(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the Heat Equation： du/dt=k(d2u/dx2), where d is a partial and d2 is the second partial. The B.C.'s are u_x(0,t)=u(0,t) and u_x(L,t)=u(L,t), where u_x is the partial of u with respect to x. The I.C is u(x,0)=f(x)

Now, consider the Boundary Value Problem X''(x)=-lambda*X(x), note the negative sign, with B.C's X'(0)=X(0) and X'(L)=X(L), where L is the length of a 1D rod(at the very end of the rod).

Find the eigenvalues and eigenfunctions for lambda>0, lambda=0, and lambda<0

3. The attempt at a solution

lambda=0 was an easy one, lambda=0, gave an eigenfunction X=0.

lambda<0, wasn't too bad either. Since lambda < 0 I get X=Acosh(sqrt(-lambda)*x)+Bsinh(sqrt(-lambda)*x), where A and B are constants. Also, note that lambda is negative, thus the square root will not give any complex numbers. Using the first initial condition X'(0)=X(0), I get that A=sqrt(-lambda)*B and plugging back in again, using the second BC, I get that lambda=-1 and thus the eigenfunction is X=Bcosh(x)+Bsinh(x).

lambda>0 is the one I'm having trouble with. If do what I did for lambda<0, and use X=Acos(sqrt(lambda)*x)+Bsin(sqrt(lambda)*x), I get that lambda is -1, again, however, lambda is suppose to be greater than 0, and the eigenfunction is complex. Any help would be appreciated. Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: PDE-Heat Equation with weird boundary conditions help!

**Physics Forums | Science Articles, Homework Help, Discussion**