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I have the following initial-boundary value problem

$$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial t}\\\dfrac{\partial^2 u_2}{\partial x^2}=A_{21}\dfrac{\partial u_1}{\partial t}+A_{22}\dfrac{\partial u_2}{\partial t}\end{cases}$$

with $A_{12}=A_{21}$

Initial conditions

$$u_1(x,0)=c_1\\u_2(x,0)=c_2$$

for each $x$ such that $0<x<L$

Boundary conditions

$$u_1(0,t)=0\\u_2(L,t)=0\\\frac{\partial u_1}{\partial x}(L,t)=0\\\frac{\partial u_2}{\partial x}(0,t)=0$$

for each $t$ such that $t>0$

What I tried.
First I tried to solve the uncoupled system when $A_{12}=A_{21}=0$, to gain some knowledge. I found by separation of variables and using Fourier Series the following solution $\left(\text{with }\omega_m=\dfrac{(2m+1)\pi}{2L}\right)$:

$$ u_1(x,t)=\sum_{m=0}^\infty\frac{2 c_1}{\omega_m L} \exp\left({-\frac{\omega_m^2}{A_{11}}t}\right)\sin(\omega_m x)\\ u_2(x,t)=\sum_{m=0}^\infty(-1)^m\frac{2 c_2}{\omega_m L} \exp\left({-\frac{\omega_m^2}{A_{22}}t}\right)\cos(\omega_m x) $$

Next, I tried to uncouple the original equations, by defining $u_i=P_{ij}v_j$, and found

$$ \frac{\partial^2 v_1}{\partial x^2}= \lambda_1\frac{\partial v_1}{\partial t}\\ \frac{\partial^2 v_2}{\partial x^2}= \lambda_2\frac{\partial v_2}{\partial t} $$

where $\lambda_i$ are the eigenvalues of $A$, and the columns of $P$ are the eigenvectors.

By separation of variables I found

$$ v_i(x,t)=\exp\left(-\frac{\mu_i^2}{\lambda_i}t\right)\left(D_i^{+}\cos(\mu_i x)+D_i^{-}\sin(\mu_i x)\right) $$

where the $\mu_i$ are positive constants to be fixed by the initial-boundary conditions.
The boundary conditions become

$$ u_1(0,t)=P_{11}v_1(0,t)+P_{12}v_2(0,t)=0\\ u_2(L,t)=P_{21}v_1(L,t)+P_{22}v_2(L,t)=0\\ \frac{\partial u_1}{\partial x}(L,t) =P_{11}\frac{\partial v_1}{\partial x}(L,t) +P_{12}\frac{\partial v_2}{\partial x}(L,t)=0\\ \frac{\partial u_2}{\partial x}(0,t) =P_{21}\frac{\partial v_1}{\partial x}(0,t) +P_{22}\frac{\partial v_2}{\partial x}(0,t)=0 $$

Writing down explicitly this system, it can be regarded as an algebraic homogeneous linear system in the $4$ unknown $D_i^\pm$, whose coefficients depend on $t$ and on the $\mu_i$. It should have a solution other than the trivial one, and this for each $t>0$.

Is this the right direction and where to go from here?
Is there a completely different approach I am missing?

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  • $\begingroup$ Could you give some background of the origin of this system? Maybe it helps if we know it. $\endgroup$ – vesszabo May 4 '14 at 17:18
  • $\begingroup$ @vesszabo: it is a problem proposed by a friend engineering. I hope to explain it correctly. It should model the unidimensional diffusion of two substances in a finite interval $[0,L]$, being known the initial constant densities ($c_1$ and $c_2$), the two sides being non permeable to the two substances, respectively (null derivatives). Cannot remember the meaning of $u_1(0,t)=u_2(L,t)=0$. $\endgroup$ – enzotib May 4 '14 at 18:48

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