duhamel

1. Duhamel's Theorem

I want to prove: Let x:y denote the partial derivative of x with respect to y. Let u,v be functions of x,t. u,v --> 0 as x goes to infinity. u:t=u:xx for x,t>0 u(x,0) = 0 u(0,t)=f(t) and v:t=v:xx for x,t>0 v(x,0)=0 v(0,t)=1 then u(x,t)=INT(t,0){f(t-s)*v:t(x,s)}ds...