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

Submitted by Structure on Sun, 11/18/2007 - 18:34.

Let consider heat equation
$P(D)u=\frac{\partial u}{\partial t}(x,t)-\frac{\partial^2 u}{\partial x^2}(x,t)=F(x,t)$
with initial condition
$u(x,0)=f(x)$ and boundary condition
$u(0,t)=\phi_1(t)\:and \frac{\partial u}{\partial x}(1,t)=\phi_2(t);$
We shall look for the solution of this mixed problem with variable boundary condition of the form
$u(x,t)=u_1(x,t)+u_2(x,t)+u_3(x,t)$
where $P(D)u_1=\frac{\partial u_1}{\partial t}(x,t)-\frac{\partial^2 u_1}{\partial x^2}(x,t)=F_1(x,t)$
with initial condition
$u_1(x,0)=f_1(x)$

are determined by the values of a function $u_1(x,t)$ chosen to satisfy
boundary condition
$u_1(0,t)=\phi_1(t)\:and \frac{\partial u_1}{\partial x}(1,t)=\phi_2(t);$

Then we try to solve a homogeneous equation with null boundary condition and Cauchy initial data modified by the effect of previous function $u_1(x,t)$
$P(D)u_2=\frac{\partial u_2}{\partial t}(x,t)-\frac{\partial^2 u_2}{\partial x^2}(x,t)=0$
with initial condition
$u_2(x,0)=f_2(x)$ where $f_2(x)=f(x)-u_1(x,0)$
and boundary condition
$u_2(0,t)=0\:$ and $\frac{\partial u_2}{\partial x}(1,t)=0$
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 $F_3(x,t)=F(x,t)-P(D)u_1(x,t)$
$P(D)u_3=\frac{\partial u_3}{\partial t}(x,t)-\frac{\partial^2 u_3}{\partial x^2}(x,t)=F_3(x,t)$
with initial condition
$u_3(x,0)=0$ and boundary condition
$u_3(0,t)=0:and \frac{\partial u_3}{\partial x}(1,t)=0;$

To find $u_3(x,t)$ using Dirichlet principle we look for a function $u_3(x,t)=\int_{0}^{t}v(x,t-\tau,\tau)d\tau$
where $v_{\tau}(x,t)=v(x,t,\tau)$ satisfies
$P(D)v=\frac{\partial v_{\tau}}{\partial t}(x,t)-\frac{\partial^2 v_{\tau}}{\partial x^2}(x,t)=F(x,t)$
with initial condition
$v_{\tau}(x,0)=F(x,\tau)$ and boundary condition
$v_{\tau}(0,t)=0\:$ and $\frac{\partial v_{\tau}}{\partial x}(1,t)=0;$
See also
" Non Homogeneous Cauchy Problem Dirichlet Principle"

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