Spectral approach to derive the representation formulae for solutions of the wave equation. (English) Zbl 1251.35042

Summary: 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.


35L15 Initial value problems for second-order hyperbolic equations
35C15 Integral representations of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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