Non-homogeneous wave equation on a cone.

The wave equation ∂ t t − c 2 Δ x u (x , t) = e − t f (x , t) in the cone { (x , t) : ∥ x ∥ ≤ t , x ∈ R d , t ∈ R + } is shown to have a unique solution if u and its partial derivatives in x are in L 2 (e − t) on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the cone, which can be combined with a solution to the homogeneous wave equation in the cone to obtain the full solution. [ABSTRACT FROM AUTHOR]