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(Solved): Non-Homogeneous Dirichlet IBVP Let T(t,x) be the solution of the Initial-Boundary Value Problem (IBV ...
Non-Homogeneous Dirichlet IBVP Let
T(t,x)be the solution of the Initial-Boundary Value Problem (IBVP) for the Heat Equation,
del_(t)T(t,x)=4del_(x)^(2)T(t,x),tin(0,\infty ),xin(0,5)with non-homogenous Dirichlet boundary conditions
T(t,0)=3,T(t,5)=5and with initial condition
T(0,x)=\tau (x)={(3,xin[0,(5)/(2))),(5,xin[(5)/(2),5]):}The solution
T(t,x)of the problem above, with the conventions given in class, has the form
T(t,x)=T_(E)(x)+\sum_(n=1)^(\infty ) c_(n)v_(n)(t)w_(n)(x)where
T_(E)(x)is the equilibrium solution and the functions
v_(n)(t),w_(n)(x)satisfy the normalization conditions
v_(n)(0)=1 and w_(n)((5)/(2n))=1Find the functions
T_(E)(x),v_(n)(t),w_(n)(x), and the constants
c_(n).
T_(E)(x)=
v_(n)(t)=\epsi lon
w_(n)(x)=
c_(n)=?