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

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Department Tadris Matematika, IAIN Bengkulu, Kota Bengkulu 38211, Indonesia
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Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India
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Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, United Arab Emirates
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Faculty of Computing and Informatics, Universiti Malaysia Sabah Labuan International Campus, Labuan F.T. 87000, Malaysia
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Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Kota Kinabalu 88400, Malaysia
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Authors to whom correspondence should be addressed.
Symmetry 2021, 13(6), 1005; https://doi.org/10.3390/sym13061005
Received: 30 April 2021 / Revised: 26 May 2021 / Accepted: 29 May 2021 / Published: 3 June 2021
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. View Full-Text
MDPI and ACS Style

Sunarto, A.; Agarwal, P.; Chew, J.V.L.; Sulaiman, J. Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation. Symmetry 2021, 13, 1005. https://doi.org/10.3390/sym13061005

AMA Style

Sunarto A, Agarwal P, Chew JVL, Sulaiman J. Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation. Symmetry. 2021; 13(6):1005. https://doi.org/10.3390/sym13061005

Chicago/Turabian Style

Sunarto, Andang, Praveen Agarwal, Jackel Vui Lung Chew, and Jumat Sulaiman. 2021. "Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation" Symmetry 13, no. 6: 1005. https://doi.org/10.3390/sym13061005

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