Application of the Sumudu Variational Iteration Method with Atangana-Baleanu-Caputo Operator for Solving Fractional-Order Heat-Like Equations with Initial Conditions
Article Sidebar
-
Fractional-order partial diffferential Heat-like equation, Atangana-Baleanu-Caputo operator, Sumudu transform, Variational iteration method
Abstract
Fractional calculus techniques are widely utilized across various engineering disciplines and applied sciences. Among these techniques is the Sumudu Variational Iteration Method (SVIM), which has not yet been tested with the Atangana-Baleanu-Caputo fractional derivative in academic literature. This work aims to explore the application of SVIM for solving fractional-order partial differential equations using the Atangana-Baleanu-Caputo derivative. The method integrates the Sumudu transform with the variational iteration method. To demonstrate the effectiveness and validity of SVIM, we apply it to solve one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) fractional-order heat-like partial differential equations. The results indicate that SVIM is both convergent and efficient for solving these types of fractional partial differential equations.
Full text article
References
Mainardi, F., (1997), Fractional calculus: Some basic problems in continuum and statistical mechanics in: A. Carpinteri, F. Mainardi (Eds.)., Fractal and Fractional Calculus in Continuum Mechanics, Springer-Verlag, New York., pp. 291–348. DOI: https://doi.org/10.1007/978-3-7091-2664-6_7
Gorenflo, R., Mainardi, F., (1997), Fractional calculus: Int and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.)., Fractals and Fractional Calculus, New York. DOI: https://doi.org/10.1007/978-3-7091-2664-6_5
Kilbas, A. A, Srivastava, H. M, Trujillo, J. J., (2006), Theory and applications of fractional differential equations., North-Holland Math, Studies: Elsevier.
Podlubny, I., (1999), Fractional Differential Equations., Academic Press, New York.
Caputo, M., (1969), Elasticita e Dissipazione., Zani-Chelli, Bologna, Italy.
Al-Refai, M., Jarrah, A. M., (2019), Fundamental results on weighted Caputo-Fabrizio fractional derivative., Chaos Solitons Fractals., 126, 7–11 . DOI: 10.1016/j.chaos.2019.05.035. DOI: https://doi.org/10.1016/j.chaos.2019.05.035
Atangana, A., Baleanu D., (2016), New fractional derivatives with non-local and non-singular kernel: theory and applications to heat transfer model., Therm Sci., 20, 763–9. DOI: https://doi.org/10.2298/TSCI160111018A
Caputo, M.,Fabrizio, M., (2015), A new definition of fractional derivative without singular kernel., Prog Fract Differ Appl., 1(2), 73–85. DOI: 10.12785/pfda/010201.
Sarwar, S., Alkhalaf,S., Iqbal, S., Zahid. M. A., (2015), A note on optimal homotopy asymptotic method for the solutions of fractional order heat- and wave-like partial differential equations., Computers and Mathematics with Applications., 70, 942–953. DOI: https://doi.org/10.1016/j.camwa.2015.06.017
Bhargave, A., Jain, D., Suthar, D. L., (2003), Applications of the Laplace variational iteration method to fractional heat like equations. ,Partial Diif Eq in App Math., 8, 1-8. DOI: https://doi.org/10.1016/j.padiff.2023.100540
Molliq, T., Noorani, M. S. M., Hashim, I., (2009), Variational iteration method for fractional heat- and wave-like equations., Nonlinear Anal, RWA., 10, 1854–1869. DOI: https://doi.org/10.1016/j.nonrwa.2008.02.026
Xu, H., Cang, J., (2008), Analysis of a time fractional wave-like equation with the homotopy analysis method., Phys, Lett., A 372, 1250–1255. DOI: 10.1016/j.physleta.2007.09.039. DOI: https://doi.org/10.1016/j.physleta.2007.09.039
Momani, S., (2005), Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method., Appl. Math. Comput., 165(2), 459–472. DOI: 10.1016/j.amc.2004.06.025. DOI: https://doi.org/10.1016/j.amc.2004.06.025
Shou, D. H., He, J. H., (2007), Beyond Adomian methods: The variational iteration method for solving heat-like and wave-like equations with variable coefficients., Phys. Lett, A., 372 (3), 223–237. DOI: 10.1016/j.physleta.2007.07.011. DOI: https://doi.org/10.1016/j.physleta.2007.07.011
Khan, H., hah, R., Kumam, P., Arif, M., (2019), Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method., Entropy., 21(6), 1-21. DOI: 10.3390/e21060597. DOI: https://doi.org/10.3390/e21060597
Mtawal, A. A. H., Maity, E. A., (2021), Exact solution for local fractional Diffusion and Wave Equations on Cantor Sets., Global Libyan Journal., 21, 1-16. DOI: https://doi.org/10.37376/glj.vi52.1730
He, J. H., (1999), Variational iteration method-A kind of non-linear analytical technique some examples., Int J Non Linear Mech., 34(4), 699-708. DOI:10.1016/S0020-7462(98)00048-1 DOI: https://doi.org/10.1016/S0020-7462(98)00048-1
He, J. H., (2000), Variational iteration method for autonomous ordinary differential systems., Appl. Math. Comput., 114, 115–123. DOI: 10.1016/S0096-3003(99)00104-6. DOI: https://doi.org/10.1016/S0096-3003(99)00104-6
Mahdy, A. M. .S., Mohamed, A. S., and Mtawal, A. A. H., (2015), Implementation of the Homotopy perturbation Sumudu Transform Method for Solving Klein-Gordon Equation., Applied Mathematics., 6 (3), 617-628. DOI: 10.4236/am.2015.63056 . DOI: https://doi.org/10.4236/am.2015.63056
Mechee, M. S and Naeemah, A. J., (2020), Astudy of double Sumudu transform for solving differential equations with some applications., International Journal of Engineering and Information Systems., 4(1), 20-27.
Mahdy, A. M. S., Mohamed, A. S., Mtawal, A. A. H., (2015), Variational homotopy perturbation method for solving the generalized time-space fractional Schrödinger equation., International Journal of Physical Sciences., 10(11), 342-350. DOI: https://doi.org/10.5897/IJPS2015.4287
Odibat, Z., Momani, S., (2008), Modified homotopy perturbation method application to quadratic riccati differential equation of fractional order., Chaos Solitons Fractals., 36(1), 167–174. DOI: https://doi.org/10.1016/j.chaos.2006.06.041
Mahdy, A.M.S., Mohamed, A.S., Mtawal, A.A.H., (2015), Sumudu decomposition method for solving fractional-order Logistic differential equation., Journal of Advances and Mathematics., 10(7), 3632-3639.
Shawagfeh,N. T., (2002), Analytical approximate solutions for linear differential equations., Appl., Math. Comput., 131 (2–3), 517–529. DOI: https://doi.org/10.1016/S0096-3003(01)00167-9
Yadav, S., Pandey, R. K., Shukla. A. K., (2019), Numerical approximations of Atangana-Baleanu Caputo derivative and its application., Chaos Solitons Fractals.,118, 58-64. DOI: https://doi.org/10.1016/j.chaos.2018.11.009
Watugala, G.K., (1993), Sumudu transform: A new integral transform to solve differential equations and control engineering problems., Int J of Math Ed in Sci and Tec., 24(1), 35-43. DOI: 10.1080/0020739930240105 DOI: https://doi.org/10.1080/0020739930240105
Belgacem, F. B. M., Karaballi, A. A., (2006), Sumudu transform fundamental properties investigations and applications., Inter. J. Appl. Math. Stoch. Anal. PP., 1-23. DOI: 10.1155/JAMSA/2006/91083. DOI: https://doi.org/10.1155/JAMSA/2006/91083
Authors
Copyright (c) 2024 Journal of Pure & Applied Sciences
This work is licensed under a Creative Commons Attribution 4.0 International License.
In a brief statement, the rights relate to the publication and distribution of research published in the journal of the University of Sebha where authors who have published their articles in the journal of the university of Sebha should how they can use or distribute their articles. They reserve all their rights to the published works, such as (but not limited to) the following rights:
- Copyright and other property rights related to the article, such as patent rights.
- Research published in the journal of the University of Sebha and used in its future works, including lectures and books, the right to reproduce articles for their own purposes, and the right to self-archive their articles.
- The right to enter a separate article, or for a non-exclusive distribution of their article with an acknowledgment of its initial publication in the journal of Sebha University.
Privacy Statement The names and e-mail addresses entered on the Sabha University Journal site will be used for the aforementioned purposes only and for which they were used.
