In this paper a fractional differential equation of the Euler-Lagrange/ Sturm-Liouville type is considered. The fractional equation with derivatives of order α ∈ (0, 1] in the finite time interval is transformed to the integral form. Next the numerical scheme is presented. In the final part of this paper examples of numerical solutions of this equation are shown. The convergence of the proposed method on the basis of numerical results is also discussed.MSC 2010 : Primary 26A33; Secondary 34A08, 65L10

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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.

This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view, the equation includes at least two fractional derivatives: spatial and temporal. In this paper we proposed a new numerical scheme for the spatial derivative, the so called RieszFeller operator. Moreover, using the finite difference method, we show how to employ this scheme in the numerical solution of fractional partial differential equations. In other words, we considered an initial-boundary value problem in one dimensional space. In the final part of this paper some numerical results and plots of simulations are shown as examples.