I want to get the fundamental solution for the following 1D nonhomogeneous wave equation:\begin{align}\left\{ \begin{aligned} &\frac{\partial^2u}{\partial^2t}-\frac{\partial^2u}{\partial^2x}-au=0,x\in R^1,t>0,a\ is\ a\ constant.\\& u(x,0)=0,u_t(x,0)=\phi(x) \end{aligned} \right.\end{align} I know the fundamental solution should statisfy the following equation:\begin{align}\left\{ \begin{aligned} &\frac{\partial^2E}{\partial^2t}-\frac{\partial^2E}{\partial^2x}-aE=0,x\in R^1,t>0,a\ is\ a\ constant.\\& E(x,0)=0,E_t(x,0)=\delta(x) \end{aligned} \right.\end{align}Use the fourier transform and then: \begin{align}\left\{ \begin{aligned} &\frac{\partial^2\hat{E}}{\partial^2t}+(\xi^2-a)\hat{E}=0,\xi\in R^1,t>0,a\ is\ a\ constant.\\& \hat{E}(\xi,0)=0,\hat{E}_t(\xi,0)=1 \end{aligned} \right.\end{align} Solve the equation then we got:$$\hat{E}(\xi,t)=\frac{sin(\sqrt{\xi^2-a}t)}{\sqrt{\xi^2-a}}$$Using the inverse Fourier transform:$$E(x,t)=\frac{1}{2\pi}\int_R\frac{sin(\sqrt{\xi^2-a}t)}{\sqrt{\xi^2-a}}e^{-ix\xi}d\xi=\frac{1}{\pi}\int_{0}^{\infty}\frac{sin(\sqrt{\xi^2-a}t)}{\sqrt{\xi^2-a}}cos(x\xi)d\xi$$I don't know how to compute this integration,or maybe I shouldn't use the fourier transform to solve this problem? Appreciate anybody who answer this question first

  • $\begingroup$ "I don't know how to compute this integration" - that may be, but it doesn't mean that the solution is wrong. Giving a solution in the form of an integral is valid, in general you can always compute the integral numerically $\endgroup$ – Yuriy S Apr 17 at 11:43

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