of iterative methods is the best linear solver [26][27][28]. For instance, the implementation of the point iteration family namely SOR [29], AOR [30] and KSOR [31] can be used to solve this linear system. In addition to this point iteration family, Evans [32] introduced the Explicit Group (EG) iterative method which is faster than the Gauss-Seidel (GS) iterative method to get the numerical solution of this linear system.…”

Section: Introductionmentioning

confidence: 99%

Redlich-Kister Finite Difference Solution for Solving Two-Point Boundary Value Problems by using Ksor Iteration Family

Suardi¹,

Sulaiman²

2021

Adv. sci. technol. eng. syst. j.

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In this paper, we are concerned to investigate the efficiency of the second-order Redlich-Kister Finite Difference (RKFD) discretization scheme together with the Four Point Explicit Group Kaudd Successive Over Relaxation (4EGKSOR) iterative method for solving twopoint boundary value problems (TPBVPs). In order to apply this block iteration to solve any linear system, firstly we discretize all derivative terms via the second-order RKFD discretization scheme over the proposed problem in order to get the second-order RKFD approximation equation. Due to the main characteristics of the coefficient matrix for the generated linear system which are large-scale and sparse, the best choice for solving this linear system is using one of the iterative methods. Therefore, the formulation of the Kaudd Successive Over Relaxation method together with the Explicit Group iteration method mainly on the Four-Point Explicit Group Kaudd Successive Over Relaxation (4EGKSOR) iterative method has been presented to solve this linear system iteratively. In order to show the efficiency of the 4EGKSOR, another two iterative methods have also been considered which are the Gauss-Seidel (GS) and the Kaudd Successive Over Relaxation (KSOR) to solve three examples of the proposed problems in which all numerical results obtained were recorded based on the number of iterations, execution time and maximum norm. Based on the performance analysis, clearly, the 4EGKSOR iterative method shows substantiated improvement in terms of the number of iterations and execution time.

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“…For solving the linear systems which have large and sparse, there are several concept iterative methods from previous researchers have been discussed [10,11]. Other than that, Rahman et al [12] studied the numerical solution of two-point FBVPs based on the iterative method where SOR method has been used and AGE method in studied Rahman et al [13]. From the studied, this paper focus to expand the iterative method to the half-sweep scheme.…”

Section: Introductionmentioning

confidence: 99%

Application of The Half-Sweep Egsor Iteration for Two-Point Boundary Value Problems of Fractional Order

Rahman¹,

Ali²,

Sulaiman³

et al. 2019

Adv. sci. technol. eng. syst. j.

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The point of this study is to explore and elucidating the performance of the four-point Half-Sweep EGSOR (4HSEGSOR) iterative method to solve fractional two-point boundary value problems by using Caputo's fractional operator and family of finite differences (FD) schemes. To apply the iterative methods, linear system needs to be constructed via the discretization process with fractional order to get the approximation equation of the linear fractional two point boundary value problem by using the Caputo's derivative operator. Then the generated linear system has been solved using the proposed 4HSEGSOR iterative method. In the addition, the formulation and application of the 4HSEGSOR method to solve the problems are also presented. Three numerical examples and comparison are used to illustrate with tested FSSOR and HSSOR methods. The numerical results reveals the effectiveness of 4HSEGSOR method compared with tested iterative methods.

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Caputo’s finite difference solution of fractional two-point boundary value problems using SOR iteration

Cited by 6 publications

References 13 publications

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

Redlich-Kister Finite Difference Solution for Solving Two-Point Boundary Value Problems by using Ksor Iteration Family

Application of The Half-Sweep Egsor Iteration for Two-Point Boundary Value Problems of Fractional Order

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