A SINGULAR FRACTIONAL DIFFERENTIAL EQUATION WITH RIESZ-CAPUTO DERIVATIVE

Dehong Ji; Yuan Ma; Weigao Ge

doi:10.11948/20220402

2024 Volume 14 Issue 2

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Dehong Ji, Yuan Ma, Weigao Ge. A SINGULAR FRACTIONAL DIFFERENTIAL EQUATION WITH RIESZ-CAPUTO DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 642-656. doi: 10.11948/20220402

Citation:

Dehong Ji, Yuan Ma, Weigao Ge. A SINGULAR FRACTIONAL DIFFERENTIAL EQUATION WITH RIESZ-CAPUTO DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 642-656. doi: 10.11948/20220402

A SINGULAR FRACTIONAL DIFFERENTIAL EQUATION WITH RIESZ-CAPUTO DERIVATIVE

1.
School of Science, Tianjin University of Technology, Tianjin 300384, China
2.
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Author Bio: Email: mayuan@163.com(Y. Ma); Email: gwei@163.com(W. Ge)
Corresponding author: Email: jdh200298@163.com(D. Ji)
Fund Project: The authors were supported by Natural Science Foundation of Tianjin (No. (19JCYBJC30700))

Abstract

Abstract

In this paper, we obtain the existence results for positive solutions of a class of boundary value problems for fractional differential equations with Riesz-Caputo derivative by using of the theory of Leray-Schauder degree. The interesting point is the nonlinear term $ f(t, u) $ may be singular at $ u=0. $ An example is also given to demonstrate the validity of the main result.
- Positive solution /
- singular boundary value problem /
- Riesz-Caputo derivative /
- Leray-Schauder degree
MSC: 34A08

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References

[1]	O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A: Math. Theor., 2007, 40, 6287–6303. doi: 10.1088/1751-8113/40/24/003 CrossRef Google Scholar
[2]	B. Azzaoui, B. Tellab and K. Zennir, Positive solutions for a fractional configuration of the Riemann-Liouville semilinear differential equation, Math. Method Appl. Sci., 2022. DOI: 10.1002/mma.8110. CrossRef Google Scholar
[3]	D. Baleanu, K. Diethelm and E. Scalas, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, 2012. Google Scholar
[4]	D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research, 2000, 36(6), 1403–1412. doi: 10.1029/2000WR900031 CrossRef Google Scholar
[5]	D. A. Benson, S. W. Wheatcraft and M. M. Meerschaert, The fractional-order governing equation of Lvy motion, Water Resources Research, (2000), 36(6), 1413–1423. doi: 10.1029/2000WR900032 CrossRef Google Scholar
[6]	F. L. Chen, D. Baleanu and G. C. Wu, Existence results of fractional differential equations with Riesz-Caputo derivative, Eur. Phys. J. Special Topics, 226, 2017, 3411–3425. doi: 10.1140/epjst/e2018-00030-6 CrossRef Google Scholar
[7]	F. L. Chen, A. P. Chen and X. Wu, Anti-periodic boundary value problems with Riesz-Caputo derivative, Adv. Differ. Equ., 2019, 119, 1–13. Google Scholar
[8]	A. M. A. El-Sayed and M. Gaber, On the finite Caputo and finite Riesz derivatives, Electronic Journal of Theoretical Physics, 2006, 3(12), 81–95. Google Scholar
[9]	G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense, Appl. Math. Comput., 2010, 217, 1023–1033. Google Scholar
[10]	C. Y. Gu, J. Zhang and G. C. Wu, Positive solutions of fractional differential equations with the Riesz space derivative, Appl. Math. Lett., 2019, 95, 59–64. doi: 10.1016/j.aml.2019.03.006 CrossRef Google Scholar
[11]	H. Jiang, F. Liu, I. Turner and K. Burrage, Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain, Journal of Mathematical Analysis and Applications, 2012, 389, 1117–1127. doi: 10.1016/j.jmaa.2011.12.055 CrossRef Google Scholar
[12]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Google Scholar
[13]	Y. Luchko, Maximum principle and its application for the time-fractional diffusion equations, Fractional Calculus and Applied Analysis, 2011, 14(1), 110–124. doi: 10.2478/s13540-011-0008-6 CrossRef Google Scholar
[14]	R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006. Google Scholar
[15]	R. Magin, M. D. Ortigueira, I. Podlubny and J. Trujillo, On the fractional signals and systems, Signal Processing, 2011, 91, 350–371. doi: 10.1016/j.sigpro.2010.08.003 CrossRef Google Scholar
[16]	F. C. Meral, T. J. Royston and R. Magin, Fractional calculus in viscoelasticity: An experi- mental study, Communications in Nonlinear Science and Numerical Simulation, 2010, 15, 939–945. doi: 10.1016/j.cnsns.2009.05.004 CrossRef Google Scholar
[17]	R. K. Pandey, O. P. Singh and V. K. Baranwal, An analytic algorithm for the space-time fractional advection-dispersion equation, Computer Physics Communications, 2011, 182, 1134–1144. doi: 10.1016/j.cpc.2011.01.015 CrossRef Google Scholar
[18]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Google Scholar
[19]	J. Ren and C. Zhai, Solvability for p-Laplacian generalized fractional coupled systems with two-sided memory effects, Math. Meth. Appl. Sci., 2020, 43, 8797–8822. doi: 10.1002/mma.6545 CrossRef Google Scholar
[20]	Q. Yang, F. Liu and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathematical Modelling, 2010, 34, 200–218. doi: 10.1016/j.apm.2009.04.006 CrossRef Google Scholar
[21]	G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 2002, 371(6), 461–580. doi: 10.1016/S0370-1573(02)00331-9 CrossRef Google Scholar
[22]	X. G. Zhang, D. Z. Kong, H. Tian, Y. H. Wu and B. Wiwatanapataphee, An upper-lower solution method for the eigenvalue problem of Hadamard-type singular fractional differential equation, Nonlinear Anal-Model, 2022. DOI: 10.15388/namc.2022.27.27491. CrossRef Google Scholar
[23]	X. Q. Zhang, Z. Y. Shao and Q. Y. Zhong, Multiple positive solutions for higher-order fractional integral boundary value problems with singularity on space variable, Fract. Calc. Appl. Anal., 2022, 25, 1507–1526. doi: 10.1007/s13540-022-00073-9 CrossRef Google Scholar
[24]	P. Zhuang, F. Liu, V. Anh and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM Journal on Numerical Analysis, 2009, 47(3), 1760–1781. doi: 10.1137/080730597 CrossRef Google Scholar