Answer to Question #157213 in Differential Equations for eli

Question

A 10-FT LONG ROD HAS BOTH OF ITS ENDS INSULATED DURING HEATING OPERATION. BUT EVEN AT ITS INITIAL OPERATION (T=0), THE ROD IS ALREADY LATERALLY INSULATED. FIND U(X,T) IF THE THERMAL DIFFUSIVITY EQUALS 1.5 (X=5.0)

Answer 1

The heat equation:

$\frac{\partial u}{\partial t}=k\frac{\partial^2u}{\partial x^2}$

We have:

$k=1.5$

$\frac{\partial u}{\partial x}_{x=0}=\frac{\partial u}{\partial x}_{x=L}=0$

$f(x)=u(x,0)=5$

The solution of heat equation:

$u(x,t)=(Acos\lambda x+Bsin\lambda x)e^{-k^2\lambda^2t}$

Applying conditions:

$u(x,0)=Acos\lambda x+Bsin\lambda x=5$

$\frac{\partial u}{\partial x}=(B\lambda cos\lambda x-A\lambda sin\lambda x)e^{-k^2\lambda^2t}$

$\frac{\partial u}{\partial x}_{x=0}=B\lambda e^{-k^2\lambda^2t}=0\implies B=0$

$\frac{\partial u}{\partial x}_{x=L}=-A\lambda sin(\lambda L)e^{-k^2\lambda^2t}=0\implies \lambda=\pi n/L$

$u(x,t)=5e^{-k^2\lambda^2t}=5e^{-k^2\pi^2n^2t/L^2}$

$u(x,t)=5e^{-2.25\pi^2n^2t/100}=5e^{-0.22n^2t}$