Spectral Approach to Derive the Representation Formulae for Solutions of the Wave Equation

Gusein Sh Guseinov

doi:10.1155/2012/761248

Abstract

Using spectral properties of the Laplace operator and some structural formula for rapidly decreasing functions of the Laplace operator, we offer a novel method to derive explicit formulae for solutions to the Cauchy problem for classical wave equation in arbitrary dimensions. Among them are the well‐known d′Alembert, Poisson, and Kirchhoff representation formulae in low space dimensions.

Highlights

The wave equation for a function u x1, . . . , xn, t the time t is given by u x, t of n space variables x1, . . . , xn and ∂2u ∂t2 Δu, Δ ∂2 ∂x12 ∂2 ∂xn21.2 is the Laplacian
Using spectral properties of the Laplace operator and some structural formula for rapidly decreasing functions of the Laplace operator, we offer a novel method to derive explicit formulae for solutions to the Cauchy problem for classical wave equation in arbitrary dimensions
In the Cauchy Journal of Applied Mathematics problem initial value problem one asks for a solution u x, t of 1.1 defined for x ∈ Rn, t ≥ 0 that satisfies 1.1 for x ∈ Rn, t > 0 and the initial conditions u x, 0 φ x

Summary

Introduction

Formulae that give the solution of the Cauchy problem in explicit form are of great significance. If n 2 and φ ∈ C3 R2 , ψ ∈ C2 R2 , the solution of problem 1.1 , 1.3 is given by Poisson’s formula

Results

Conclusion