Theory of discrete fractional Sturm–Liouville equations and visual results

Erdal Bas; Ramazan Ozarslan; Erdal Bas; Ramazan Ozarslan

doi:10.3934/math.2019.3.593

AIMS Mathematics

2019, Volume 4, Issue 3: 593-612. doi: 10.3934/math.2019.3.593

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Research article

Theory of discrete fractional Sturm–Liouville equations and visual results

Erdal Bas ,
Ramazan Ozarslan ^,

Department of Mathematics, Firat University, 23119, Elazig, Turkey

Received: 11 March 2019 Accepted: 09 May 2019 Published: 03 June 2019
MSC : Primary: 34A08, 39A70, 34B24, 28A75; Secondary: 44A10

In this article, we study discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grünwald-Letnikov fractional operators with both delta and nabla operators. Self-adjointness of the DFSL operator is analyzed and fundamental spectral properties are proved. Besides, we get sum representation of solutions for DFSL problem by means of Laplace transform for nabla fractional difference equations and find the analytical solutions of the problem. Moreover, the results for DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, and fractional Sturm-Liouville (FSL) problem are compared with the second order classical Sturm-Liouville (CSL) problem. We display the results comparatively by tables and figures.
- fractional difference equations,
- Laplace transform,
- fractional differential equations,
- Sturm-Liouville,
- eigenfunctions, eigenvalues
Citation: Erdal Bas, Ramazan Ozarslan. Theory of discrete fractional Sturm–Liouville equations and visual results[J]. AIMS Mathematics, 2019, 4(3): 593-612. doi: 10.3934/math.2019.3.593

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Abstract

In this article, we study discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grünwald-Letnikov fractional operators with both delta and nabla operators. Self-adjointness of the DFSL operator is analyzed and fundamental spectral properties are proved. Besides, we get sum representation of solutions for DFSL problem by means of Laplace transform for nabla fractional difference equations and find the analytical solutions of the problem. Moreover, the results for DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, and fractional Sturm-Liouville (FSL) problem are compared with the second order classical Sturm-Liouville (CSL) problem. We display the results comparatively by tables and figures.

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