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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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