Iterative Processes Methods for Solving Boundary Value Problem for the Caputo Fractional Differential Equations
DOI:
https://doi.org/10.37375/sjfssu.v4i1.2598Keywords:
Caputo Fractional Differential equations, Boundary value problem, Fixed-point theorem, iterative processes methods.Abstract
In this article, we introduce a study of approximate solutions for the Caputo fractional differential equations with boundary conditions in Banach space. We transformed given equations into equivalent integral equations for constructing of contraction mapping and other compact mapping, both of which allow for the proof of the existence solution. The ultimate goal of this study is to present a comparison of the speed convergence of the approximate solutions of Caputo Fractional differential equations obtained by using the processes repetitiveness of the Picard, Mann, Picard-Mann hybrid, Picard -Krasnoselskii hybrid, and Ishikawa methods to the general solutions
References
Abdeljawad, T.; Ullah K., Ahmad, J.: “On Picard–Krasnoselskii hybrid iteration process in Banach spaces.” Jour. of Mathematics, 2020 (2020): 1-5.
Berinde, V.: “Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators.” Fixed Point Theory and Appl., 2004.2 (2004):1-9.
Berinde, V., Mădălina, B.: “The fastest Krasnose-lskij iteration for approximating fixed points of strictly pseudo-contractive mappings.” Carpathian Journal of Mathematics (2005): 13-20.
Chipot, M.: Handbook of differential Equations: Stationary Partial Differential Equations. 6(2008): 503-583.
Chidume, C.E.; Chidume, C.O.; Djitté, N., Minjibir, M.S.: “Convergence theorems for fixed points of multivalued strictly pseudo contractive mappings in Hilbert spaces.” Abstract Appl. Anal., vol. 2013, (2013, January) Hindawi.
Chidume, C.E.: “Strong convergence and stability of Picard iteration sequences for a general class of contractive-type mappings.” Fixed Point Theory Appl., 2014(1), (2014): 1-10.
Chidume, C.E.; “Picard iteration process for a general class of contractive mappings.” Journal of the Nigerian Mathematical Society, 33(1-3), (2014) 19-23.
Diethelm, K., “The analysis of fractional differen-tial equations.” Lect. Notes in Math., Springer (2010).
Furati, K. M., Tatar, N. E.; “Power-type estimates for a nonlinear fractional differential equation.” Nonlinear Analysis: Theory Methods & Applications, 62.6 (2005): 1025-1036.
Granas, A., Dugundji J.; “Fixed point theory.” Vol. 14. New York: Springer (2003).
Ghiura, A.: “Convergence of modified Picard-Mann hybrid iteration process for nearly nonexpansive mappings.” arXiv preprint arXiv:2101.04567 (2021).
Ishikawa, S.: “Fixed points by a new iteration method.” Proceedings of the American Mathematical Society 44.1 (1974): 147-150.
Krasnoselskii, M.A.; “Two observations about the method of successive approximations.” Uspekhi Matematicheskikh Nauk, 10, (1955): 123-127
Kilbas, A.A., Srivastava H.M., Trujillo J.J.,: “Theory and applications of fractional differential equations.” Elsevier (2006) Vol. 204.
Khan, S. H.: “A Picard-Mann hybrid iterative process.” Fixed Point Theory and Applications 2013.1 (2013): 1-10.
Lyons, R., Aghalaya S.V., Ross A. C.: “Picard’s iterative method for Caputo fractional differential equations with numerical results.” Mathematics 5.4 (2017): 65.
Mann, W. R.: “Mean value methods in iteration.” Proceedings of the American Math. Society 4.3 (1953): 506-510.
Osilike, M.O., Aniagbosor, S.C.: “Weak and strong convergence theorems for fixed points of asymptotically nonexpensive mappings.” Math. and Comp. Modelling 32.10 (2000): 1181-1191.
Okeke, G. A., Mujahid, A.: “A solution of delay differential equations via Picard–Krasnoselskii hybrid iterative process.” Arabian Journal of Mathematics 6 (2017): 21-29.
Picard, E.: Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. Jour. Math. Pures Appl. 6(1890). 145–210.
Wang, J., Michal F., Yong Z., “Weakly Picard operators method for modifed fractional iterative functional differential equations.” Fixed Point Theory 15.1 (2014): 297-310.
Şoltuz, Ş.M., Diana, O.: “Classical results via Mann-Ishikawa iteration.” Revue d'analyse numérique et de théorie de l'approximation 36.2 (2007): 193-197.
Zhang, S.: “Positive solutions for boundary-value problems of nonlinear fractional differential equations." Electronic Journal of Differential Equations (EJDE) [electronic only] 2006 (2006): Paper-No.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Scientific Journal for Faculty of Science-Sirte University
This work is licensed under a Creative Commons Attribution 4.0 International License.
