Numerical method for the time-fractional porous medium equation
This work provides a numerical method for solving a challenging class of fractional nonlinear PDEs, but the results are incremental as they extend existing techniques to a specific equation.
The authors developed a convergent finite difference scheme for the time-fractional porous medium equation, which exhibits nonlocal and nonlinear behavior. They proved convergence for a large subset of the parameter space and illustrated the method with a midpoint quadrature example.
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.