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

 P: 11 I am trying to solve a transport problem which in its most general form is a diffusion-advection equation with variable coefficients: $$\frac{\partial f(x,t)}{\partial t}=a(x,t)\frac{\partial^2 f(x,t)}{\partial x^2}+b(x,t)\frac{\partial f(x,t)}{\partial x}+c(x,t)f(x,t)$$ 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 $$f(x,0)=\alpha\delta(x)$$ by manipulating the equation in "Fourier space" I arrive at a solution of the form $$f(x,t)=\frac{\alpha e^{-\frac {(x+B(x,t))^{2}} {4A(x,t)} +C(x,t)}}{\sqrt{4\pi A(x,t)}}$$ where $$A(x,t)=\int_{0}^{t}a(x,\tau)d\tau$$ $$B(x,t)=\int_{0}^{t}b(x,\tau)d\tau$$ $$C(x,t)=\int_{0}^{t}c(x,\tau)d\tau$$ 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!!!
 P: 11 Is there a general method for solving advection-diffusion equations? Thank you for the reply, greatly appreciated. The other thread I am referring to is http://www.physicsforums.com/showthread.php?t=492001 What I have done is substitute only $$f(x,t)=\frac{1}{2\pi}\int e^{ikx}\tilde{f}(k,t)dk$$ 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".