A Comparative Study of Partial Differential Equation Solving Methods and Their Applications

In this review, we undertake an in-depth survey of the traditional as well as modern methods used in finding solutions for partial differential equations (henceforth PDEs). We categorise these equations into three main kinds: elliptic, parabolic, and hyperbolic. We also give illustrative examples of these PDEs and discuss the applications of them in a range of fields. This range extends from fluid dynamics (hydrodynamics), as well as thermal (heat) conduction, to quantum mechanics. Our exploration features a number of analyses used in this regard such as variable splitting or defactorising in addition to the transforms invented by Fourier and Laplace. Not only this but also this survey takes in numerical methods ranging from grid-based (finite difference), mesh-based (finite element) to spectral. Also discussed in this paper is a range of special techniques that ranges from the variational techniques, Green's (fundamental solution) functions to perturbation (also known as Asymptotic expansion in addition to sketching the latest developments with respect to computational methods. This review also sheds light on current challenges that confront addressing complicated PDEs especially those nonlinear and multi-variable. In this regard, the paper calls for more research in order to develop more effective methods. The paper maps out the importance of PDE usages in real-life and its potential for more related discoveries in the future particularly with respect to areas such as machine learning and quantum computing.