DISCONTINUOUS FRACTIONAL STURM-LIOUVILLE PROBLEMS WITH EIGEN-DEPENDENT BOUNDARY CONDITIONS

Jing Fu; XiaoLing Hao; Kun Li; Siqin Yao

doi:10.11948/20200308

2021 Volume 11 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(4): 2037-2051

Next Article Previous Article

Jing Fu, XiaoLing Hao, Kun Li, Siqin Yao. DISCONTINUOUS FRACTIONAL STURM-LIOUVILLE PROBLEMS WITH EIGEN-DEPENDENT BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 2037-2051. doi: 10.11948/20200308

Citation:

Jing Fu, XiaoLing Hao, Kun Li, Siqin Yao. DISCONTINUOUS FRACTIONAL STURM-LIOUVILLE PROBLEMS WITH EIGEN-DEPENDENT BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 2037-2051. doi: 10.11948/20200308

DISCONTINUOUS FRACTIONAL STURM-LIOUVILLE PROBLEMS WITH EIGEN-DEPENDENT BOUNDARY CONDITIONS

1.
Department of Mathematics, Inner Mongolia University, Hohhot, 010021, China
2.
Department of Mathematics, Qufu Normal University, Qufu, 276826, China

Corresponding author: Email: xlhao1883@163.com(X. Hao)

Abstract

Abstract

In this paper, a fractional discontinuous Sturm-Liouville type boun-dary-value problem with eigenparameter-dependent boundary conditions and with two fractional transmission conditions is investigated. Using operator theory, a new inner product is defined by combining the parameters in the boundary and transmission conditions, then the boundary value transmission problem is transferred to an operator in a new Hilbert space such that the eigenvalues and eigenfunctions of the main problem coincide with those of this operator. Moreover, the fundamental solutions are constructed, and then the characteristic function whose zeros are the eigenvalues of the problem is established.
- Fractional Sturm-Liouville problem /
- fractional eigen-dependent boundary conditions /
- fractional transmission conditions
MSC: 34B24, 34L20

FullText(HTML)

References

[1]	Z. Akdoğan, M. Demirci and O. Sh. Mukhtarov, Discontinuous Sturm-Liouville problem with eigen parameter-dependent boundary and transmission conditions, Acta Applicandae Mathematica, 2005, 86, 329-344. doi: 10.1007/s10440-004-7466-3 CrossRef Google Scholar
[2]	Z. Akdoğan, M. Demirci and O. Sh. Mukhtarov, Sturm-Liouville problem with eigen dependent boundary and transmissions conditions, Acta Mathematica Scientia, 2005, 25B(4), 731-740. Google Scholar
[3]	K. Aydemir1 and O. Sh. Mukhtarov, Variational principles for spectral analysis of one Sturm-Liouville problem with transmission conditions, Advances in Difference Equations, 2016, 2016, 1-14. doi: 10.1186/s13662-016-0800-z CrossRef Google Scholar
[4]	Q. M. Al-Mdallal, On the numerical solution of fractional Sturm-Liouville problem, International Journal Computer Mathematics., 2010, 87, 183-189. Google Scholar
[5]	Z. Akdoğan, A. Yakar and M. Demirci, Discontinuous fractional Sturm-Liouville problems with transmission conditions, Applied Mathematics and Computation, 2019, 350, 1-10. Google Scholar
[6]	D. Baleanu and E. Uğurlu, Regular fractional dissipative boundary value problems, Advances in Difference Equation, 2016, 2016, 1-6. doi: 10.1186/s13662-016-0883-6 CrossRef Google Scholar
[7]	M. Dehghan and A. B. Mingarelli, Fractional Sturm-Liouville eigenvalue problems, I, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales-Serie A: Matematicas, 2020, 114, 46. Google Scholar
[8]	C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1977, 77A, 293-308. Google Scholar
[9]	A. A. Kilbass, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Netherlands, Amsterdam, 2006. Google Scholar
[10]	J. Li and J. Qi, Note on a nonlocal SturmšCLiouville problem with both right and left fractional derivatives, Applied Mathematics Letters, 2019, 97, 14-19. doi: 10.1016/j.aml.2019.05.011 CrossRef Google Scholar
[11]	J. Li and J. Qi, Eigenvalue problems for fractional differential equations with right and left fractional derivatives, Applied Mathematics and Computation, 2015, 256, 1-10. doi: 10.1016/j.amc.2014.12.146 CrossRef Google Scholar
[12]	O. Sh. Mukhtarov, Discontinuous boundary-value problem with spectral parameter in boundary conditions, Turkish Journal of Mathematics. 1994, 18, 183-192. Google Scholar
[13]	O. Sh. Mukhtarov and K. Aydemir, Minimization principle and generalized Fourier series for discontinuous Sturm-Liouville systems in direct sum spaces, Journal of Applied Analysis and Computation, 2018, 8(5), 1511-1523. doi: 10.11948/2018.1511 CrossRef Google Scholar
[14]	O. Sh. Mukhtarov and K. Aydemir, Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Mathematica Scientia, 2015, 35(3), 639-649. doi: 10.1016/S0252-9602(15)30010-2 CrossRef Google Scholar
[15]	R. L. Magin, Fractional calculus in bioengineering, Begell House Incorporated, Redding, CT, 2004, 32, dol: 10.1615/CritRevBiomedEng. v32. i1.10. Google Scholar
[16]	M. A. Naimark, Linear Differential Operators, English Transl, New York, 1968. Google Scholar
[17]	M. A. Naimark, Linear Differential Operators Part Ⅱ, London, 1968. Google Scholar
[18]	H. Olğar and O. Sh. Mukhtarov, Weak eigenfunctions of two-interval Sturm-Liouville problems together with interaction conditions, Journal of Mathematical Physics, 2017, 58(4), 042201-1-042201-13. Google Scholar
[19]	I. Podlubny, Fractional Differential Equations, Academic Press, London, 1999. Google Scholar
[20]	J. Qi and S. Chen, Eigenvalue problems of the model from nonlocal continuum mechanics, Journal of Mathematical Physics, 2011, 52(7), 537-546. Google Scholar
[21]	M. Rivero, J. J. Trujillo and M. P. Velasco, A fractional approach to the Sturm-Liouville problem, Central European Journal of Physics, 2013, 11, 1246-1254. doi: 10.2478/s11534-013-0216-2 CrossRef Google Scholar
[22]	B. J. West, M. Bologna and P. Grigolini, Physics of fractal operators, Springer Verlag, New York, 2003. Google Scholar
[23]	A. Yakar and Z. Akdoğan, On the fundamental solutions of a discontinuous fractional boundary boundary value problem, Advances in Difference Equations, 2017, 2017, doi: 10.1186/s13662-017-1433-6. CrossRef Google Scholar
[24]	A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, American Mathematical Society, 2005. Google Scholar
[25]	M. Zayernouri and G. E. Karniadakis, Fractional Sturm Liouville eigen-problems: theory and numerical approximation, Journal of Computational Physics, 2013, 252, 495-517. doi: 10.1016/j.jcp.2013.06.031 CrossRef Google Scholar