Семинар Отдела динамических систем
ИММ УрО РАН
26.10.2022

Towards some advances in the numerical analysis for nonlinear fractional order partial differential equations and their applications

Hendy A.

We are dealing with some advances in the numerical analysis of reaction diffusion equations and their applications especially with time Caputo fractional order or Riesz space fractional derivatives in space or both included. These advances starting from the ability to construct numerical methods that can inherit energy preserving conservation or dissipation on the discrete scale as in its continuous style. The other advance is to construct combined numerical schemes (Finite difference/Galerkin Legendre spectral) that can cope better near the initial singularity (t = 0) for the Caputo time fractional derivatives. Also, the possibility of constructing a nonuniform Rothe scheme that can be helpful in the reconstruction of source terms of time Caputo fractional diffusion problems. We finally fill a gap of the nonexistence of a discrete fractional Grönwall-type inequality to be applied in the numerical analysis of high order finite difference schemes of Alikhanov type for multiterm time Caputo fractional diffusion problems with delay.