锥上的非齐次wave equation
Non-homogeneous wave equation on a cone
摘要：
证明了波动方程$\left(\partial_{tt} - c^2 Δ_x\right) u(x,t) = e^{-t} f(x,t)$有唯一解，如果$u$及其在$x$中的偏导数在锥上的$L^2(e^{-t})$中，并且解可以用锥上正交多项式的傅里叶级数显式给出.这为圆锥上非齐次波动方程的边值问题提供了一个特解，该特解可以与圆锥上齐次波动方程的解相结合以获得完整解。
英文摘要：
The wave equation $\left(\partial_{tt} - c^2 Δ_x\right) u(x,t) = e^{-t} f(x,t)$ 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.
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