Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations

Badawi Hamza Elbadawi Ibrahim; Qixiang Dong; Zhengdi Zhang

doi:doi:10.11648/j.ajam.20190706.13

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Boundary Value Problems of Nonlinear Variable Coefficient Fractional Differential Equations

Badawi Hamza Elbadawi Ibrahim, Qixiang Dong, Zhengdi Zhang

Published in American Journal of Applied Mathematics (Volume 7, Issue 6)

Received: 24 October 2019 Accepted: 19 November 2019 Published: 30 December 2019

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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: ^cD^αu(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
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Fractional Derivatives, Fixed Point Theorem, Boundary Value Problem

References

[1]	Miller. KS. Ross. B, An Introductionto the Fractional Calculusand Fractional Differential Equations. Wiley, NewYork, (1993).
[2]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic press, New York, 1999.
[3]	R. Hilfer(Ed.), Applications of Fractional Calculus in Physics, World Scientific publishing Co, Singapore, 2000.
[4]	K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathmatics 2004, Springer-Verlag, Berlin, 2010.
[5]	K. Diethelm, A. D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, Sci. Comput. Chem. Eng. II, Springer Berlin Heidelberg, (1999), 217-224.
[6]	M. A. Krasnoselskii, Two remarks on the method of successive approximations, (Russian) Uspehi Mat. Nauk (N.S.), 10(1955), 123-127.
[7]	N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta, 45(2006), 765-772.
[8]	F. Metzler, W. Schick, H. G. Kilian, T. F. Nonnenmacher, Relaxation in filled polymers. A fractional calculus approach, J. Chem. Phys, 103(1995), 7180-7186.
[9]	Ge. F, Kou. C, Stability analysis by Krasnoselskiis fixed point theorem for nonlinear fractional differential equations. Applied Mathematics and Computation 257(2015), 308-316.
[10]	Su. X, Zhang. S. Unbounded solutions to aboundary value problem of fractional order on the half-line. Computers and Mathematics with Applications 61(2011), 1079-1087.
[11]	Wang. G. Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval. Applied Mathematics Letters 47(2015), 1-7.
[12]	Yang. L. Application of Avery-Peterson fixed point theorem to nonlinear boundary value problem of fractional differential equation with the Caputos derivative. Communications in Nonlinear Science and Numerical Simulation 17 (2012), 4576-4584.
[13]	X. Zhang, L. Liu, Y. Wu, The Multiple positive solution of a singular fractional differential equations with negatively perturbed term, Math. Comput. Modelling 55(2012), 1263-1274.
[14]	Y. Cui. Uniqueness of solution for boundary value problems for fractional differetial equation J. Appl. Math. Letters. 51(2016), 48-54.
[15]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud., vol 204, Elsevier Science B.V., Amsterdam, 2006.
[16]	K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.

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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

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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

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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

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@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}
}

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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  -

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Author Information

Badawi Hamza Elbadawi Ibrahim

Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, China
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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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

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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

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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

Copy | Download

@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}
}

Copy | Download

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  -

Copy | Download