How to Solve Heat Equations with Generation Terms?

[NoobixCube]

Hey,

I am wondering how to solve this heat equation with a 'generation' term included. In one instance I am adding the generation term, in the other I am subtracting it.

$$a\frac{\partial^{2}f }{\partial x^{2}}-\frac{\partial f}{\partial t} - \lambda f = 0$$

$$a\frac{\partial^{2}f }{\partial x^{2}}-\frac{\partial f}{\partial t} + \lambda f = 0$$


Any information on how to solve these/links to a table would be great :D

 

[LCKurtz]

Of course, you need some initial/boundary conditions to have a well posed problem. I would try separation of variables.

 

[elibj123]

Is the system bounded or is it in the whole space (x-axis in our case)?

A problem with boundary condition would be best solved with variable separation.
An infinite problem would be best solved with a Fourier Transform.

Both will give you an implicit form solution (as a sum or an integral) unless the initial conditions are specifically chosen.

 

[clustro]

Look at Partial Differential Equations: Sources and Solutions by AD Snider.

He talks about how to solve these sorts of problems.
I think the solution is to use a combination of Green's functions and Laplace transforms, but I do not recall exactly.