# PDE11

 $u_{{xx}}=a^{{-2}}u_{t}\,$ $u(0,t)=10\,$ $u(10,t)=30\,$ $u(x,0)=0\,$ $00\,$

Write the solution as a sum of two functions, one independent of time.

$u(x,t)=s(x)+v(x,t)\,$

The new boundary conditions are:

$s(0)=10,\,\,\,s(10)=30\,$

$v(0,t)=0,\,\,\,v(10,t)=0\,$

$u(x,0)=s(x)+v(x,0)=0,\,\,\,v(x,0)=-s(x)\,$

Substituting the sum of functions into the DE gives a simple ODE and a PDE:

$s''=0\,$

$s(x)=Ax+B\,$

$s(0)=B=10\,$

$s(10)=10A+10=30,\,\,\,A=2\,$

• $s(x)=2x+10\,$

And the PDE

$v_{{xx}}=a^{{-2}}v_{t}\,$

Seperate variables to get two ODEs. Set them equal to a constant that will simplify calculations later:

$v(x,t)=X(x)T(t)\,$

$X''T=a^{{-2}}XT'\,$

${X'' \over X}=a^{{-2}}{T' \over T}=-\lambda ^{2}\,$

$X''+\lambda ^{2}X=0\,$

$X(x)=c_{1}\cos(\lambda x)+c_{2}\sin(\lambda x)\,$

$X(0)=c_{1}=0\,$

$X(10)=c_{2}\sin(\lambda 10)=0\,$

$\lambda _{n}={n\pi \over 10},\,\,\,n=1,2,3,...\,$

And

$T'+a^{2}\lambda ^{2}T=0\,$

$T(t)=c_{3}e^{{-a^{2}\lambda ^{2}t}}\,$

• $v(x,t)=\sum _{{n=1}}^{\infty }A_{n}\sin(\lambda _{n}x)e^{{-a^{2}\lambda ^{2}t}}\,$

$v(x,0)=\sum _{{n=1}}^{\infty }A_{n}\sin({n\pi x \over 10})=-2x-10\,$

This is a Fourier sine series for the function on the right. The coefficients are then given by:

$A_{n}={-1 \over 5}\int _{0}^{{10}}(2x+10)\sin({n\pi x \over 10})\,dx,\,\,\,n=1,2,3,...\,$

$={20(3(-1)^{n}-1) \over n\pi }\,$

The solution is

• $u(x,t)=2x+10+{20 \over \pi }\sum _{{n=1}}^{\infty }{3(-1)^{n}-1 \over n}\sin({n\pi x \over 10})e^{{-a^{2}{n^{2}\pi ^{2} \over 10^{2}}t}}\,$