Is it possible to solve this equation analytically? $$ u_t = k u_{xx} + \frac{k}{c} u_{xt} $$

I attempted to solve it for a finite domain and homogeneous B.C with separation of variables but it got very ugly, with complex eigenvalues. I'm wondering if the equation could be solved with the method of characteristics? Or if there is a coordinate transformation which converts this to an easier PDE?

I am interested in solving the equation for a sine function initial condition. Either infinite or finite domain would be okay, whichever is easier.

  • $\begingroup$ Have you tried the Fourier transform in the $x$ variable? Or the Laplace transform in the $t$ variable? $\endgroup$ – user8960 Oct 17 '16 at 22:56
  • $\begingroup$ So I write my solution in the form $u=T(t)X(x)$ and found that the $T$ variable is just simple exponential decay, but when I tried doing Fourier series for the $X$ variable it got too messy when trying to find the eigenvalues. $\endgroup$ – Llouis Oct 18 '16 at 2:32
  • $\begingroup$ Once $T(t)$ is known, substituting it back into the PDE gives: $$ X(x) = {k \over T'(t)} X''(x) + {k \over c} X'(x), $$ which, for each fixed $t$, is a 2nd-order linear homogeneous ODE in $X(x)$ with constant coefficients. Can you attack this ODE without resorting to Fourier series? $\endgroup$ – user8960 Oct 18 '16 at 16:31
  • $\begingroup$ After separation of variables, I have two ODE's: $T'=-\lambda k T$ and $X''=-\lambda X + \lambda \frac{k}{c} X'$. The Fourier series is necessary to figure out what $\lambda$ is. If I have boundary conditions of $X(0)=X(L)=0$, then I know $\lambda^2 v^2 - 4 \lambda$ must be less than zero in order to satisfy the boundary conditions. Typically in these problems, you then find a list of $\lambda$ values that satisfy the B.C's. Here I had trouble finding the $\lambda$ 's. $\endgroup$ – Llouis Oct 18 '16 at 16:46
  • $\begingroup$ I was a bit imprecise in the last comment, the Fourier series is how you write the solution in terms of $\lambda$, not how you find $\lambda$. $\endgroup$ – Llouis Oct 18 '16 at 16:52

I realized that the coordinate transformation $x=x_*$ and $t=t_*-\frac{1}{2c} x_*$ transforms this equation into the damped wave equation, which has well-known solutions which can be obtained through separation of variables.


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