The equation set consists of three equations, two ODEs and a PDE \begin{equation*} \begin{cases} \dfrac {d^{2}u} {dx^{2}}+au-bv=0\\ \dfrac {d^{2}v} {dx^{2}}+av+bu=0\\ c\dfrac {\partial w} {\partial t}=d\dfrac {\partial ^{2}w} {\partial x^{2}}+e\left( u^{2}+v^{2}\right) \end{cases} \end{equation*}
Boundary conditions when $x=0$ : \begin{equation*} \begin{cases} \dfrac {du} {dx}-fv=g\\ \dfrac {dv} {dx}+fu=g\\ \dfrac {\partial w} {\partial x}=hw\\ \end{cases} \end{equation*}
Boundary conditions when $x=1$ : \begin{equation*} \begin{cases} \dfrac {du} {dx}+fv=k\\ \dfrac {dv} {dx}-fu=k\\ \dfrac {\partial w} {\partial x}=-hw\\ \end{cases} \end{equation*}
The initial condition is $w=m$ and $a, b, c, d, e, f, g, h, k, m$ are all constant values.
I'm now trying to using MATLAB's ode45 and pdetool to solve this but boundary conditions in my problem is not fit in pde45 and it is difficult to couple these two functions together.
Best regards.