Analytical Method for Solving the One-Dimensional Wave Equation With Moving Boundary - Abstract

Abstract

It is shown that a large class of dynamic systems with non-stationary dimensions can be solved analytically. The corresponding mathematical models of these systems are initial-boundary-value-problems for the wave equation with moving boundary conditions. The method given in a previous paper is generalized and used for solving such problems. It consists of transforming the wave equation for the domain with moving boundary into a form-invariant wave equation for a domain with fixed boundary. The transformation is accomplished through a formal change of variable, and an analytical function F(W) of a complex variable W, permitting a conformal mapping between the two domains. For the transformation to exist, F(W) has to satisfy the condition F*(W) = F(W*). We show that the solution of such systems is obtained in the form of a functional Fourier series,, and the modes are dynamical, each having an instantaneous frequency. The problem of initial conditions is also treated. Furthermore, it is shown that all possible transformation functions permitting the covariant transformation of the wave equation have to satisfy two fundamental relations. As applications, some examples of F(W) and transformation functions are given.