An efficient fourth order Hermite spline collocation method for time fractional diffusion equation describing anomalous diffusion in two space variables

Anomalous diffusion of particles in fluids is better described by the fractional diffusion models. A robust hybrid numerical algorithm for a two-dimensional time fractional diffusion equation with the source term is presented. The well-known L1 scheme is considered for semi-discretization of the diffusion equation. To interpolate the semi-discretized equation, orthogonal collocation with bi-quintic Hermite splines as the basis is chosen for the smooth solution. Quintic Hermite splines interpolate the solution as well as its first and second order derivatives. The technique reduces the proposed problem to an algebraic system of equations. Stability analysis of the implicit scheme is studied using \(\mathcal {\tilde{H}}_{1}^m\) ${\tilde{\mathcal{H}}}_1^m$ -norm defined in Sobolev space. The optimal order of convergence is found to be of order \(O(h^4)\) $O(h^4)$ in spatial direction and is of order \(O(\Delta t)^{2-\alpha }\) $O{(\mathrm{\Delta }t)}^{2-\alpha }$ in the temporal direction where h is the step size in space direction and \(\Delta t\) $\mathrm{\Delta }t$ is the step size in time direction and \(\alpha \) $\alpha$ is the fractional order of the derivative. Numerical illustrations have been presented to discuss the applicability of the proposed hybrid numerical technique to the problems having fractional order derivative.