Exact and numerical solutions of the fractional Sturm–Liouville problem

Abstract

In the paper, we discuss the regular fractional Sturm-Liouville problem in a bounded domain, subjected to the homogeneous mixed boundary conditions. The results on exact and numerical solutions are based on transformation of the differential fractional Sturm-Liouville problem into the integral one. First, we prove the existence of a purely discrete, countable spectrum and the orthogonal system of eigenfunctions by using the tools of Hilbert-Schmidt operators theory. Then, we construct a new variant of the numerical method which produces eigenvalues and approximate eigenfunctions. The convergence of the procedure is controlled by using the experimental rate of convergence approach and the orthogonality of eigenfunctions is preserved at each step of approximation. In the final part, the illustrative examples of calculations and estimation of the experimental rate of convergence are presented.

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

“…All the above considerations lead to the proposition on convergence of the series associated with the intermediate fractional integral eigenvalue problem given in (50) and (A5). Analogous convolutions' series were also studied on the C[a, b] and L 2 (a, b) function spaces for FSLPs with homogeneous mixed and Robin boundary conditions, respectively [13,14].…”

“…Now, we shall quote the general formulation of the fractional eigenvalue problem, introduced and investigated in papers [7,[11][12][13][14]21].…”

“…In our earlier works [7,[11][12][13][14], we focused on the construction of a fractional version of operator L q , which includes standard fractional derivatives. The characteristic feature of the proposed approach is the mixture of the left and right fractional derivatives in the fractional Sturm-Liouville operator (FSLO).…”

Numerical approximation of tempered fractional Sturm‐Liouville problem with application in fractional diffusion equation

“…Therefore to study FSLPs in bounded domain is beneficial for fractional diffusive process. Such process appears in fields of engineering and science, for example, fluid pressure in porous media, heat movement in materials, human migration, motility of bacteria, etc 47,48 …”

Numerical approximation to Prabhakar fractional Sturm–Liouville problem

“…This type of fractional problem is related to finding the solutions of partial fractional differential equations using the method of separation of variables. In analytical and numerical aspects, many researchers studied FSLP (2) with different types of fractional derivatives, such as the Riemann-Liouville, Caputo, Weyl, Hilfer and Weber fractional derivatives (Ansari 2015;Derakhshan and Ansari 2019;Kilbas et al 2004;Agrawal 2012, 2013;Klimek et al 2014Klimek et al , 2016Klimek et al , 2018. They showed that the eigenfunctions of regular and singular FSLPs construct a set of basis functions for the approximations of solutions of partial fractional differential equations in finite and infinite domains, see (Al-Mdallal 2009; Ansari 2015; Bas and Metin 2013; Derakhshan and Ansari 2019; Erturk 2011; Ansari 2016, 2017;Hilfer 2000;Hilfer et al 2009;Klimek et al 2014Klimek et al , 2016Klimek and Agrawal 2012;Klimek 2009;Luchko and Yamamoto 2016;Rivero et al 2013;Zayernouri and Karniadakis 2013;Zayernouri and Karniadakis 2014a, b, c).…”