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<x<\infty \,




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<x<\infty \,



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)\,

0<x<\infty ,\,\,\,\,\,0<y<\infty ,\,\,\,\,\,t>0\,



  • 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)\,

0<x<\infty ,\,\,\,\,\,0<y<\infty ,\,\,\,\,\,t>0\,



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



solutionu_{{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)\,

0<x<\infty ,\,\,\,\,\,0<y<\infty ,\,\,\,\,\,t>0\,



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

Partial Differential Equations


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