## Non homogeneous Cauchy problem for mixed parabolic equation

We try to solve non 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

We shall solve a family of parameter dependent homogeneous equation for the same mixed Cauchy problem

with initial condition

and boundary condition

From "Homogeneous Equation for Mixed Cauchy Problem" we know
From we have

Now we can verify that
is the desired solution for the non homogeneous equation.