Solving modified heat equation

[psie]

Homework Statement

Find a solution of the following problem \begin{align} u_t&= u_{xx} - hu,\qquad &0<x<\pi, \ t>0; \\ u(0,t)&=0,u(\pi,t)=1,\qquad &t>0; \\ u(x,0)&=0,\qquad &0<x<\pi.\end{align} Here $h>0$ is a constant.

Relevant Equations

The heat equation in the form $u_t= u_{xx}$ and its solution $u(x,t)=\sum_{n=1}^\infty b_n e^{-n^2t}\sin nx$.

In my Fourier analysis book, the author introduces some basic PDE problems and how one can solve these using Fourier series. I know how to solve basic heat equation problems, but the above one is different from the previous problems I've worked in terms of the boundary conditions. Using $u(x,t)=v(x,t)e^{-ht}$ I can transform the equation into the heat equation, i.e. $v_t= v_{xx}$ , however, the boundary conditions become $$v(0,t)=0,\quad v(\pi,t)=e^{ht}.$$ I don't know how to deal with non-constant boundary conditions...any ideas on how to proceed?

 

[pasmith]

Think about the long-term steady state. Find a function u_\infty(x) which satisfies u_\infty&#039;&#039; - hu_\infty = 0 with u_\infty(0)= 0 and u_\infty(\pi) = 1. Then f(x,t) = u(x,t) - u_\infty(x) must satisfy f_t = f_{xx} - hf subject to the self-adjoint boundary condition f(0,t) = f(\pi,t) = 0 and the initial condition f(x,0) = -u_\infty(x).

 

[psie]

Ok. I found $$u_{\infty}(x)=\frac{\sinh \sqrt{h}x}{\sinh \sqrt{h}\pi}.$$ I still think I have to use the trick $f(x,t)=e^{-ht}v(x,t)$ to turn $f_t = f_{xx} - hf$ into $v_t = v_{xx}$. The solution to the latter will be $$v(x,t)=\sum_{n=1}^\infty b_n e^{-n^2t}\sin nx.$$ But I'm stuck at how to solve for $b_n$ in $$f(x,0)=v(x,0)=\sum_{n=1}^\infty b_n \sin nx=-\frac{\sinh \sqrt{h}x}{\sinh \sqrt{h}\pi}.$$

 

[pasmith]

Do you not know how to determine the coefficients b_n?

 

[psie]

Do you not know how to determine the coefficients b_n?

Right, I don't. I don't see any connection between $\sinh$ and $\sin$ that's useful here.

 

[pasmith]

If \sum_{n=1}^\infty b_n \sin nx = f(x), \quad x \in [0, \pi] then <br /> b_n = \frac{2}{\pi} \int_0^\pi f(x) \sin nx \,dx. This should be derived in any decent textbook on fourier series.

Integrals of the form \int \sin ax \sinh bx\,dx can be done by integratin by parts twice or by expressing everything in terms of exponentials.