You should starting by taking the Fourier transform of with respecto to , this is and apply the properties of the Fourier transform of the derivative.
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 t 0
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
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)
and eventually you arrive at a First Order Ordinary Differential Equation for treating as constant. In the Homogeneus problem . In your problem this will not be true.
You solve both problems, one for each domain of and finally paste both solutions by continuity.
The point of using Foutier Transform is that your Heat Problem became an EDO, way easier to solve.