# Thread: Heat Conduction, Finding Cn

1. ## Heat Conduction, Finding Cn

Hello, I'm having difficulty finding the Cn for a heat conduction problem, and any help would be appreciated.

The problem is #8, Section 10.5 in Boyce and DiPrima if you have it.

Uxx = 4Ut, 0 < x < 2, t>0
U(0,t) = 0 = U(2,t), t>0
U(x,0) = 2sin(pi x / 2) - sin(pi x) + 4sin(2pi x)

Thanks.

2. ## General solution

To solve $\displaystyle \dfrac{\partial^{2}U}{\partial x^2} = 4 \dfrac{\partial U}{\partial t}$, you use separation of variables $\displaystyle U(x,t)=X(x)T(t)$ to get
$\displaystyle \dfrac{4}{T}\dfrac{dT}{dt}=\dfrac{1}{X}\dfrac{d^{2 }X}{dx^2}=-\dfrac{1}{\lambda^2}$ with lambda a constant (of the same units as x) and the negative sign chosen to make the solution finite for all t>0. Solving these separately, one gets an exponential for T(t) and a sine wave function for X(x). Using the x boundary conditions, one determines the discrete values $\displaystyle \lambda_n$ allowed for $\displaystyle \lambda$ to get the $\displaystyle U_{n}(x,t)$, and

$\displaystyle U(x,t)=\sum_{n=1}^\infty c_{n}U_{n}(x,t)$

Then we compare this at t=0 with the boundary condition $\displaystyle U(x,0)=2\sin (\tfrac{\pi}{2}x) - \sin (\pi x) + 4\sin (2\pi x)$. The $\displaystyle c_n$ should then become quite apparent.

--Kevin C.

3. No need to solve this problem from first principles. There is a general formula for the coefficients that is derived from seperation of variables. But when I solve a PDE I do not do it from first principles, I identity what equation it it ands its BVP and apply the known formula. It saves a lot of time.