SOLVABILITY OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR A CLASS OF FRACTIONAL ADVECTION-DISPERSION EQUATIONS THROUGH VARIATIONAL APPROACH

Dandan Min; Fangqi Chen

doi:10.11948/20210265

2022 Volume 12 Issue 2

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Dandan Min, Fangqi Chen. SOLVABILITY OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR A CLASS OF FRACTIONAL ADVECTION-DISPERSION EQUATIONS THROUGH VARIATIONAL APPROACH[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 676-691. doi: 10.11948/20210265

Citation:

Dandan Min, Fangqi Chen. SOLVABILITY OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR A CLASS OF FRACTIONAL ADVECTION-DISPERSION EQUATIONS THROUGH VARIATIONAL APPROACH[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 676-691. doi: 10.11948/20210265

SOLVABILITY OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR A CLASS OF FRACTIONAL ADVECTION-DISPERSION EQUATIONS THROUGH VARIATIONAL APPROACH

Dandan Min^1,2,,
Fangqi Chen^1,2, ,

1.
Deparement of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
2.
Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA), MIIT, Nanjing 211106, China

Author Bio: Email: mindan123@nuaa.edu.cn(D. Min)
Corresponding author: Email: fangqichen1963@126.com(F. Chen)
Fund Project: The authors were supported by National Natural Science Foundation of China (Nos. 11872201, 12172166)

Abstract

Abstract

In this paper, we probe into the solvability of Sturm-Liouville problem for fractional advection-dispersion equations without traditional Amb-rosetti-Rabinowitz conditions. Some existence results of infinitely many small negative energy and large energy solutions are obtained by employing variant fountain theorems. The nonlinearity f and l_i (i=1, 2, m, m) are considered under certain appropriate assumptions which are distinct from those assumed in previous articles. In addition, the main result is confirmed by an example which is provided.
- Sturm-Liouville boundary conditions /
- variant fountain theorems /
- fractional advection-dispersion equations /
- variational method
MSC: 26A33, 35A15, 34B15, 34A08

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References

[1]	G. Bonanno and G. D'Agui, A critical point theorem and existence results for a nonlinear boundary value problem, Nonlinear Anal., 2010, 72, 1977-1982. doi: 10.1016/j.na.2009.09.039 CrossRef Google Scholar
[2]	G. Chai, Infinitely many solutions for nonlinear fractional boundary value problems via variational methods, Adv. Difference Equ., 2016, 2016, 1-23. Google Scholar
[3]	J. Chen and X. Tang, Infinitely many solutions for boundary value problems arising from the fractional advection dispersion equation, Appl. Math., 2015, 6, 703-724. Google Scholar
[4]	V. Ervin and J. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differ. Equ., 2006, 22, 558-576. doi: 10.1002/num.20112 CrossRef Google Scholar
[5]	D. Gao and J. Li, Infinitely many solutions for impulsive fractional differential equations through variational methods, Quaest. Math., 2019, 2019, 1-17. Google Scholar
[6]	J. Graef, L. Kong and Q. Kong, Multiple solutions of systems of fractional boundary value problems, Appl. Anal., 2015, 94, 1288-1304. doi: 10.1080/00036811.2014.930822 CrossRef Google Scholar
[7]	J. Graef, L. Kong and M. Wang, A variational framework for a second order discrete boundary value problem with mixed periodic boundary conditions, Results Math., 2021, 76, 1-12. DOI:10.1007/s00025-021-01406-5. CrossRef Google Scholar
[8]	F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 2011, 62, 1189-1199. Google Scholar
[9]	A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006. Google Scholar
[10]	L. Kong, Existence of solutions to boundary value problems arising from the fractional advection dispersion equation, Electron. J. Differential Equations., 2013, 2013, 1-15. Google Scholar
[11]	D. Li, F. Chen and Y. An, Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p-Laplacian via critical point theorey, Math. Meth. Appl. Sci., 2018, 41, 3197-3212. doi: 10.1002/mma.4810 CrossRef Google Scholar
[12]	Z. Liu, H. Chen and T. Zhou, Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem, Comput. Math. Appl., 2011, 61, 1687-1699. doi: 10.1016/j.camwa.2011.01.042 CrossRef Google Scholar
[13]	S. Lu, F. Molz and G. Fix, Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation equation to natural porous media, Water Resour. Res., 2002, 38, 1165-1171. Google Scholar
[14]	D. Ma, L. Liu and Y. Wu, Existence of nontrivial solutions for a system of fractional advection-dispersion equations, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 2019, 113, 1041-1057. doi: 10.1007/s13398-018-0527-7 CrossRef Google Scholar
[15]	M. Meerschaert, D. Benson and B. Bäumer, Multidimensional advection and fractional dispersion, Phys. Rev. E, 1999, 59, 5026-5028. Google Scholar
[16]	D. Min and F. Chen, Three solutions for a class of fractional impulsive advection-dispersion equations with Sturm-Liouville boundary conditions via variational approach, Math. Meth. Appl. Sci., 2020, 43, 9151-9168. doi: 10.1002/mma.6608 CrossRef Google Scholar
[17]	D. Min and F. Chen, Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with sturm-Liouville boundary-value problem, Fract. Calc. Appl. Anal., 2021, 24, 1069-1093. doi: 10.1515/fca-2021-0046 CrossRef Google Scholar
[18]	N. Nyamoradi, Existence and multiplicity of solutions for impulsive fractional differential equations, Mediterr. J. Math., 2017, 14, 1-17. doi: 10.1007/s00009-016-0833-2 CrossRef Google Scholar
[19]	N. Nyamoradi and E. Tayyebi, Existence of solutions for a class of fractional boundary value equations with impulsive effects via critical point theory, Mediterr. J. Math., 2018, 15, 1-25. doi: 10.1007/s00009-017-1047-y CrossRef Google Scholar
[20]	N. Nyamoradi and S. Tersian, Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theorey, Fract. Calc. Appl. Anal., 2019, 22, 945-967. doi: 10.1515/fca-2019-0051 CrossRef Google Scholar
[21]	N. Nyamoradi and Y. Zhou, Existence results to some damped-like fractional differential equations, Int. J. Nonlinear Sci. Numer. Simul., 2017, 18, 233-243. doi: 10.1515/ijnsns-2016-0093 CrossRef Google Scholar
[22]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Google Scholar
[23]	H. Sun and Q. Zhang, Existence of solutions for a fractional boundary value problem via the mountain pass method and an iterative technique, Comput. Math. Appl., 2012, 64, 3436-3443. doi: 10.1016/j.camwa.2012.02.023 CrossRef Google Scholar
[24]	Y. Tian and J. Nieto, The applications of critical-point theory to discontinuous fractional-order differential equations, Proc. Edinb. Math. Soc., 2017, 60, 1021-1051. doi: 10.1017/S001309151600050X CrossRef Google Scholar
[25]	Y. Wang, Y. Li and J. Zhou, Solvability of boundary value problems for impulsive fractional differential equations via critical point theory, Mediterr. J. Math., 2016, 13, 4845-4866. doi: 10.1007/s00009-016-0779-4 CrossRef Google Scholar
[26]	Y. Wang, Y. Liu and Y. Cui, Multiple solutions for a nonlinear fractional boundary value problem via critical point theory, J. Funct. Space, 2017, 2017, 1-8. Google Scholar
[27]	T. Xue, F. Kong and L. Zhang, Research on Sturm-Liouville boundary value problems of fractional p-Laplacian equation, Bound. Value Probl., 2021, 2021, 1-20. doi: 10.1186/s13661-020-01478-2 CrossRef Google Scholar
[28]	X. Zhang, L. Liu and Y. Wu, Variational structure and multiple solutions for a fractional advection-dispersion equations, Comput. Math. Appl., 2014, 68, 1794-1805. doi: 10.1016/j.camwa.2014.10.011 CrossRef Google Scholar
[29]	X. Zhang, L. Liu, Y. Wu and B. Wiwatanapataphee, Nontrivial solutions for a fractional advection-dispersion equation in anomalous diffusion, Appl. Math. Lett., 2017, 66, 1-8. doi: 10.1016/j.aml.2016.10.015 CrossRef Google Scholar
[30]	Z. Zhang and R. Yuan, Two solutions for a class of fractional boundary value problems with mixed nonlinearities, Bull. Malays. Math. Sci. Soc., 2018, 41, 1233-1247. doi: 10.1007/s40840-016-0386-3 CrossRef Google Scholar
[31]	W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 2001, 104, 343-358. doi: 10.1007/s002290170032 CrossRef Google Scholar