Well-posedness of time-fractional advection-diffusion-reaction equations

Numerical Solution of Advection-Diffusion-Reaction Equations

Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations

We establish the well-posedness of an hp-version time-stepping discontinuous Galerkin (DG) method for the numerical solution of fractional super-diffusion evolution problems. In particular, we prove the existence and uniqueness of approximate solutions for generic hp-version finite element spaces featuring non-uniform time-steps and variable approximation degrees. We then derive new hp-version ...

Positivity-preserving nonstandard finite difference Schemes for simulation of advection-diffusion reaction equations

Systems in which reaction terms are coupled to diffusion and advection transports arise in a wide range of chemical engineering applications, physics, biology and environmental. In these cases, the components of the unknown can denote concentrations or population sizes which represent quantities and they need to remain positive. Classical finite difference schemes may produce numerical drawback...

Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations.

We have revisited the problem of anomalously diffusing species, modeled at the mesoscopic level using continuous time random walks, to include linear reaction dynamics. If a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction-diffusion equation with a fractional order temporal derivative o...

An explicit high order method for fractional advection diffusion equations

We propose a high order explicit finite difference method for fractional advection diffusion equations. These equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order α with 1 < α ≤ 2. This operator is defined by a combination of the left and right Riemann–Liouville fractional derivatives. We stu...