## Mixed Cauchy problem for non homogenous parabolic equation (Heat transfer)

Let consider heat equation

with initial condition
and boundary condition

We shall look for the solution of this mixed problem with variable boundary condition of the form

where
with initial condition

are determined by the values of a function chosen to satisfy
boundary condition

Then we try to solve a homogeneous equation with null boundary condition and Cauchy initial data modified by the effect of previous function

with initial condition
where
and boundary condition
and
For a solution, see
"Homogeneous Equation for Mixed Cauchy Problem"

Finally we try to solve the perturbed problem with null initial Cauchy condition and null data on the boundary. Here

with initial condition
and boundary condition

To find using Dirichlet principle we look for a function
where satisfies

with initial condition
and boundary condition
and