Solving Diffusion equation with Convection

[PiRho31416]

The problem is as follows:
$$\frac{\partial u}{\partial t}=k\frac{\partial^{2}u}{\partial x^{2}}+c\frac{\partial u}{\partial x},$$

$$-\infty<x<\infty$$

$$u(x,0)=f(x)$$

Fourier Transform is defined as:

$$F(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(x)e^{i\omega x}dx$$

So, I took the Fourier Transform which brought me to

$$\frac{\partial F}{\partial t}=-\omega^{2}F-ci\omega F=-F(\omega^{2}+ci\omega)$$

Solving the first order differential equation brought me to

$$F(\omega)=e^{-\frac{1}{6}(3ic+2\omega)\omega^{2}}$$

When I try to integrate using the inverse Fourier transform

$$f(x)=\int_{-\infty}^{\infty}F(\omega)e^{-i\omega x}\, d\omega$$

I get stuck. Did I do the steps right?

 

[JJacquelin]

As usual, it is not dificult to find particular solutions and more general solution of the PDE.
The real difficulty is encountered when we have to fit the general solution to the boundary conditions so that the solution of the problem should be derived.
In the present wording, the bondary condition is specified by a function f(x) which is not explicit. So, we cannot say if the problem can be analytically solved.

 

[jmk9]

I had a similar equation which I would like to solve but with variable coefficients; I have produced a solution which I feel is wrong and have posted it - twice - but nobody seemed to be willing to offer their insight.

Anyway, significant progress can be made for your problem with Fourier transforms, since c and k are constants.

Given the original equation

$$\frac{\partial{u(x,t)}}{\partial{t}}=k\frac{\partial^2{u(x,t)}}{\partial{x^2}}+c\frac{\partial{u(x,t)}}{\partial{x}}$$

taking the f.t. in the space dimension and canceling out any shared factors you obtain

$$\frac{\partial{U(\omega,t)}}{\partial{t}}=-k\omega^2U(\omega,t)+ic\omega U(\omega,t)$$

which when solved results to

$$U(\omega,t)=U(\omega,0)e^{-(k\omega^2-ic\omega)t}$$

so, assuming you can determine

$$U(\omega,0)=\int{u(x,0)e^{-i\omega x}dx}$$

the solution will be given by

$$u(x,t)=\frac{1}{2\pi}\int{U(\omega,0) e^{-kt(\omega^2-\frac{i(c+x)\omega}{k})} d\omega}$$

I believe this is correct. What is your f(x)? If you can expand U(w,0) in powers of w then you can complete the square in the exponential and solve the resulting gaussian integrals.
If anyone can spot any horrible mistakes in the above, please shout

 

[JJacquelin]

If anyone can spot any horrible mistakes in the above, please shout

Only a very, very small shout for a mistake at the end.

 

[blue_raver22]

I cannot provide a definitive response without fully understanding the context and assumptions of the problem. However, I can offer some general thoughts and suggestions.

Firstly, it seems like you have correctly applied the Fourier transform to the diffusion equation with convection. This is a common approach in solving partial differential equations, as it can simplify the equation and make it more manageable.

However, it is important to note that the Fourier transform is only applicable to linear problems, and it may not be appropriate for non-linear diffusion equations. Additionally, the boundary conditions and initial conditions must also be taken into consideration when applying the Fourier transform. It is possible that these factors may be causing difficulties in your integration step.

If you are stuck on the integration step, it may be helpful to consult with a colleague or a textbook for guidance. You can also try using numerical methods to approximate the solution, such as finite differences or finite element methods.

In summary, while your steps may be correct, it is important to carefully consider all aspects of the problem and to seek out additional resources or assistance if necessary.