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

[jmk9]

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?

Advice will be much appreciated.

 

[gtoguy97]

Yes, this is an acceptable method to solve a convective heat equation. You have obtained a reasonable solution to the problem, and you can find different solutions by changing the initial conditions, boundary conditions, or parameters of the equation. You may also need to use other methods, such as separation of variables or Fourier series, to obtain more accurate solutions.