{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,19]],"date-time":"2025-03-19T16:32:31Z","timestamp":1742401951427,"version":"3.37.3"},"reference-count":24,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,3]],"date-time":"2021-06-03T00:00:00Z","timestamp":1622678400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"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\u2019s 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\u201314 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.<\/jats:p>","DOI":"10.3390\/sym13061005","type":"journal-article","created":{"date-parts":[[2021,6,4]],"date-time":"2021-06-04T01:03:32Z","timestamp":1622768612000},"page":"1005","source":"Crossref","is-referenced-by-count":6,"title":["Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4556-602X","authenticated-orcid":false,"given":"Andang","family":"Sunarto","sequence":"first","affiliation":[{"name":"Department Tadris Matematika, IAIN Bengkulu, Kota Bengkulu 38211, Indonesia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7556-8942","authenticated-orcid":false,"given":"Praveen","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India"},{"name":"Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, United Arab Emirates"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1195-0955","authenticated-orcid":false,"given":"Jackel Vui Lung","family":"Chew","sequence":"additional","affiliation":[{"name":"Faculty of Computing and Informatics, Universiti Malaysia Sabah Labuan International Campus, Labuan F.T. 87000, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9538-6588","authenticated-orcid":false,"given":"Jumat","family":"Sulaiman","sequence":"additional","affiliation":[{"name":"Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Kota Kinabalu 88400, Malaysia"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"013119","DOI":"10.1063\/1.5074099","article-title":"New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives","volume":"29","author":"Veeresha","year":"2019","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.chaos.2019.06.027","article-title":"A fractional mathematical model of breast cancer competition model","volume":"127","author":"Atangana","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2813","DOI":"10.3934\/math.2020181","article-title":"A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model","volume":"5","author":"Fatmawati","year":"2020","journal-title":"Aims Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"359","DOI":"10.18280\/mmep.070305","article-title":"Caputo-Fabrizio Fractional Derivative to Solve the Fractional Model of Energy Supply-Demand System","volume":"7","author":"Noeiaghdam","year":"2020","journal-title":"Math. Model. Eng. Probl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"110007","DOI":"10.1016\/j.chaos.2020.110007","article-title":"Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic","volume":"138","author":"Higazy","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1743","DOI":"10.1016\/j.jcp.2011.11.008","article-title":"Crank\u2013Nicolson method for the fractional diffusion equation with the Riesz fractional derivative","volume":"231","author":"Duman","year":"2012","journal-title":"J. Comput. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/j.jcp.2018.03.032","article-title":"A fourth-order scheme for space fractional diffusion equations","volume":"373","author":"Guo","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"012014","DOI":"10.1088\/1742-6596\/1179\/1\/012014","article-title":"Investigation of fractional diffusion equation via QSGS iterations","volume":"1179","author":"Sunarto","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1090\/S0025-5718-2015-02917-2","article-title":"A class of second order difference approximations for solving space fractional diffusion equations","volume":"84","author":"Tian","year":"2015","journal-title":"Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s10915-012-9661-0","article-title":"Quasi-compact finite difference schemes for space fractional diffusion equations","volume":"56","author":"Zhou","year":"2013","journal-title":"J. Sci. Comput."},{"key":"ref_11","first-page":"331","article-title":"Fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation","volume":"350","author":"Li","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.matcom.2019.11.007","article-title":"Kronecker product based preconditioners for boundary value method discretizations of space fractional diffusion equations","volume":"170","author":"Chen","year":"2020","journal-title":"Math. Comput. Simul."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.jcp.2019.03.030","article-title":"A finite volume method for two-dimensional Riemann-Liouville space-fractional diffusion equation and its efficient implementation","volume":"388","author":"Fu","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_14","first-page":"465","article-title":"Numerical solution of space fractional diffusion equation by the method of lines and splines","volume":"336","author":"Salehi","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_15","unstructured":"Hacksbusch, W. (2016). Iterative Solution of Large Sparse Systems of Equations, Springer International Publishing. [2nd ed.]."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Saad, Y. (2003). Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics. [2nd ed.].","DOI":"10.1137\/1.9780898718003"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1080\/00207169508804447","article-title":"Solving the two dimensional diffusion equation by the four point explicit decoupled group (EDG) iterative method","volume":"58","author":"Ibrahim","year":"1995","journal-title":"Int. J. Comput. Math."},{"key":"ref_18","unstructured":"Caputo, M. (2003). Diffusion with space memory modelled with distributed order space fractional differential equations. Ann. Geophys., 46."},{"key":"ref_19","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience. [1st ed.]."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"139","DOI":"10.17576\/jsm-2015-4401-19","article-title":"Quarter-sweep iteration concept on conjugate gradient normal residual method via second order quadrature-finite difference schemes for solving Fredholm integro-differential equations","volume":"44","author":"Aruchunan","year":"2015","journal-title":"Sains Malays."},{"key":"ref_21","first-page":"33","article-title":"On quarter-sweep finite difference scheme for one-dimensional porous medium equations","volume":"2020","author":"Lung","year":"2020","journal-title":"Int. J. Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sunarto, A., Sulaiman, J., and Saudi, A. (2014, January 20\u201321). Solving the time fractional diffusion equations by the half-sweep SOR iterative method. Proceedings of the 2014 International Conference of Advanced Informatics: Concept, Theory and Application (ICAICTA), Bandung, Indonesia.","DOI":"10.1109\/ICAICTA.2014.7005953"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0024-3795(91)90376-8","article-title":"Modified iterative methods for consistent linear systems","volume":"154\u2013156","author":"Gunawardena","year":"1991","journal-title":"Linear Algebra Appl."},{"key":"ref_24","unstructured":"Sunarto, A., and Sulaiman, J. (2020, January 10\u201314). Application half-sweep preconditioned SOR method for solving time-fractional diffusion equations. Proceedings of the International Conference on Industrial Engineering and Operations Management, Detroit, MI, USA."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1005\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,13]],"date-time":"2024-07-13T19:27:41Z","timestamp":1720898861000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1005"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,3]]},"references-count":24,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["sym13061005"],"URL":"https:\/\/doi.org\/10.3390\/sym13061005","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,6,3]]}}}