A matrix approach for partial differential equations with Riesz space fractional derivatives
Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Lecce, Italy
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Received: 3 June 2013
Revised: 6 August 2013
Published online: 7 October 2013
Fractional partial differential equations are emerging in many scientific fields and their numerical solution is becoming a fundamental topic. In this paper we consider the Riesz fractional derivative operator and its discretization by fractional centered differences. The resulting matrix is studied, with an interesting result on a connection between the decay behavior of its entries and the short memory principle from fractional calculus. The Shift-and-Invert method is then applied to approximate the solution of the partial differential equation as the action of the matrix exponential on a suitable vector which mimics the given initial conditions. The numerical results confirm the good approximation quality and encourage the use of the proposed approach.
© EDP Sciences, Springer-Verlag, 2013