# 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

See also

" Non Homogeneous Cauchy Problem Dirichlet Principle"