The Steepest descent method and the Conjugate gradient method to minimize nonlinear functions have been studied in this work. Algorithms are presented and implemented in Matlab software for both methods. However, a comparison has been made between the Steepest descent method and the Conjugate gradient method. The obtained results in Matlab software has time and efficiency aspects. It is shown that the Conjugate gradient method needs fewer iterations and has more efficiency than the Steepest descent method. On the other hand, the Steepest descent method converges a function in less time than the Conjugate gradient method.

The variable telegraph partial differential equation depend on initial boundary value problem has been studied. The coefficient constant time-space telegraph partial differential equation is obtained from the variable telegraph partial differential equation throughout using Cauchy-Euler formula. The first and second order difference schemes were constructed for both of coefficient constant time-space and variable time-space telegraph partial differential equation. Matrix stability method is used to prove stability of difference schemes for the variable and coefficient telegraph partial differential equation. The variable telegraph partial differential equation and the constant coefficient time-space telegraph partial differential equation are compared with the exact solution. Finally, approximation solution has been found for both equations. The error analysis table presents the obtained numerical results.

In this study, Laplace partial differential equations with initial boundary conditions has been studied. A numerical method has been proposed for the solution of the IBVP Laplace equation. The technique based on finite difference methods. The stability of the difference schemes are guaranteed. Approximation solution of the problem was achieved. For testing the accuracy of the proposed method, two different initial boundary value problems are provided. Moreover, a comparison between the numerical solution and analytical solution has been done. MATLAB software implemented for calculation of absolute errors. Illustration graphs presented. It has been demonstrated that the results of the comparison guarantee the accuracy and reliability of the provided method.

This study presents numerical solutions for initial boundary value problems of homogeneous and nonhomogeneous Helmholtz equations using first- and second-order difference schemes. The stability of these methods is rigorously analyzed, ensuring their reliability and convergence for a wide range of problem instances. The proposed schemes’ robustness and applicability are demonstrated through several examples, accompanied by an error analysis table and illustrative graphs that visually represent the accuracy of the solutions obtained. The results confirm the effectiveness and efficiency of the proposed schemes, making them valuable tools for solving Helmholtz equations in practical applications.