Fractional view of heat‐like equations via the Elzaki transform in the settings of the Mittag–Leffler function

What is it about?

In this article, the Elzaki homotopy perturbation transform method (EHPTM) is profusely employed to discover the approximate solutions of fractional-order (FO) heat-like equations. To show this, we first establish the Elzaki transform in the context of the Atangana–Baleanu fractional derivative in the Caputo sense (ABC) and then extend it to heat-like equations. Our suggested approach has been reinforced by convergence and error analysis. The validity of the novel technique is tested with the aid of some illustrative examples. Comparative analysis has been established for both fractional and integer-order solutions. EHPTM is considered to be an appropriate and convenient approach for solv- ing FO time-dependent linear and nonlinear partial differential problems. Plots and tables are being used to reveal the findings. The relatively high validity and reliability of the present approach are also reflected by the comparative solution analysis by means of statistical analysis.

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Photo by Sean Sinclair on Unsplash

Photo by Sean Sinclair on Unsplash

Why is it important?

The current research work is based on a fractional view of the evaluation of heat-like equations by employing an effective approach by taking into account statistical tests. To begin, we developed the ET of the ABC operator and implemented it to solve the heat-like equations. Taking into consideration a new iterative method, EHPTM, the approximate solutions of certain heat-like equations for both fractional and integer-order are formulated. EHPTM yields small errors and fast convergence. Throughout the analysis, the acquired solutions are depicted in the form of plots and tables with varying parameters of time and space-scale variables, which highlight the behavior of the complicated problem under considera- tion. Moreover, EHPTM has the ability to deal with nonlinearites with homotopies and can be employed to manage the convergence of the series solution. The findings show that the outcomes acquired by means of EHPTM are more general than the natural transform decomposition method, Laplace homotopy perturbation transform method, Laplace iterative transform method, and the iterative Sumudu transform method. The EHPTM interprets and supervises the series solu- tion, which swiftly converges to the exact solution in a limited time frame. Finally, we can deduce from the acquired outcomes that the suggested strategies are highly efficient and may be used to investigate a wide range of nonlinear FO mathematical models in order to comprehend the nature of complicated occurrences.