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

Posted November 18th, 2007 by Structure

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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