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Nonhomogeneous Heat Equation

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Homework Statement



So I'm trying to solve Evans - PDE 2.5 # 12...

"Write down an explicit formula for a solution of

[tex]u_t - \Delta u + cu=f[/tex] with [tex] (x,t) \in R^n \times (0,\infty)[/tex]
[tex]u(x,0)=g(x)[/tex]"

Homework Equations



The Attempt at a Solution



I figure if I can a fundamental solution for

[tex]u_t - \Delta u + cu = 0[/tex],

The rest follows through straight forward. I've tried multiplying the solution to the heat equation by a number of terms such as [itex]e^{-ct}[/itex], [itex]e^{v(x,t)}[/itex], but everything I've tried so far either gives me a non-fundamental solution, or a non-linear pde.

I've also tried mimicking the two ways they give to finding the solution to the original equation but neither seem to work. Looking for solutions of the form [itex]u(x,t) = v(x^2/t)[/itex] and looking for solutions of the form [itex]u(x,t) = 1/t^\beta v(x/t^\alpha)[/itex] but neither seems to reduce the equation in the problem to a single variables

Any hints?
 

Answers and Replies

  • #2
68
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yay, figured it out.

letting [itex]u(x,t)=e^{-ct}v(x,t)[/itex], with [itex]v(x,t)[/itex] solving

[tex]v_t - \Delta v = f(x,t) e^{ct}[/tex]
[tex]v(x,0) = g(x)[/tex]

Solves the original equation. I guess I was just thinking too hard.
 

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