Modified spline collocation for linear fractional differential equations. (English) Zbl 1311.65094
Summary: We propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi-term fractional differential equations involving Caputo-type fractional derivatives. Using an integral equation reformulation of the initial value 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. Theoretical results are verified by some numerical examples.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 34A30 | Linear ordinary differential equations and systems |
| 34A08 | Fractional ordinary differential equations |
| 65R20 | Numerical methods for integral equations |
| 45D05 | Volterra integral equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
Keywords:
fractional differential equation; Caputo derivative; Volterra integral equation; smoothing transformation; spline collocation method; initial value problem; global convergence; supercomvergence; numerical exampleReferences:
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