# PDE:Fourier Transforms

solution Find the Fourier transform of $\displaystyle f(t) = e^{-|t|}\,$

solution Find the Fourier transform of $\displaystyle f(t) = \begin{cases}1&|t|<1\\0&|t|>1\end{cases}\,$

 solution $\displaystyle u_t=ku_{xx}\,$ $\displaystyle u(0,t) = 0\,$ $\displaystyle u(x,0) = f(x)\,$ $\displaystyle t>0,\,\,0

 solution $\displaystyle u_{xx}+u_{yy}=0\,$ $\displaystyle u(0,y) = 0\,$ $\displaystyle u(1,y) = 0\,$ $\displaystyle u(x,0) = 0\,$ $\displaystyle u(x,1) = B x(1-x)\,$ $\displaystyle t>0,\,\,0

 solution $\displaystyle u_t=-u_{xxxx}\,$ $\displaystyle u(x,0) = f(x)\,$ $\displaystyle t>0,\,\,x\isin\mathbb{R},$

 solution $\displaystyle u_{tt}=c^2\,u_{xx}\,$ $\displaystyle u(x,0) = f(x)\,$ $\displaystyle u_t(x,0)=g(x)\,$ $\displaystyle t>0,\,\,x\isin\mathbb{R},$

 solution $\displaystyle u_{xx}+u_{yy}+u_{zz}=0\,$ $\displaystyle u(x,y,0) = f(x,y)\,$ Auxiliary condition: $\displaystyle u$ is bounded. $\displaystyle t>0,\,\,x,y\isin\mathbb{R},\,\,\,z>0\,$

• $\displaystyle u(x,y,z) = \int\!\!\!\int_\Re e^{i \lambda x + i \mu y} B(\lambda,\mu) e^{-\sqrt{\lambda^2+\mu^2}\,z}\,d\lambda d\mu\,$

 [Quick Answer] Write the form of the solution: $\displaystyle u_{tt}=c^2(u_{xx}+u_{yy})\,$ $\displaystyle u(x,0,t) = g(x)\,$ $\displaystyle u(0,y,t) = h(y)\,$ $\displaystyle u(x,y,0) = 0\,$ $\displaystyle u_t(x,y,0) = f(x,y)\,$ $\displaystyle 00\,$

• $\displaystyle u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \sin(\mu y)\,d\lambda\,d\mu\,$

 [Quick Answer] Write the form of the solution: $\displaystyle u_{tt}=c^2(u_{xx}+u_{yy})\,$ $\displaystyle u_y(x,0,t) = g(x)\,$ $\displaystyle u(0,y,t) = h(y)\,$ $\displaystyle u(x,y,0) = 0\,$ $\displaystyle u_t(x,y,0) = f(x,y)\,$ $\displaystyle 00\,$

• $\displaystyle u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \cos(\mu y)\,d\lambda\,d\mu\,$

 solution$\displaystyle u_{tt}=c^2(u_{xx}+u_{yy})\,$ $\displaystyle u_y(x,0,t) = g(x)\,$ $\displaystyle u(0,y,t) = h(y)\,$ $\displaystyle u(x,y,0) = 0\,$ $\displaystyle u_t(x,y,0) = f(x,y)\,$ $\displaystyle 00\,$

• $\displaystyle u(x,y,t) = \int_0^\infty \int_0^\infty U(\lambda,\mu,t) \sin(\lambda x) \cos(\mu y)\,d\lambda\,d\mu\,$