Meshless numerical solution for a fractional PDE in the electroanalytical chemistry

In this paper a numerical technique based on a meshless method is proposed for solving a fractional PDE in the electroanalytical chemistry. The fractional derivative is described in the Riemann-Liouville sense with order $\gamma$. Firstly, we obtain a time discrete scheme based on a finite difference scheme, then we use the meshless collocation method, to approximate the spatial derivatives and obtain a full discrete scheme. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method. We show convergence order of the time discrete scheme is $\mathcal{O}(\tau^{\gamma})$ in which $\tau$ is the time-step size. We solve the mentioned equation on irregular domains. Numerical examples confirm the efficiency and method, to approximate the spatial derivatives and obtain a full discrete scheme. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy accuracy of the proposed scheme