Is this an acceptable method to solve a convectihe heat equation???

I am trying to solve the following PDE:

$$\frac{\partial f(x,t)}{\partial t}=a(x,t)\frac{\partial^2 f(x,t)}{\partial x^2}+b(x,t)\frac{\partial f(x,t)}{\partial x}+c(x,t)f(x,t)$$

where a, b and c are known functions of x and t with an impulse function initial condition

$$f(x,0)=\alpha\delta(x)$$

This actually describes a specific mass transport problem. I would like to know whether the solution I obtain, described below, is at all reasonable.

By letting

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

($$\tilde{f}$$ being the Fourier transform of $$f$$) and substituting to the above, one obtains

$$\int e^{ikx}\frac{\partial \tilde{f}(k,t)}{\partial t}dk=-\int a(x,t)e^{ikx}k^{2}\tilde{f}(k,t)dk+i\int b(x,t)e^{ikx}k\tilde{f}(k,t)dk+\int c(x,t)e^{ikx}\tilde{f}(k,t)dk$$

which is satisfied if

$$\frac{\partial \tilde{f}}{\partial t}=-a(x,t)k^{2}\tilde{f}+ib(x,t)k\tilde{f}+c(x,t)\tilde{f}$$

Solving this 1st order ODE and subsequently taking the inverse Fourier transform one gets:

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

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? Or have I done something horribly wrong? Is it a reasonable solution to the transport problem that I am examining? How can I find different solutions to this problem?