A Comprehensive Literature Review: Numerical Solutions for Differential, Integral and Mathematical Equations

Abstract

In this paper, we systematically review the literature on numerical methods for solving differential equations, integral equations and integro-differential equations. This paper investigates classical numerical techniques, including finite difference methods, finite element methods, spectral methods, collocation methods, and wavelet-based approaches, emphasising the mathematical fundamentals, the convergence behaviour, and computational efficiency. Modern developments, especially on spectral collocation methods, wavelet approaches, fractional calculus techniques and adaptive numeric schemes for boundary value and fractional-order problems are emphasized.

It also explores new machine learning–based methods, especially Physics-Informed Neural Networks (PINNs), Deep Operator Networks, and hybrid computational methods that merge numerical analysis with artificial intelligence. The resulting methods have received much attention since they show better flexibility, faster computation, and more powerful for solving high-dimensional and nonlinear partial differential equations. It also covers recent progress on tensor-network methods and quantum-enhanced PINNs, before moving to require more involved scientific and engineering applications and discussing quantum-enhanced methods based on stochastic differential equation solvers.

This study further discusses the advantages and disadvantages of the different numerical methods based on the accuracy, convergence process, stability, computational complexity, and suggestions for applications to real problems. This review suggests that the computational mathematics of the future is at an exciting juncture as it increasingly shifts towards the use of hybrid and adaptive frameworks combining classical numerical methods with machine learning techniques targeted to efficiently solving large-scale, multi-physics and data-driven problems. Numerical solution of equations arising in science and engineering: Outlook and directions AbstractThis paper provides an informative guide to what is happening now and what is expected to happen in the field of numerical solutions of mathematical equations in the future.

Keywords: Numerical Methods, Differential Equations,Spectral Collocation Methods, Finite Element Method (FEM), Integro-Differential Equations, Physics-Informed Neural Networks (PINNs) ,Fractional Calculus