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In the present research paper, an iterative approach named the iterative Shehu transform method is implemented to solve time-fractional hyperbolic telegraph equations in one, two, and three dimensions, respectively. These equations are the prominent ones in the field of physics and in some other significant problems. The efficacy and authenticity of the proposed method are tested using a comparison of approximated and exact results in graphical form. Both 2D and 3D plots are provided to affirm the compatibility of approximated-exact results. The iterative Shehu transform method is a reliable and efficient tool to provide approximated and exact results to a vast class of ODEs, PDEs, and fractional PDEs in a simplified way, without any discretization or linearization, and is free of errors. A convergence analysis is also provided in this research.

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In the current paper, a review of the Homotopy perturbation method is offered thoroughly to fetch the analytical solution of coupled 1D non-linear Burgers’ equation. The exact solution of the coupled 1D Burgers’ equation is attained in the system of a power series, (convergent in nature). A suitable optimal of the initial condition leads towards the vital exact solution after some iterative phases.

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<abstract> <p>Present research deals with the time-fractional Schr<italic>ö</italic>dinger equations aiming for the analytical solution via Shehu Transform based Adomian Decomposition Method [STADM]. Three types of time-fractional Schr<italic>ö</italic>dinger equations are tackled in the present research. Shehu transform ADM is incorporated to solve the time-fractional PDE along with the fractional derivative in the Caputo sense. The developed technique is easy to implement for fetching an analytical solution. No discretization or numerical program development is demanded. The present scheme will surely help to find the analytical solution to some complex-natured fractional PDEs as well as integro-differential equations. Convergence of the proposed method is also mentioned.</p> </abstract>

In the present study, time-fractional Schrödinger equations are dealt with for the analytical solution using an integral transform named Shehu Transform. Three kinds of time-fractional Schrödinger equations are discussed in the present study. Shehu transform is utilized to reduce the time-fractional PDE along with the fractional derivative in the Caputo sense. The present method is easy to implement in the search for an analytical solution. As no discretization or numerical program is required, the present scheme will surely be helpful in finding the analytical solution to some complex-natured fractional PDEs.

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