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Finite difference method for a fractional porous medium equation

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Abstract

We formulate a numerical method to solve the porous medium type equation with fractional diffusion

$$\begin{aligned} \frac{\partial u}{\partial t}+(-\Delta )^{1/2} (u^m)=0. \end{aligned}$$

The problem is posed in \(x\in {\mathbb {R}}^N\), \(m\ge 1\) and with nonnegative initial data. The fractional Laplacian is implemented via the so-called Caffarelli–Silvestre extension. We prove existence and uniqueness of the solution of this method and also the convergence to the theoretical solution of the equation. We run numerical experiments on typical initial data as well as a section that summarizes and concludes the proposed method.

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Acknowledgments

The author partially supported by the Spanish Project MTM2011-24696 and by a FPU grant from Ministerio de Educación, Ciencia y Deporte, Spain.

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  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 , Madrid, Spain

    Félix del Teso

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del Teso, F. Finite difference method for a fractional porous medium equation. Calcolo 51, 615–638 (2014). https://doi.org/10.1007/s10092-013-0103-7

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  • DOI: https://doi.org/10.1007/s10092-013-0103-7

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