How to Solve Heat Equations with Generation Terms?

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Discussion Overview

The discussion centers on solving heat equations that include generation terms, specifically examining two forms of the equation with differing signs for the generation term. Participants explore methods for addressing these equations, including the need for initial and boundary conditions.

Discussion Character

  • Technical explanation, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant inquires about solving heat equations with generation terms, presenting two variations of the equation.
  • Another participant emphasizes the necessity of initial and boundary conditions to properly pose the problem and suggests separation of variables as a potential method.
  • A different participant questions whether the system is bounded or extends over the entire x-axis, noting that boundary conditions would favor separation of variables while an infinite domain would be better suited for Fourier Transform methods.
  • Another contribution references a specific text, suggesting that Green's functions and Laplace transforms may be relevant to finding solutions, though the participant is uncertain about the details.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate methods for solving the equations, with no consensus on a single approach or solution method. The discussion remains unresolved regarding the best techniques to apply.

Contextual Notes

Participants mention the importance of initial and boundary conditions, as well as the implications of the system being bounded versus unbounded, but do not resolve these aspects or provide specific conditions.

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

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