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) = B x(1-x)\,
t>0,\,\,0<x<\infty\,



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



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