Exact Solution of Time-Fractional Cauchy Reaction-Diffusion Equation Using Jafari Transform
DOI:
https://doi.org/10.37376/ajhas.vi4.7492Keywords:
Fractional Cauchy reaction-diffusion equation, Caputo fractional operator, Jafari transform, New iterative methodAbstract
Fractional calculus techniques are extensively employed in science and engineering including the new Jafari transform iterative method (NJTIM), which has not been studied by researchers using the Caputo fractional derivative. The new methodology demonstrates how two strong methods, the new iterative method and the Jafari transform method, may be combined and applied to provide exact solutions to fractional partial differential equations. Three distinct examples are also provided to demonstrate the accuracy and efficacy of my methodology
Downloads
References
Kilbas, AA., Samko, SG., Marichev, OI., (1993)., Fractional integrals and derivatives theory and applications., Gordon and Breach., New York.
Podlubny, I., (1999)., Fractional Differential Equations., Academic Press., New York.
Oldham, KB., Spanier, J., (1974)., The fractional calculus., Academic Press., New York.
Kilbas AA , Srivastava HM , Trujillo JJ . Theory and applications of fractional differential equations. North-Holland Math. Studies: Elsevier; 2006 .
He, JH., (1999)., Homotopy perturbation technique., Comput Method Appl Mech Eng., 178(3):257–262. https://doi.org/10.1016/S0045-7825(99)00018-3.
Mtawal,AAH., Alkaleeli, SA., (2020)., A new modified homotopy perturbation method for fractional partial differential equations with proportional delay., Journal of Advances In Mathematics., 19: 58-73. https://doi.org/10.24297/jam.v19i.8876
Liao, SJ., (2004)., On the homotopy analysis method for nonlinear problems., Appl Math Comput., 147:499–513. https://doi.org/10.1016/S0096-3003(02)00790-7
Wazwaz, AM., (1999)., A reliable modification of Adomian decomposition method., Appl Math Comput., 102:77–86. https://doi.org/10.1016/S0096-3003(98)10024-3
Safari, M., Ganji, DD., Moslemi, M., (2009)., Application of He’s variational iteration method and Adomian’s decomposition method to the fractional KdV–Burgers–Kuramoto equation., Comput Appl., 58:2091–2097. https://doi.org/10.1016/j.camwa.2009.03.043
Mtawal, AAH., Muhammed, S., Almabrok, A., (2020)., Application of the alternative variational iteration method to solve delay differential equations., International Journal of Physical Sciences., 15(3): 112-119. https://doi. org/10.5897/IJPS2020.4879
Mahdy, AMS.; Mohamed, AS., Mtawal, AAH.,( 2015)., Variational homotopy perturbation method for solving the generalized time-space fractional Schrödinger equation., International Journal of Physical Sciences., 10(11): 342-350. https://doi org/10.5897/IJPS2015.4287
Mahdy, AMS., Mohamed, AS., Mtawal, AAH.,( 2015)., Implementation of the Homotopy perturbation Sumudu Transform Method for Solving Klein-Gordon Equation., Applied Mathematics., Vol.06 No.03: 617-628. https://doi.org/10.4236/am.2015.61014
Mtawal, AAH.,(2024)., Application of the Sumudu Variational Iteration Method with Atangana-Baleanu-Caputo Operator for Solving Fractional-Order Heat-Like Equations with Initial Conditions., Journal of Pure & Applied Sciences., 23(2): 50-60. https://doi.org/10.51984/jopas.v23i2.3151
Mittal, RC., Jiwari, R., (2011)., Numerical study of two-dimensional reaction–diffusion Brusselator system., Appl Math Comput., 217(12):5404–5415. https://doi.org/10.1016/j.amc.2010.12.010
Jiwari, R., Yuan, JY., Jiwari, R., (2014)., A computational modeling of two dimensional reaction–diffusion Brusselator system arising in chemical processes., J Math Chem., 52(6):1535–1551. https://doi.org/10.1186/s40064-016-2426-8
Verma, A., Jiwari, R., Koksal, ME., (2014)., Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions., Adv Differ Eq., 2014:229. https://doi.org/10.1186/1687-1847-2014-229
Britton, NF., (1998)., Reaction–diffusion equations and their applications to biology., Academic Press/Harcourt Brace Jovanovich Publishers., New York.
Kumar, S., (2013)., A new fractional modeling arising in engineering sciences and its analytical approximate solution., Alex Eng J., 52(4):813–819. https://doi.org/10.1016/j.aej.2013.09.005
Wang, K., Liu, S., (2016)., A new Sumudu transform iterative method for time fractional Cauchy reaction–diffusion equation., SpringerPlus., 5(865): 1-20. DOI 10.1186/s40064-016-2426-8.
Gejji, VD., Jafari, H., (2006)., An iterative method for solving non linear functional equations., J Math Anal Appl., 316:753–763. https://doi.org/10.1016/j.jmaa.2005.05.009
Jafari, H., (2021)., A new general integral transform for solving integral equations., J. Adv. Res., 32, 133-138. https://doi.org/10.1016/j.jare.2020.08.016
M. Caputo, Elasticita e Dissipazione. ( Zani-Chelli, Bologna, Italy, 1969).
Momani, S., Yildirim, A., (2010)., Analytical approximate solutions of the fractional convection diffusion equation with nonlinear source term by He homotopy perturbation method., Int J Comput Math., 87(5):1057–1065. https://doi.10.1080/00207160903023581
Yang, XJ., (2011)., Local Fractional Functional Analysis and its Applications., Asian Academic Publisher., Hong Kong.
Yang, XJ., (2012)., Advanced Local Fractional Calculus and Its Applications., World Science Publisher., New York, NY, USA.
Wang, F., Fang, Q., Hu, Y., (2025)., Homotopy Analysis Transform Method for Solving Systems of Fractional-Order Partial Differential Equations., Fractal and Fractional., 9 (253):1-20. https://doi.org/10.3390/fractalfract9040253
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Afaq Journal for Human and Applied Studies

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.



