Heat eqtn with Generation term

[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.

 

[blue_raver22]

Hi there,

The heat equation with a generation term is a common problem in many fields of science and engineering. The generation term represents a source or sink of heat within the system, which can arise from various physical processes such as chemical reactions, nuclear reactions, or external sources of heat.

To solve these equations, you can use a variety of analytical and numerical methods. For the first equation (with a positive generation term), the solution will depend on the boundary conditions and initial conditions of the system. One possible approach is to use separation of variables and solve for the steady-state solution first, and then use that solution as a starting point for solving the time-dependent solution. Another approach is to use numerical methods such as finite difference or finite element methods to solve the equation directly.

For the second equation (with a negative generation term), the solution will also depend on the boundary and initial conditions. In this case, the negative generation term will act as a heat sink, which can lead to a decrease in temperature over time. The same methods mentioned above can be applied to solve this equation as well.

I recommend checking out some resources on heat equation with generation terms, such as textbooks or online tutorials, to get a better understanding of the different methods and techniques used to solve these types of equations. I hope this helps and good luck with your problem-solving!