Finite element method for drifted space fractional tempered diffusion equation

Off-late many models in viscoelasticity, signal processing or anomalous diffusion equations are formulated in fractional calculus. Tempered fractional calculus is the generalization of fractional calculus and in the last few years several important partial differential equations occurring in different field of science have been reconsidered in this terms like diffusion wave equations, Schr\(\ddot{o}\)dinger equation and so on. In the present paper, a time dependent tempered fractional diffusion equation of order \(\gamma \in (0,1)\) with forcing function is considered. Existence, uniqueness, stability, and regularity of the solution has been proved. Crank–Nicolson discretization is used in the time direction. By implementing finite element approximation a priori space–time estimate has been derived and we proved that the convergent order is \(\mathcal {O}(h^2+\varDelta t ^2)\) where h is the space step size and \(\varDelta t\) is the time difference. A couple of numerical examples have been presented to confirm the accuracy of theoretical results. Finally, we conclude that the studied method is useful for solving tempered fractional diffusion equations.