- #1
Hendrick
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Homework Statement
Solve the heat flow problem using the method of separation of variables:
Homework Equations
PDE:[tex]\frac{\partial u}{\partial t}=k\frac{\partial^{2} u}{\partial t^{2}}[/tex]
for 0<x<L, 0<t<[tex]\infty[/tex]
BC's:[tex]\frac{\partial u}{\partial x}(0,t)=0,\frac{\partial u}{\partial x}(L,t)=0[/tex]
for 0<t<[tex]\infty[/tex]
IC's: [tex]u(x,0)=[/tex]
{0, [tex]0<x<L/4[/tex]
{1, [tex]L/4<x<3L/4[/tex]
{0, [tex]3L/4<x<L[/tex]
(Piecewise IC)
The Attempt at a Solution
I have separated the variables, then applied the boundary conditions. I am stuck on applying the initial conditions.
I have come up with a general product solution of [tex]u_{n}=F_{n}cos(\frac{n \pi x}{L}) e^{-k(\frac{n \pi x}{L})^{2}t}[/tex]
Trying to combine all product solutions and match the initial data:
[tex]u(x,0)=f(x)[/tex]
[tex]\sum^{\infty}_{n=1}u_{n}(x,0)=f(x)[/tex]
[tex]\sum^{\infty}_{n=1}F_{n}cos(\frac{n \pi x}{L})=f(x)[/tex]
I don't know how to apply the piecewise initial condition, any help would be appreciated. Thank you