Investigation of Fractional Diffusion Equation via QSGS iterations

Sunarto, Andang; Sulaiman, Jumat

doi:10.1088/1742-6596/1179/1/012014

Cited by 3 publications

(5 citation statements)

References 7 publications

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“…Based on our short review of the existing solution methods, many researchers have suggested the finite-difference method. The method of finite difference can be implicit or explicit, depending on its stability to obtain the solution [6][7][8][9][10]. Since SFDE involves changing the time and space of a continuous variable, numerical treatment such as finite difference discretization is necessary to transform the differential equation into a finite system of linear equations that a computer can solve.…”

Section: Introductionmentioning

confidence: 99%

“…Since the sparse and large-scale linear system's coefficient matrix is difficult to be solved analytically, an iterative method is employed as an alternative. In terms of the efficiency of the iterative processes, many authors [8,15,16] have suggested and debated over several numerical iterative methods. Besides that, the block iteration has been presented [17] to show the computation cost efficiency improvement by separating systems of equations.…”

Section: Introductionmentioning

confidence: 99%

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Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation

et al. 2021

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In this study, the numerical solution of a space-fractional parabolic partial differential equation was considered. The investigation of the solution was made by focusing on the space-fractional diffusion equation (SFDE) problem. Note that the symmetry of an efficient approximation to the SFDE based on a numerical method is related to the compatibility of a discretization scheme and a linear system solver. The application of the one-dimensional, linear, unconditionally stable, and implicit finite difference approximation to SFDE was studied. The general differential equation of the SFDE was discretized using the space-fractional derivative of Caputo with a half-sweep finite difference scheme. The implicit approximation to the SFDE was formulated, and the formation of a linear system with a coefficient matrix, which was large and sparse, is shown. The construction of a general preconditioned system of equation is also presented. This study’s contribution is the introduction of a half-sweep preconditioned successive over relaxation (HSPSOR) method for the solution of the SFDE-based system of equation. This work extended the use of the HSPSOR as an efficient numerical method for the time-fractional diffusion equation, which has been presented in the 5th North American International Conference on industrial engineering and operations management in Detroit, Michigan, USA, 10–14 August 2020. The current work proposed several SFDE examples to validate the performance of the HSPSOR iterative method in solving the fractional diffusion equation. The outcome of the numerical investigation illustrated the competence of the HSPSOR to solve the SFDE and proved that the HSPSOR is superior to the standard approximation, which is the full-sweep preconditioned SOR (FSPSOR), in terms of computational complexity.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation

et al. 2021

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“…The partial differential equation (1) arises, for example, in flow of incompressible fluids, gas flow in porous media, nonlinear process of heat transfer and image processing. Equation 1itself represents a nonlinear diffusion process.…”

Section: Introductionmentioning

confidence: 99%

“…Since the introduction of Quarter-Sweep approach by Othman and Abdullah [10], it has been studied by few researchers which can be referred in Eng et al [5], Sunarto and Sulaiman [1], Suardi et al [11], Aruchunan et al [4], and Nusi and Othman [12]. In contrast to these works, in the present article, we attempt to apply the Quarter-Sweep into the formulation of finite difference approximation for 1D PME.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we present the formulation of the implicit finite difference approximation equation using Quarter-Sweep for equation (1). Then, we show the derivation of the Newton-Gauss-Seidel method for equation (1) from the formulated approximation equation. Section 3 illustrates the numerical results from the implementation of QSNGS on some examples of 1D PME problems.…”

Section: Introductionmentioning

confidence: 99%

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On Quarter-Sweep Finite Difference Scheme for One-Dimensional Porous Medium Equations

Lung¹,

Sulaiman²

2020

Int. J. of Appl. Math.

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In this article, we introduce an implicit finite difference approximation for one-dimensional porous medium equations using Quarter-Sweep approach. We approximate the solutions of the nonlinear porous medium equations by the application of the Newton method and use the Gauss-Seidel iteration. This yields a numerical method that reduces the computational complexity when the spatial grid spaces are reduced. The numerical result shows that the proposed method has a smaller number of iterations, a shorter computation time and a good accuracy compared to Newton-Gauss-Seidel and Half-Sweep Newton-Gauss-Seidel methods.

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Pathfinding algorithm based on rotated block AOR technique in structured environment

Dahalan

Saudi

2022

MATH

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<abstract> <p>Harmonic potential fields are commonly used as guidance in a global approach for self-directed robot pathfinding. These harmonic potentials are generated using Laplace's equation solutions. The computation of these harmonic potentials often requires the use of immense amounts of computing resources. This study introduces a numerical technique called Rotated Block Accelerated Over-Relaxation (AOR), also known as Explicit Decoupled Group AOR (EDGAOR), to deal with pathfinding problem. Several robot navigation simulations were performed in a static, structured, known indoor environment to validate the efficiency of the suggested approach. The paths generated by the simulations are shown using several different starting and target positions. The performance of the proposed approach in computing harmonic potentials for solving pathfinding problems is also discussed.</p> </abstract>

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Investigation of Fractional Diffusion Equation via QSGS iterations

Cited by 3 publications

References 7 publications

Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation

Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation

On Quarter-Sweep Finite Difference Scheme for One-Dimensional Porous Medium Equations

Pathfinding algorithm based on rotated block AOR technique in structured environment

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