Hello
I have one problem with solve heat equation with integral Fourier,
This is problem
du/dt-d(2)u/dx^2= 2t when 0<x<, 1 when x>0
u(x,0)=2x when o<x<1 , -1 when x>1
u(0,t)=t-1 when t0
I write u(x,t)=F(x)G(t) and try to find answer!
but I can not find a(w) and b(w) and also G(t)
I cannot understand what it is you are trying to solve. It is very hard to read.Originally Posted by Hamed
Hello
I have one problem with solve heat equation with integral Fourier,
This is problem
du/dt-d(2)u/dx^2= 2t when 0<x<
u(x,0)=2x when o<x<1 , -1 when x>1
u(0,t)=t-1 when t
I write u(x,t)=F(x)G(t) and try to find answer!
but I can not find a(w) and b(w) and also G(t)
Do you mean
Where
So I am assuming that the function is not defined for x < 0. If that is the cases you should use the Fourier sine transform since you do not know the value of the normal derivative at x=0. Also note that the transform of the right hand side will be a distribution as the function is not in
The procedure is similar. You can divide the problem in two parts, one for each domain of.
When the problem is homogeneous yo take the Fourier transform of the equation, wich means just (using TheEmptySet notation for the right hand side)
Heat Equation
=\mathcal{F}\left[\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2 }\right](\xi,t)=\mathcal{F}[g](\xi,t)](http://latex.codecogs.com/png.latex?](\xi,t)=\mathcal{F}\left[\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2 }\right](\xi,t)=\mathcal{F}[g](\xi,t))
and eventually you arrive at a First Order Ordinary Differential Equation fortreating
as constant. In the Homogeneus problem
. In your problem this will not be true.
You solve both problems, one for each domain of
The point of using Foutier Transform is that your Heat Problem became an EDO, way easier to solve.
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