A Method for Obtaining Exact Solutions to Partial Differential Equations with Variable Coefficients

Abstract

This paper describes techniques that can be used to transform PDEs with variable coefficients into equations with constant coefficients. The techniques are illustrated by calculating shear flows over quite general surfaces, by solving the signaling problem for diffusive processes in inhomogeneous materials, and by solving the signaling problem for acoustical waves when the sound speed varies with distance. The techniques may also be used to solve equations governing processes in inhomogeneous, anisotropic materials.

Resonant oscillations of an inhomogeneous gas between concentric spheres

“…The effect of the geometry is contained in the term (L + x) 2 , so that specifying a form of s(R) defines a particular class of gas stratification through c(R). This can be done in many ways (see Varley & Seymour 1988), but here one simple form is selected where all integrations at O(3) can be performed and the effect of the stratification can be calculated explicitly:…”

Resonant hydraulic sloshing in a tank of variable depth

“…Such solutions are implicit in the work of Varley and Seymour [10], and we exploit these here. It can readily be checked that these solutions lead to incommensurate eigenvalues.…”

“…The fundamental normal mode is a solution of a wave equation with variable coefficients determined by the depth profile. Using a technique due to Varley and Seymour [10] we find a class of depth profiles for which the wave equation and associated eigenvalue problem can be solved explicitly, yielding non-commensurate eigenvalues. Then the solution to the resonant problem, found by a Duffing-type expansion, is continuous with an amplitude that depends on the detuning from resonance.…”

Diffusion problems in bonded nonhomogeneous materials with an interface cut

“…It is clear that solutions for ti(y) with grcatcr number of arbitrary constants may be obtained from (3.5) by considering cases for which V > 1 [10]. Technically the greater number of arbitrary constants would provide a better representation of the given shear modulus.…”