In this paper, we construct exact analytical solutions, where they exist, of some nonlinear fractional Schrödinger equations in the sense of Caputo-Hadamard. Our results are obtained using an improved version of the SBA method.

SBA plus method, Hadamard integral, Caputo-Hadamard fractional derivative, Caputo-Hadamard fractional differential equation.

Received: March 1, 2024; Accepted: April 18, 2024; Published: December 12, 2024

How to cite this article: Germain KABORE, Windjiré SOME, Abdarahman Abakar Himeda, Bakari Abbo, Ousséni SO and Blaise SOME, Use of the SBA plus method to solve some nonlinear fractional Schrödinger equations in the sense of Caputo-Hadamard, Far East Journal of Dynamical Systems 38(1) (2025), 31-46. https://doi.org/10.17654/0972111825002

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

References:
[1] Blaise SOME, Méthode SBA de résolution des modèles mathématiques en environnement, Éditions Universitaires Européennes, 2018.
[2] B. Abbo, Nouvel algorithme numérique de résolution des équations différentielles ordinaries (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.
[3] C. Besse, B. Bidégaray and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 40(1) (2002), 26-40.
[4] C. Besse, Christophe BESSE Méthodes numériques et conditions aux limites artificielles pour les équations de Schrödinger linéaires et non linéaires et modélisation d’irrégularités du plasma ionosphérique terrestre, Thesis defended on 08/12/2004, Université Paul Sabatier - Toulouse III, France.
[5] F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2012. DOI: 10.1186/1687-1847-2012-142.
[6] 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.
[7] Germain KABORE, Abakar Mahamat SEID, Bakari Abbo, Ousséni SO and Blaise SOME, Application of the SBA method to the solution of some nonlinear fractional equations in the sense of Caputo Hadamard, Universal Journal of Mathematics and Mathematical Sciences 19(2) (2023), 103-117.
http://dx.doi.org/10.17654/2277141723019.
[8] J. Hadamarad, Essai sur l’etude des fonctions donnes par leur développment de Taylor, J. Pure Appl. Math. 4(8) (1892), 101-186.
[9] J. B. Yindoula, Y. A. S. Wellot, B. Hamadou, F. Bassono and Y. Paré, Application of the Some Blaise Abbo (SBA) method to solving the time-fractional Schrödinger equation and comparison with the homotopy perturbation method, Asian Research Journal of Mathematics 18(11) (2022), 271-286.
Article no. ARJOM.92257. DOI: 10.9734/ARJOM/2022/v18i11601.
[10] 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), Article ID 535793, 1-12.
[11] J. T. Katsikadelis, Nonlinear dynamic analysis of viscoelastic membranes described with fractional differential models, J. Theorical. Appl. Mech. 50(3) (2012), 743-753.
[12] J. R. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl. 12 (2011), 262-272.
[13] B. C. Reed, Numerical Solution of Schrödinger’s Equation, Quantum Mechanics, Springer, Cham, 2022. https://doi.org/10.1007/978-3-031-14020-4-10.