I'm currently confused about how to solve this two-dimensional coupled PDE. I'm currently using Matlab to solve it using the $pdepe$ function, but I don't have related initial and boundary conditions for the $w$ function shown in the third equation. If there's a way to solve this manually or with another program, instructions would be very helpful.
Thank you in advance.
$$ \ \\ Equations \\ \rule{5cm}{0.4pt} $$
$$
\frac{\partial u}{\partial t} = -w \ \cdot \ u \ + \ \frac{\partial^2 u}{\partial x^2}
$$
$$
\frac{\partial v}{\partial t} = -w \ \cdot \ u \ + \ \frac{\partial^2 v}{\partial x^2}
$$
$$
\frac{\partial w}{\partial x} = -w \ \cdot \ (u \ + \ v)\\
\ \\
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$$
$$
Boundary \ \ Conditions \\
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$$
$$
u(x,0) = u_0 \\
v(x,0) = 0 \\
w(0,t) = w_0 \\
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$$
$$
\frac{\partial u}{\partial x}(0,t) = \frac{\partial v}{\partial x}(0,t) = 0 \\
\frac{\partial u}{\partial x}(x_{max},t) = \frac{\partial v}{\partial x}(x_{max},t) = 0 \\
\ \\
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$$
I'm not sure if pdepe is suitable to solve this system because pdepe was designed to solve parabolic-elliptic PDEs and your third equation is hyperbolic.
You can check this discussion: https://www.mathworks.com/matlabcentral/answers/402158-system-of-pdes-which-is-tricky-for-pdepe