An Overview of Various Analytical Methods for Solving One Dimensional Wave Equation

Abstract

Partial Differential Equations has many physical applications in various fields such as Hydraulics, Mechanics and Theory of elasticity and so on. They have much wider range of application than Ordinary Differential equations which can model only the simplest physical system. Laplace equation, Navier-Stokes, Wave& Heat equations play a vital role in Fluid Dynamics & Electromagnetism. Schrodinger’s equation constitutes a fundamental part of Quantum Physics. In Partial Differential Equations, one dimensional wave equation is one of the major mathematical problems whose governing equation signifies transverse vibrations of an elastic string. To get the solution of wave equation, various analytical as well as numerical methods are available..In the present article, we take an overview of some of the analytical methods. Separation of Variables is most commonly used method for wave equations. In which, given function is expressed as a product of two single variable functions which reduces the partial differential equation to two ordinary differential equations. Determining the solution of these ordinary differential equations with boundary conditions defines the general solution of wave equation. D’Alembert’s method is transforming the partial differential equation by introducing two new independent variables corresponding to an explicit solution of wave equation along with boundary conditions. In Laplace transform method, the transform is used with respect to one of the variables. This converts to an ordinary differential equation, which gives the solution by boundary conditions. Another approach to find the solution using finite Fourier sine transform is also cited here.