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The fractional sub-diffusion equation, which is obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order ϑ with 0<ϑ<1$0 < \vartheta < 1$, has appeared in numerous complex system. In this paper, we suggest an efficient and accurate iterative method based on coupling the variational iteration method (VIM) with an auxiliary parameter for solving three-dimensional FDEs described in the Riemann–Liouville sense. However, though the standard VIM is often invalid on large domains, the VIM with an auxiliary parameter is highly efficient in approximating the solution of complex systems even on large domains. The procedure of obtaining an optimal auxiliary parameter is illustrated through some examples, while the theoretical analysis confirms the convergence of the proposed method. Comparing the results of standard VIM and modified VIM by the auxiliary parameter confirms the effectiveness of using the new technique on the magnitude of the convergence region.

An efficient iterative approach for three-dimensional modified anomalous fractional sub-diffusion equations on a large domain