im sorry, I am obviously knew to this forum...
For this problem, I am trying to use the identity as follows
2u((\partialu/\partialt)-(\partial^{2}u/\partialx^{2})) = (\partialu^{2}/\partialt)-(\partial/\partialx)*(u*(\partialu/\partialx))+2*(\partialu/\partialx)^{2}
Homework Statement
Let u(x,t) satisfy
Homework Equations
(\partialu/\partialt) = (\partial^{2}u/\partialx^{2})...(0<x<1,t>0)
u(0,t)=u(1,t)=0...(t\geq0)
u(x,0)=f(x)...(o\leqx\leq1),
where f\inC[0.1] show that for any T\geq0
\int from 0..1 (u(x,T))^{2}dx \leq \int from...
Homework Statement
Formally solve the following boundary value problem using Fourier Transforms.
Homework Equations
(\partial^{2}u/\partialx^{2})+(\partial^{2}u/\partialy^{2}) = 0
(-\infty<x<\infty,0<y<1)
u(x,0)= exp^{-2|x|}
(-\infty<x<\infty)
u(x,1)=0...
Homework Statement
Formally solve the following boundary value problem using Fourier Transforms.
Homework Equations
\partialu/\partialt = (\partial^{2}u/\partialx^{2})+(\partialu/\partialx)
(-\infty<x<\infty,t>0)
u(x,0)=\Phi(x)
(-\infty<x<\infty)
u(x,t) is bounded for...