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Numerical approximations for fractional diffusion equations via splines. (English) Zbl 1228.65153
Summary: A one-dimensional fractional diffusion model is considered, where the usual second order derivative gives place to a fractional derivative of order \(\alpha \), with \(1<\alpha \leq 2\). We consider the Caputo derivative as the space derivative, which is a form of representing the fractional derivative by an integral operator. An implicit numerical method is derived which uses a spline approximation for the Caputo derivative. The consistency and stability of the method are examined and numerical results are presented.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
45K05 Integro-partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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