Diffusion Equation with source term

[iva.mn87]

I have been asked to solve a diffusion equation with a source term using finite differences method. I need to numerically integrate the following equation either in MATLAB or C++.

The equation is

dT/dt = d2T/dx2 + S(x)

The form of S(x) is some function given by a Gaussian profile.

Could anyone have a solution to the problem!

Thanks!

 

[pafcu]

What exactly is the problem? How far have you gotten?

I've been working on the same problem, and I solved it by using the Crank-Nicolson method. Look it up on e.g. Wikipedia.

A worse, but far simpler method (especially since the source is non-linear) is to use the Forward Euler method:
$$\frac{T(x,t+\Delta t)-T(x,t)}{\Delta t} = \frac{T(x-\Delta x,t)-2T(x,t)+T(x+\Delta x,t)}{(\Delta x)^2} + S(x)$$

Solve for $$T(x,t+\Delta t)$$ and you get
$$T(x,t+\Delta t) = \frac{T(x-\Delta x,t)-2T(x,t)+T(x+\Delta x,t)}{(\Delta x)^2}\Delta t + S(x)\Delta t+T(x,t)$$

So if you know the value of T(x,t) for every x at some time t, you can calculate the value for a future time $$t+\Delta t$$

 

[charng]

I understand your question and can offer some guidance on how to approach solving this diffusion equation with a source term using finite differences method. First, it is important to understand the physical meaning of the equation. The term dT/dt represents the rate of change of temperature over time, while d2T/dx2 represents the spatial variation of temperature. The source term, S(x), represents a function that adds or removes heat from the system at a given position x.

To solve this equation numerically, you will need to discretize the equation in both time and space. This means dividing the domain into a grid and approximating the derivatives using finite difference approximations. Once the equation is discretized, you can use a numerical solver in MATLAB or C++ to solve the resulting system of equations.

One approach to solving this type of problem is to use an explicit finite difference scheme, where the temperature at each grid point is updated based on the temperature at neighboring grid points and the source term. Another approach is to use an implicit scheme, where the temperature at each grid point is updated based on the temperature at the previous time step and the source term.

It is also important to consider the stability and accuracy of your numerical solution. The stability of the solution will depend on the time and space discretization parameters, while the accuracy can be improved by using a finer grid or higher-order finite difference approximations.

In conclusion, solving a diffusion equation with a source term using finite differences method can be achieved by discretizing the equation and using a numerical solver. However, it is important to carefully consider the discretization parameters and the stability and accuracy of the solution to ensure reliable results.