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In this paper, we aim to present a powerful approach for the approximate results of multi-dimensional diffusion problems with time-fractional derivatives. The fractional order is considered in the view of the Caputo fractional derivative. In this analysis, we develop the idea of the Yang homotopy perturbation transform method (YHPTM), which is the combination of the Yang transform (YT) and the homotopy perturbation method (HPM). This robust scheme generates the solution in a series form that converges to the exact results after a few iterations. We show the graphical visuals in two-dimensional and three-dimensional to provide the accuracy of our developed scheme. Furthermore, we compute the graphical error to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem. [ABSTRACT FROM AUTHOR]

This paper presents the concept of a new strategy that examines the analytical solution of the Harry Dym equation (HDEq) with fractional derivative in the Caputo sense. This new approach is called the Mohand homotopy perturbation transform scheme (MHPTS) which is constructed on the basis of the Mohand transform (MT) and the homotopy perturbation method (HPM). The implementation of MT produces the recurrence relation without any assumption and hypothesis theory whereas HPM is additionally used to overcome the nonlinearity in differential problems. Our primary focus is to handle the error analysis in the recurrence relation and generates the solution in the order of series. These obtained results yield the exact solution very rapidly due to its fast convergence. Some graphical representations are demonstrated to show the high efficiency and performance of this approach. [ABSTRACT FROM AUTHOR]