Eigenvalue problem with fractional differential operator: Chebyshev cardinal spectral method

Document Type : Research Article

Authors

1 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran

2 Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran

Abstract

In this paper, we intend to introduce the Sturm-Liouville fractional problem and solve it using the collocation method based on Chebyshev cardinal polynomials. To this end, we first provide an introduction to the Sturm-Liouville fractional equation. Then the Chebyshev cardinal functions are introduced along with some of their properties and the operational matrices of the derivative, fractional integral, and Caputo fractional derivative are obtained for it. Here, for the first time, we solve the equation using the operational matrix of the fractional derivative without converting it to the corresponding integral equation. In addition to efficiency and accuracy, the proposed method is simple and applicable. The convergence of the method is investigated, and an example is presented to show its accuracy and efficiency.

Keywords

Main Subjects

References

[1] S. Abbasbandy, A. Shirzadi, A new application of the homotopy analysis method: solving the Sturm-Liouville problems, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 112–126.
[2] A. Afarideh, F. Dastmalchi Saei, M. Lakestani, B.N. Saray, Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions, Phys. Scripta, 96 (2021) 125267.
[3] M.A. Al-Gwaiz, Sturm-Liouville Theory and its Applications, Springer-Verlag, London, 2008.
[4] M. Alquran, K. Al-Khaled, Approximations of Sturm-Liouville eigenvalues using sinc-Galerkin and differential transform methods, Appl. Math. 5 (2010) 128–147.
[5] R. Agarwal, S. Grace, D. O'Regan, Oscillation Theory for Second-Order Dynamic Equations, Taylor and Francis, London, 2003.
[6] R. Agarwal, S. Grace, D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer, Dordrecht, 2000.
[7] N. Bujurke, C. Salimath, S. Shiralashetti, Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets, J. Comput. Appl. Math. 219 (2008) 90–101.
[8] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.
[9] M. El-gamel, M. Abd El-hady, Two very accurate and efficient methods for computing eigenvalues of Sturm-Liouville problems, Appl. Math. Model. 37 (2013) 5039–5046.
[10] V.S. Ertürk, Computing eigenelements of Sturm-Liouville problems of fractional order via fractional differential transform method, Math. Comput. Appl. 16 (2011) 712-720.
[11] P. Ghelardoni, Approximations of Sturm-Liouville eigenvalues using Boundary Value Methods, Appl. Numer. Math. 23 (1997) 311–325.
[12] B. Jin, L. Raytcho, J. Pasciak, W. Rundell, A Finite Element Method for the Fractional Sturm-Liouville Problem, arXiv preprint arXiv:1307.5114 (2013).
[13] A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[14] M. Klimek, A.B. Malinowska, T. Odzijewicz, Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain, Fract. Calc. Appl. Anal. 19 (2016) 516-550.
[15] D. Min, 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. 43 (2020) 9151–9168.
[16] B.M. Levitan, I.S. Sargsjan, Sturm-Liouville and Dirac Operators, Springer Dordrecht, 2012.
[17] M.A. Naimark, Linear Differential Operators, Ungar, New York, 1968.
[18] R. Ozarslan, E. Bas, D. Baleanu, Representation of solutions for Sturm-Liouville eigenvalue problems with generalized fractional derivative, Chaos 30 (2020) 033137.
[19] R. Ozarslan, A. Ercan, E. Bas, β-type fractional Sturm-Liouville Coulomb operator and applied results, Math. Meth. Appl. Sci. 42 (2019) 6648–6659.
[20] U. Yücel, Approximations of Sturm-Liouville eigenvalues using differential quadrature (DQ) method, J. Comput. Appl. Math. 192 (2006) 310–319.
Journal of Mathematical Modeling
Volume 11, Issue 2
July 2023
Pages 343-355

Files

Share

How to cite

Statistics