Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations
American Journal of Applied Mathematics
Volume 7, Issue 6, December 2019, Pages: 157-163
Received: Oct. 24, 2019; Accepted: Nov. 19, 2019; Published: Dec. 30, 2019
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Authors
Badawi Hamza Elbadawi Ibrahim, Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, China; Department of Mathematics, Faculty of Education, University of Khartoum, Sudan
Qixiang Dong, School of Mathematical Sciences, Yangzhou University, Yangzhou, China
Zhengdi Zhang, Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, China
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Abstract
It is recognized that the theory of boundary value problems for fractional order-differential equations is one of the rapidly developing branches of the general theory of differential equations. As far as we know, most of the papers studied the fractional Riemann-Liouville derivative with respect to boundary values that are zero. However, for the purpose of this study, we concern ourselves with Captou type derivative of the order α∈(2, 3), with respect to boundary values that are nonzero. We establish sufficient conditions for the existence of solutions for boundary value problem of nonlinear variable coefficient of fractional order. On the other hand, the boundary value problem is formulated as follows: cDαu(t) + p(t)f(t, u(t)) + q(t) = 0, u(0) = a, u'(0) = b, u(1) = d. Where a, b, dR are constants. In this paper, we investigate the existence and uniqueness of solutions for a class of boundary value problem of the nonlinear variable coefficient of fractional differential equations. The existence of solutions involving Captuo fractional derivatives is discussed under the assumption that the bounded conditions are constants. By means of the Banach contraction mapping principle and Larry- Schauder alternative, the existence of solutions are obtained. Finally, some examples are discussed to illustrate the results, which are generalized to nonlinear fractional derivatives with variable coefficients.
Keywords
Fractional Derivatives, Fixed Point Theorem, Boundary Value Problem
To cite this article
Badawi Hamza Elbadawi Ibrahim, Qixiang Dong, Zhengdi Zhang, Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations, American Journal of Applied Mathematics. Vol. 7, No. 6, 2019, pp. 157-163. doi: 10.11648/j.ajam.20190706.13
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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