A Study of Numerical Solutions for Some Differential, Integral and other Mathematical Equations

Abstract

Mathematical models arising in science and engineering are often represented by differential, integral, or other complex equations that rarely admit closed-form analytical solutions. In such cases , numerical methods provide effective tools to approximate solutions desired accuracy. This study investigates a range of numerical techniques for solving ordinary and partial differential equations, integral equations, and related mathematical problems. Emphasis is placed on finite difference methods, finite element methods, iterative approximation schemes, and quadrature-based approaches. The convergence, stability, and computational efficiency of these methods are examined through theoretical analysis and numerical experiments. Case studies include boundary value problems, Volterra and Fredholm integral equations, and nonlinear algebraic systems derived from discretization. The results highlight the trade-off between accuracy and computational cost while demonstrating the applicability of each method across different problem classes. This work contributes to a broader understanding of numerical analysis by presenting a comparative evaluation of methods and by outlining potential directions for further research in the efficient solution of mathematical equations.