Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems

Abstract We construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form ( D ∗ α y ) ( t ) = f ( t , y ( t ) ) with Caputo type fractional derivatives D ∗ α y of order α > 0 . Using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation. After that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid. Optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established. To illustrate the reliability of the proposed algorithms two numerical examples are given.

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