Numerical schemes for integro-differential equations with Erd\xe9lyi-Kober fractional operator

Łukasz Płociniczak,Szymon Sobieszek

doi:10.1007/s11075-016-0247-z

Abstract

This work investigates several discretizations of the Erdélyi-Kober fractional operator and their use in integro-differential equations. We propose two methods of discretizing E-K operator and prove their errors asymptotic behaviour for several different variants of each discretization. We also determine the exact form of error constants. Next, we construct a finite-difference scheme based on a trapezoidal rule to solve a general first order integro-differential equation. As is known from the theory of Abel integral equations, the rate of convergence of any finite-different method depends on the severity of kernel’s singularity. We confirm these results in the E-K case and illustrate our considerations with numerical examples.

Highlights

Fractional calculus constitutes a very vast area in which many interesting mathematical and physical objects reside
Numer Algor (2017) 76:125–150 most successful is anomalous diffusion [34, 36, 37], which can be observed in variety of situations such as moisture percolation in porous media [39], protein random walks in cells [53], telomere motion [7, 23] and diffusion of cosmic rays across the magnetic fields [11]
We introduce two types of discretization of the E-K operator, find theirs truncation errors with exact error constants and apply those results to construct a second-order finite difference scheme which approximates the solution of the first order integro-differential equation with E-K operator Ia,b,c, namely y = f (x, y, Ia,b,cy)

Summary

Introduction

Numer Algor (2017) 76:125–150 most successful is anomalous diffusion [34, 36, 37], which can be observed in variety of situations such as moisture percolation in porous media [39], protein random walks in cells [53], telomere motion [7, 23] and diffusion of cosmic rays across the magnetic fields [11]. A review of numerical methods (as well as analytical results) for ordinary fractional differential equations has been given in [6], where a modern overview and practical algorithms are given. Paper is a first step in deriving a more optimal numerical method which is constructed for the self-similar ordinary rather than original partial differential equation. We introduce two types of discretization of the E-K operator, find theirs truncation errors with exact error constants and apply those results to construct a second-order finite difference scheme which approximates the solution of the first order integro-differential equation with E-K operator Ia,b,c, namely y = f (x, y, Ia,b,cy). The objective for future work will be to extend these results to the self-similar nonlinear time-fractional diffusion

Discretization of the Erdelyi-Kober operator

Finite difference scheme

Conclusion