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Non homogeneous Cauchy problem for mixed parabolic equation

Posted November 18th, 2007 by Structure

We try to solve non homogeneous equation for mixed Cauchy problem with null boundary condition of a special form for the parabolic 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)=0 $
and boundary condition
$ u(0,t)=0\: $ and $ \frac{\partial u}{\partial x}(1,t)=0 $

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

$ P(D)v_{\tau}(x,t)=\frac{\partial v_{\tau}}{\partial t}(x,t)-\frac{\partial^2 v_{\tau}}{\partial x^2}(x,t)=0 $
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 $
From "Homogeneous Equation for Mixed Cauchy Problem" we know $ v_{\tau}(x,t)=\sum_{n=1}^{+\infty}T_n(t)X_n(x)=\sum_{n=1}^{+\infty}A_ne^{-\frac{(2n+1)^2\pi^2t}{4}}\sin \frac{(2n+1)\pi}{2}x $
From $ u(x,0)=\sum_{n=1}^{+\infty}A_n\sin \frac{(2n+1)\pi}{2}x=F(x,\tau) $ we have
$ A_n=2\int_{0}^{1}F(x,\tau)\sin\frac{(2n+1)\pi x}{2}dx $
Now we can verify that
$ u(x,t)=\int^{t}_{0}v_{\tau}(x,t-\tau)d\tau $ is the desired solution for the non homogeneous equation.

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