Nonpolynomial collocation approximation of solutions to fractional differential equations

Abstract

We propose a non-polynomial collocation method for solving fractional differential equations. The construction of such a scheme is based on the classical equivalence between certain fractional differential equations and corresponding Volterra integral equations. Usually, we cannot expect the solution of a fractional differential equation to be smooth and this poses a challenge to the convergence analysis of numerical schemes. In this paper, the approach we present takes into account the potential non-regularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. An error analysis is provided for the linear case and several examples are presented and discussed.MSC 2010 : Primary 65L05; Secondary 34A08, 26A33

Fractional Pennes’ Bioheat Equation: Theoretical and Numerical Studies

“…Note that the convergence order depends on the fractional order α. For a method presenting optimal order convergence without the need to impose inconveniente smoothness conditions on the solution, see the work by Ford et al [12].…”

Higher order numerical methods for solving fractional differential equations

“…In [2], a high order numerical method for solving (1.1)-(1.2) is obtained where a quadratic interpolation polynomial was used to approximate the integral. Ford, Morgado and Rebelo recently (see [16]) used a nonpolynomial collocation method to achieve good convergence properties without Higher order numerical methods for solving fractional differential equations 3 assuming any smoothness of the solution. There are also several works that are related to the fixed memory principle and the nested memory concept for solving (1.1)-(1.2), see, e.g., [15], [12], [3], [4], [5], etc.…”

A non-polynomial collocation method for fractional terminal value problems

“…We refer, for example, to the work of Diethelm and Ford ([10]) where the authors, by considering the equivalence of the problem with a Volterra integral equation, established sufficient conditions for the existence and uniqueness of the solution, and investigated the sensitivity of the solution to changes in the parameters of the problem, namely, in the initial value, in the order of the derivative and in the right-hand side function f . Concerning the numerical solution of (1.1)-(1.2) with a = 0 we refer, for instance, to the articles [6], [8], [12], [13], [15], [17] among many others.…”