Is there a general method for solving advection-diffusion equations?

[jmk9]

I am trying to solve a transport problem which in its most general form is a diffusion-advection equation with variable coefficients:

<br /> <br /> <br /> \frac{\partial f(x,t)}{\partial t}=a(x,t)\frac{\partial^2 f(x,t)}{\partial x^2}+b(x,t)\frac{\partial f(x,t)}{\partial x}+c(x,t)f(x,t)<br /> <br /> <br />


I am wondering what methods are available for solving such a problem and whether a general solution exists. I have described in another thread the derivation of what appears to be a general solution when we use an initial condition of

<br /> <br /> <br /> f(x,0)=\alpha\delta(x)<br /> <br /> <br />

by manipulating the equation in "Fourier space" I arrive at a solution of the form

<br /> <br /> <br /> f(x,t)=\frac{\alpha e^{-\frac {(x+B(x,t))^{2}} {4A(x,t)} +C(x,t)}}{\sqrt{4\pi A(x,t)}}<br /> <br /> <br />

where

<br /> <br /> <br /> A(x,t)=\int_{0}^{t}a(x,\tau)d\tau<br /> <br /> <br />

<br /> <br /> <br /> B(x,t)=\int_{0}^{t}b(x,\tau)d\tau<br /> <br /> <br />

<br /> <br /> <br /> C(x,t)=\int_{0}^{t}c(x,\tau)d\tau<br /> <br /> <br />


Is this correct? I have a feeling that it is incorrect as I have not found something similar in the literature for the case when a, b and c are functions of x and t. This solution would be correct if they where constants, but I am uncertain about the variable coefficient case.

I really really need help with this one, it's been bothering me for many weeks!

 

[jmk9]

Can anyone comment if this solution is correct? I have a feeling it is not but I need to know why!

thanks in advance

 

[JJacquelin]

I have described in another thread the derivation of what appears to be a general solution...

In what thread ?
In order to express the Fourier transform of the PDE, one need to express the Fourier transforms of a(x,t)d²f(x,t)/dx², of b(x,t)df(x,t)/dx and of c(x,t)f(x,t)
I cannot see how you do that and find the relationship with the F(w,t) without the respective convolution products with the Fourier transforms of a(x,t), b(x,t) and c(x,t).
Of course, not problem if a, b, c are not functions of x.

 

[jmk9]

Thank you for the reply, greatly appreciated. The other thread I am referring to is

https://www.physicsforums.com/showthread.php?t=492001"

What I have done is substitute only

<br /> <br /> <br /> f(x,t)=\frac{1}{2\pi}\int e^{ikx}\tilde{f}(k,t)dk<br /> <br /> <br />

and then proceed from there; the method seemed coherent but I am not sure about it as I haven't encountered this formula in any handbooks or textbooks that I have looked (unless of course a, b and c are constant in which case it is correct).

It is good to know that this is a solution in the case where a, b and c are functions of t only. But I am also curious to know whether a general method exists for solving such equations given that a, b and c are continuous and generally "well behaved".