Abstract
We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.
Highlights
It is currently well established that differential and integral equations with derivatives of fractional order have great importance in the modeling of real-life processes.For details, including basic theory of fractional calculus and references to some applications, see the monographs [1,2,3,4] and review papers [5,6,7]
In order to take into account the potential non-smoothness of the exact solution y = y(t) of (8) and (9) at the origin t = 0, we introduce on the interval [0, b] a graded grid
We have introduced and analyzed a high order numerical method for solving a wide class of linear multi-term fractional weakly singular integro-differential equations with Caputo fractional derivatives for local or non-local boundary conditions
Summary
It is currently well established that differential and integral equations with derivatives of fractional (non-integer) order have great importance in the modeling of real-life processes.For details, including basic theory of fractional calculus and references to some applications, see the monographs [1,2,3,4] and review papers [5,6,7]. It is currently well established that differential and integral equations with derivatives of fractional (non-integer) order have great importance in the modeling of real-life processes. When working with problems stemming from real-world applications, it is only rarely possible to find the solution of a given fractional differential or integral equation in closed form, and even if such an analytic solution is available, it is typically too complicated to be used in practice. In general, numerical methods are required for solving fractional differential and integral equations. A comprehensive survey of the most important methods for fractional initial value problems, along with a detailed introduction to the subject and a brief summary about the convergence behaviour of the methods is given in the monograph [32]
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