System of first order PDE

Question

I have a system of first-order nonlinear partial differential equations. $$A(x,t)\frac{\partial u}{\partial t}(x,t) + B(x,t)\frac{\partial u}{\partial x}(x,t) + c(x,t) = 0$$ $$x \in \mathbb{R}, \quad u(x,0) = u_0(x), \quad u(0,t) = \varphi(t)$$ $A$, $B$ are $n \times n$ matrices. How to solve it numerically using matlab? Is there any way to transform it to system of ODEs in general case? Most of the matlab functions(pdepe, pdenonlin) seems to be inappropriate, because they can solve only 2nd order systems.

Answer 1

You cannot try to solve (systems of ) PDEs before having a vague idea of its type. Because the type determines the kind of problem you can solve (boundary-value problem, or Cauchy problem). It also tells you that some numerical approaches will be relevant and some will not.

Here, I presume that $A$ and $B$ are $n\times n$ matrices. Even if they are constant, your system can be elliptic (for instance, the Cauchy-Riemann system of holomorphic functions) or hyperbolic (for instance acoustics). If $n\ge3$, it can even be of mixed type, where ellipticity is coupled to hyperbolicity.

Therefore, in absence of more information, your question cannot receive an answer.