A robust computational framework for analyzing the Bloch–Torrey equation of fractional order

Abstract

In the last decade, the analysis of time-space fractional Bloch–Torrey equation has been considered by different studies due to its applications in many fields. Since the analytical solution of this equation is difficult or impossible, numerical solutions can be helpful and sometimes are the only choice. Therefore, in this work, a numerical-based solution is shown by virtue of the Crank–Nicolson weighted shifted Grunwald difference method. The stability, as well as solvability of this method, are also investigated. It is shown that the method for time-space fractional Bloch–Torrey equation is of order $${\mathcal {O}}({\tau ^{2 - \alpha }},{h^2})$$ , where $$0<\alpha <1$$ . Also, $$\tau $$ and h are the time step and space step, respectively. At the end, numerical applications are presented and the thrust of the present study is compared with other sophisticated schemes in the literature. The main advantage of the proposed scheme is that, it is more efficient in terms of accuracy and CPU time in comparing with the existing ones in open literature.