Analytical solution of fuzzy heat problem in two-dimensional case under Caputo-type fractional derivative.

This work aims to investigate the analytical solution of a two-dimensional fuzzy fractional-ordered heat equation that includes an external diffusion source factor. We develop the Sawi homotopy perturbation transform scheme (SHPTS) by merging the Sawi transform and the homotopy perturbation scheme. The fractional derivatives are examined in Caputo sense. The novelty and innovation of this study originate from the fact that this technique has never been tested for two-dimensional fuzzy fractional ordered heat problems. We presented two distinguished examples to validate our scheme, and the solutions are in fuzzy form. We also exhibit contour and surface plots for the lower and upper bound solutions of two-dimensional fuzzy fractional-ordered heat problems. The results show that this approach works quite well for resolving fuzzy fractional situations.

Approximate solution of Newell-Whitehead-Segel model with time-fractional derivative.

In the current analysis, we developed a significant approach for deriving the approximate solution of the Newell-Whitehead-Segel model with Caputo derivatives. This scheme is developed based on Sumudu transform and the residual power series method (RPSM) that generates the solution in the form of a series. First, we apply the Sumudu transform to decompose the fractional order and obtain a recurrence relation. Secondly, we utilize the RPSM to the recalescence relation and then we can derive the series solution with successive iterations using the initial conditions. We observe that this approach demonstrates a high accuracy and validity to the proposed fractional model. In our developed scheme, we do not face any huge calculation and restriction of elements that diverse the significance of the results. In addition, we display 2D and 3D graphical visuals to show the physical nature of the fractional model.