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For the second-order linear and nonlinear hyperbolic Telegraph Equation, we develop a comparison study of Adomian decomposition method and Homotopy perturbation approach in this research. The HPM is a powerful mathematical tool despite its simplicity and conciseness. Its comparison with the ADM demonstrates this. We've included some real-world examples to show how capable and dependable the methods are. The sample computations demonstrate how HPM is simpler and requires less manipulation than ADM. In order to verify the mathematical model, 3-dimensional graphical representations of the second-order nonlinear hyperbolic Telegraph Equation are also provided The telegraph equation's 3- and 2-dimensional graphical solutions are represented and analyzed using MATLAB software.