Analytical and Numerical Solutions of Higher-Order Boundary Value Problems in Mathematical Physics

Abstract

Mathematical physics also has a wide range of higher-order boundary value problems (BVPs) with applications in beam deformation, fluid mechanics, and quantum mechanics, where the solution is important and requires stability. The research paper unveils an analytic-numerical methodology to the problem of the linear and nonlinear higher-order BVPs. The analytical part is built by means of Variational Iteration Method (VIM) that builds very fast convergent approximations with the help of correction functional and the numerical part is built by the help of the discretization and computational validation with the help of finite difference method (FDM). A convergence analysis is done in detail to determine the reliability and stability of the suggested solution. Multiple benchmark problems using both linear and nonlinear cases, whose exact or reference solutions are known, are used to evaluate the framework. Analysis of error quantitatively proves that the suggested techniques are extremely precise with minimal error of approximation as compared to the traditional ones including shooting and standard finite difference methods. Moreover, the VIM method has a quicker convergence towards smooth solutions, whereas the FDM offers sturdy performance of both stiff and nonlinear systems. The findings affirm that hybrid analytical-numerical strategy is an efficient and reliable solution methodology of higher order different equations. This work offers a scalable basis in generalizing solution methods to more complicated and real-world mathematical physics models.