Let denote a solution to the heat equation on [0,1] under the Dirichlet boundary conditions of (0)= and (1)= and IVP of ( ,0)= ( ), and let denote the corresponding steady state solution. Show that the lim as t approaches infinity of ( , )= ( ).
Let denote a solution to the heat equation on [0,1] under the Dirichlet boundary conditions of (0)= and (1)= and IVP of ( ,0)= ( ), and let denote the corresponding steady state solution. Show that the lim as t approaches infinity of ( , )= ( ).