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Numerical solution of fractional integro-differential equations by collocation method. (English) Zbl 1100.65126

The paper concerns the numerical solution of fractional integro-differential equations of the type \(D^qy(t)=p(t)y(t)+f(t)+\int_0^tK(t,s)y(s)ds, t\int=[0,1]\) by polynomial spline functions. Specific existing literature relating to fractional differential equations is cited. A definition of the Riemann-Liouville fractional differential operator of order \(q, q \notin \mathbb{N},\) is given.
After introducing the polynomial spline space the author derives a collocation spline method. The solution of an \(m \times m\) system of linear equations on each of \(N\) subintervals is involved. A theorem by L. Blank [Nonlinear World 4, No. 4, 473–491 (1997)] is stated and used and a new theorem is stated and proved. Two numerical examples are presented to illustrate the results. Each concerns a fractional integro-differential equation with a given initial condition and a known exact solution. Computed absolute errors in the numerical solution are given for three values of \(t\).
Reviewer: Pat Lumb (Chester)

MSC:

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
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