Łukasz Płociniczak

We consider a one-dimensional nonlinear diffusion equation with nonlocal time, i.e. the temporal derivative is calculated via the Riemann-Liouville operator. This problem set on a half-line models moisture percolation in certain porous media such as construction materials.

The efficient and convergent numerical method can be constructed as follows. We cast our equation into the self-similar form with the use of the Erdelyi-Kober operator. Then, by a suitable transformation the free-boundary problem is transformed into the initial-value one. Lastly, the resulting integro-differential equation can be written as a nonlinear Volterra equation. In this way, a nonlinear PDE with free boundary can be solved by finding a solution of an ordinary Volterra equation. This quickens the calculations tremendously.

We also prove that a certain family of numerical schemes is convergent to the solution of the aforementioned Volterra equation. The main difficulty lies in the non-Lipschitzian character of the governing nonlinearity. In that case we cannot use the classical theory and have to proceed in other ways. In the talk we will describe the details of the proof.