Investigating numerical solutions and ensuring stability in nonlinear pseudo hyperbolic telegraph equations

This research explores the application of an explicit finite difference scheme to solve the nonlinear pseudo-hyperbolic telegraph equation (NPHTE). We develop a novel first-order difference approach tailored to this complex problem. Utilising MATLAB, we implement the explicit finite difference scheme to obtain approximate numerical solutions. Our study demonstrates the stability of this difference scheme for the NPHTE through rigorous mathematical proof. To evaluate the accuracy of our method, we conduct a comparative analysis between analytical and numerical solutions, quantifying the associated errors. We present graphical representations of the solutions to enhance understanding of the problem's behavior. Furthermore, we provide comprehensive simulation results for the NPHTE using our approach, including visual representations and error assessments. This thorough investigation and validation of our methodology significantly contribute to the understanding and application of the NPHTE in mathematical modeling and simulation across various scientific and engineering domains.