PDE:Fourier Transforms

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

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

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

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

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

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

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

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

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

• $u(x,y,t)=\int _{0}^{\infty }\int _{0}^{\infty }U(\lambda ,\mu ,t)\sin(\lambda x)\sin(\mu y)\,d\lambda \,d\mu \,$

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

• $u(x,y,t)=\int _{0}^{\infty }\int _{0}^{\infty }U(\lambda ,\mu ,t)\sin(\lambda x)\cos(\mu y)\,d\lambda \,d\mu \,$

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

• $u(x,y,t)=\int _{0}^{\infty }\int _{0}^{\infty }U(\lambda ,\mu ,t)\sin(\lambda x)\cos(\mu y)\,d\lambda \,d\mu \,$