Question

Consider the second order partial differential equation du/dt= d^2u/dx^2 +2du/dx+u over the domain x in [0,l) and t>=0. It is given that u(0,t)=u(l,t)=0. Use the method of separation of variables to prove that the general solution with the given boundary condition is u(x,t)= infinity series n=1 b_nsin(npix/l)exp(-x-((npi/l)^2)t) where b_n is a constant for every n N  

Hint u(x,t)=X(x)T(t)

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Answer 1

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Answer 2

d^2u/dt^2=c^2d^2u/dx^2. 00 Boundary conditions. u(0.t)=u(l.t)=0 Initial conditions: u (x.0)=sin(πx/l) du/dt(x.0)=0 Solve the equation using the separation of variables method.