Homogeneous Equation for Mixed Cauchy Problem
Submitted by Structure on Sun, 11/18/2007 - 19:13.
We try to solve homogeneous equation for mixed Cauchy problem with null boundary condition of a special form for the parabolic heat equation
with initial condition
and boundary condition
and
Look for a nonzero solution of the form u(x,t)=X(x)T(t).
We get T'(t)X(x)-T(t)X"(x)=0 or
(*)We get
(**)
Second equation has a solution of the form
So
Equation (*) has solution
or
Look now for a solution of our equation
From we have
In a particular case all you have to do is to evaluate for the given function.