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: ^{c}D^{α}u(t) + p(t)f(t, u(t)) + q(t) = 0, u(0) = a, u'(0) = b, u(1) = d. Where a, b, d ∈ R 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.
Published in | American Journal of Applied Mathematics (Volume 7, Issue 6) |
DOI | 10.11648/j.ajam.20190706.13 |
Page(s) | 157-163 |
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Fractional Derivatives, Fixed Point Theorem, Boundary Value Problem
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APA Style
Badawi Hamza Elbadawi Ibrahim, Qixiang Dong, Zhengdi Zhang. (2019). Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations. American Journal of Applied Mathematics, 7(6), 157-163. https://doi.org/10.11648/j.ajam.20190706.13
ACS Style
Badawi Hamza Elbadawi Ibrahim; Qixiang Dong; Zhengdi Zhang. Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations. Am. J. Appl. Math. 2019, 7(6), 157-163. doi: 10.11648/j.ajam.20190706.13
AMA Style
Badawi Hamza Elbadawi Ibrahim, Qixiang Dong, Zhengdi Zhang. Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations. Am J Appl Math. 2019;7(6):157-163. doi: 10.11648/j.ajam.20190706.13
@article{10.11648/j.ajam.20190706.13, author = {Badawi Hamza Elbadawi Ibrahim and Qixiang Dong and Zhengdi Zhang}, title = {Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations}, journal = {American Journal of Applied Mathematics}, volume = {7}, number = {6}, pages = {157-163}, doi = {10.11648/j.ajam.20190706.13}, url = {https://doi.org/10.11648/j.ajam.20190706.13}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190706.13}, 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, d ∈ R 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.}, year = {2019} }
TY - JOUR T1 - Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations AU - Badawi Hamza Elbadawi Ibrahim AU - Qixiang Dong AU - Zhengdi Zhang Y1 - 2019/12/30 PY - 2019 N1 - https://doi.org/10.11648/j.ajam.20190706.13 DO - 10.11648/j.ajam.20190706.13 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 157 EP - 163 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20190706.13 AB - 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, d ∈ R 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. VL - 7 IS - 6 ER -