We solve parabolic PDEs of fractional order in the Caputo sense using a numerical method called Some Blaise Abbo (SBA). The SBA method uses an algorithm that converges faster to the exact solution, when it exists in the appropriate functional space. The present article complements recent work of parabolic PDEs in the Riemann-Liouville sense of order α with 0 < α ≤ 1.

Some Blaise Abbo (SBA) method, fractional derivative in the   Caputo’s sense, parabolic PDEs.

Received: October 6, 2023; Accepted: November 29, 2023; Published: December 29, 2023

How to cite this article: DJERAYOM Luc, Germain KABORE, Bakari Abbo, Ousséni SO and Blaise SOME, On solutions of some fractional order parabolic equations by the SBA plus method, International Journal of Numerical Methods and Applications 24(1) (2024),  45-61. http://dx.doi.org/10.17654/0975045224004

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] A. Kadem and D. Baleanu, Analytical method based on Walsh function combined with orthogonal polynomial for fractional transport equation, Commun. Nonlinear Sci. Numer. Simul. 15(3) (2010), 491-501.
[2] A. Kadem and D. Baleanu, Homotopy perturbation method for the coupled fractional Lotka-Volterra equations, Rom. J. Phys. 56(3) (2011), 332-338.
[3] A. Kadem and D. Baleanu, On fractional coupled Whitham-Broer-Kaup equations, Rom. J. Phys. 56(5) (2011), 629-635.
[4] A. R. Nabulsi, The fractional white dwarf hydrodynamical nonlinear differential and emergence of quark stars, Appl. Math. Comput. 218(6) (2011), 2837-2849.
[5] Blaise SOME, Méthode SBA de résolution des modèles mathématiques en environnement, Éditions Universitaires Européennes, 2018.
[6] BRAHIM Tellab, Résolution des équations différentielles fractionnaires, Thèse soutenue le 11/01/2018, Université des Frères Mentouri Constantine-1, 2018.
[7] B. Abbo, Nouvel algorithme numérique de résolution des équations différentielles ordinaires (EDO) et des équations aux dérivées partielles (EDP) non linéaires, Thèse de Doctorat unique, Université de Ouagadougou, UFR/SEA, Département Mathématique et Informatique (Burkina Faso), 2007.
[8] B. ABBO, O. SO, G. BARRO and B. SOME, A new numerical algorithm for solving nonlinear partial differential equations with initial and boundary conditions, Far East J. Appl. Math. 28(1) (2007), 37-52.
[9] B. Zheng, -expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun. Theor. Phys. 58 (2012), 623-630.
[10] Fatemeh NOROUZI and Gaston M. N’GUÉRÉKATA, A new study of fractional order financial system via homotopy analysis, Analele Universitatii Oradea. Fasc. Matematica, Tom XXVII(1) (2020), 141-152.
[11] G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Math. Comput. Modelling 13(7) (1990), 17-43.
[12] Germain KABORE, KÉRÉ Moumini, Windjiré SOME, Ousséni SO and Blaise SOME, Solving some fractional equations, in the sense of Riemann-Liouville, of Navier-Stokes by the numerical method SBA plus, International Journal of Numerical Methods and Applications 23(2) (2023), 209-228. http://dx.doi.org/10.17654/0975045223012.
[13] Germain KABORE, Windjiré SOME, Moumini KÉRÉ, Ousséni SO and Blaise SOME, Solving some fractional ordinary differential equations by SBA method, Journal of Mathematics Research 15(1) (2023). doi :10.5539/jmr.v15n1pxx.
[14] Houmor Tarek, Analyse du chaos dans un système d’équations différentielles fractionnaires, Thèse soutenue le 30/09/2014, Université Constantine 1, 2014.
[15] H. Jafari, A. Kadem, D. Baleanu and T. Yilmaz, Solutions of the fractional Davey-Stewartson equations with variational iteration method, Rom. Rep. Phys. 64(2) (2012), 337-346.
[16] H. Jafari, A. Kadem, D. Baleanu and T. Yilmaz, Variational iteration method for a fractional-order Brusselator System, Abstr. Appl. Anal. (2014), 1-6, Article ID 496323.
[17] J. Singh, D. Kumar and A. Kilicman, Numerical solutions of nonlinear, fractional partial differential equations arising in spatial diffusion of biological populations, Abstr. Appl. Anal. (2014), 1-12, Article ID 535793.
[18] J. T. Katsikadelis, Nonlinear dynamic analysis of viscoelastic membranes described with fractional differential models, Journal of Theoretical and Applied Mechanics 50(3) (2012), 743-753.
[19] J. R. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl. 12 (2011), 262-272.
[20] MENACER Youssaf Ammar, Système parabolique non-linéaire issu d’un modèle biologique: existence et comportement asymptotique, Université Abou Bekr Belkaid Tlemcen, Vol. 20, 2016.
[21] LOCKO BIBILA Frecinet Baranchelie, Résolution de quelques problèmes paraboliques et hyperboliques par la methode de Laplace-Adomian, Université Marien Ngouabi, 2019, pp. 28-58.
[22] M. G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer Science & Business Media, 2013, p. 225.
[23] Michel Mehrenberger, Introduction à la méthode des éléments finis pour les équations elliptiques, Aix-Marseille Université, France, 2020.
[24] Abdoul Wassiha Nebie, Bere Frédéric, Bakari Abbo and Youssouf Pare, Solving some derivative equations fractional order nonlinear partials using the Some Blaise Abbo method, Journal of Mathematics Research 13(2) (2021), 101 115. doi: 10.5539/jmr.v13n2p101.