PDE initial-boundary-value problem?
PDE ut=uxx 0
by means of Laplace transform(transform t)
u(x,t)=X(x)*T(t)=X*T
ut=X*T’
uxx=X”*T => X*T’=X”*T => T’/T=X”/X=-k^2
then T’+k^2*T=0 and X”+k^2*X=0
then T=A*sin(k^2*t)+B*cos(k^2*t)
u(x,0)=0 => T(0)=0 => B=0
X=C*sin(k*t)+D*sinh(k*t)
E*cos(k*t)+F*cosh(k*t)
u(0,t)=sin(t) => X(0)=1
and you need to another boundary condition (must be two boundaries)
Mod-2 Lec-9 Boundary Value Problems
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