Numerical method for the time-fractional porous medium equation

Abstract

This papers deals with a construction and convergence analysis of a finite difference scheme for solving time-fractional porous medium equation. The governing equation exhibits both nonlocal and nonlinear behaviour making the numerical computations challenging. Our strategy is to reduce the problem into a single one-dimensional Volterra integral equation for the self-similar solution and then to apply the discretization. The main difficulty arises due to the non-Lipschitzian behaviour of the equation's nonlinearity. By the analysis of the recurrence relation for the error we are able to prove that there exists a family of finite difference methods that is convergent for a large subset of the parameter space. We illustrate our results with a concrete example of a method based on the midpoint quadrature.

References in corpus (2)

• Ten equivalent definitions of the fractional Laplace operator
• Finite difference method for a Volterra equation with a power-type nonlinearity

Cited by in corpus (6)

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• Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line
• On a discrete composition of the fractional integral and Caputo derivative
• How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations
• Alternative probabilistic representations of Barenblatt-type solutions