ABSTRACT

The wave equation $\left({\mathrm{\partial }}_{tt}-c^2{\mathrm{\Delta }}_x\right)u(x,t)=e^{-t}f(x,t)$ in the cone $\left\{\right(x,t):\parallel x\parallel \le t,x\in {\mathbb{R}}^d,t\in {\mathbb{R}}_{+}\}$ is shown to have a unique solution if u and its partial derivatives in x are in $L^2\left(e^{-t}\right)$ 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.

中文翻译：

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圆锥上的非齐次波动方程

摘要

波动方程 $\left({\mathrm{\partial }}_{吨吨}-C^2{\mathrm{\Delta }}_X\right)你(X,吨)={\mathrm{电子}}^{-吨}F(X,吨)$ 在锥体中 $\left\{\right(X,吨)：\parallel X\parallel \le 吨,X\in {\mathbb{电阻}}^d,吨\in {\mathbb{电阻}}_{+}\}$如果u及其在x中的偏导数在${升}^2\left({\mathrm{电子}}^{-吨}\right)$在锥体上，解可以在锥体上的正交多项式的傅立叶级数中明确给出。这为锥上非齐次波动方程的边值问题提供了特解，可以结合锥上齐次波动方程的解得到全解。