Version 1
: Received: 25 March 2021 / Approved: 26 March 2021 / Online: 26 March 2021 (15:42:47 CET)
How to cite:
Ozarslan, R.; Bas, E. Delta Fractional Sturm--Liouville Problems: From Discrete to Continuous. Preprints2021, 2021030674. https://doi.org/10.20944/preprints202103.0674.v1
Ozarslan, R.; Bas, E. Delta Fractional Sturm--Liouville Problems: From Discrete to Continuous. Preprints 2021, 2021030674. https://doi.org/10.20944/preprints202103.0674.v1
Ozarslan, R.; Bas, E. Delta Fractional Sturm--Liouville Problems: From Discrete to Continuous. Preprints2021, 2021030674. https://doi.org/10.20944/preprints202103.0674.v1
APA Style
Ozarslan, R., & Bas, E. (2021). Delta Fractional Sturm--Liouville Problems: From Discrete to Continuous. Preprints. https://doi.org/10.20944/preprints202103.0674.v1
Chicago/Turabian Style
Ozarslan, R. and Erdal Bas. 2021 "Delta Fractional Sturm--Liouville Problems: From Discrete to Continuous" Preprints. https://doi.org/10.20944/preprints202103.0674.v1
Abstract
In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.
Computer Science and Mathematics, Discrete Mathematics and Combinatorics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
