Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line
arXiv:2106.05138 · doi:10.1016/j.amc.2022.127033
Abstract
Many physical, biological, and economical systems exhibit various memory effects due to which their present state depends on the history of the whole evolution. Combined with the nonlinearity of the process these phenomena pose serious difficulties in both analytical and numerical treatment. We investigate a time-fractional porous medium equation that has proved to be important in many applications, notably in hydrology and material sciences. We show that solutions of the free boundary Dirichlet, Neumann, and Robin problems on the half-line satisfy a Volterra integral equation with a non-Lipschitz nonlinearity. Based on this result we prove existence, uniqueness, and construct a family of numerical methods that solve these equations outperforming the usual finite difference approach. Moreover, we prove the convergence of these methods and support the theory with several numerical examples.
arXiv commentThis is an accepted version of the manuscript published in Applied Mathematics and Computation
References in corpus (11)
- An extension problem related to the fractional Laplacian
- On an explicit finite difference method for fractional diffusion equations
- A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems
- Non-diffusive transport in plasma turbulence: a fractional diffusion approach
- Analysis of -Galerkin FEMs for time-fractional nonlinear parabolic problems
- Nonlinear porous medium flow with fractional potential pressure
- Nonlocal porous medium equation: Barenblatt profiles and other weak solutions
- Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications
- Numerical method for the time-fractional porous medium equation
- Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting
- Finite difference method for a Volterra equation with a power-type nonlinearity