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This article will give the residual power series method (RPSM) for solving pseudo hyperbolic partial differential equations with nonlocal conditions, RPSM is essentially based on general formula of Taylor series with residual error function. A new analytical solution is investigated. The analytical solution is designed to find the approximation solutions by RPSM and compare the obtained results from the current method with the exact solution that detects the precision, reliability, and rapid convergence of the proposed method. Finally at different times through the graphical representation of obtained results are given.

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The Pseudo-Hyperbolic Telegraph partial differential equation (PHTPDE) based on the Caputo fractional derivative is investigated in this paper. The modified double Laplace transform method (MDLTM) is constructed for the proposed model. The MDLTM was used to obtain the analytic solution for the pseudo-hyperbolic telegraph equation of fractional order defined by the Caputo derivative. The proposed method is a highly effective analytical method for the fractional-order pseudo-hyperbolic telegraph equation. A test problem was presented as an example. Based on the results, it is clear that this method is more convenient and produces an analytic solution in fewer steps than other methods that require more steps to have an identical analytical solution. This paper claims to provide an analytic solution to the fractional order pseudohyperbolic telegraph equation using the MDLTM. An analytical solution directs to an exact, closed-form solution that can be expressed in mathematical functions or known operations. Obtaining analytic solutions to PDEs is often challenging, especially for fractional order equations, making this achievement noteworthy.