# Examples of system of one dimensional first order hyperbolic PDEs with source term

I am writing an algorithm to solve a system of one- dimensional first order hyperbolic PDEs with constant coefficient matrices and I require some example problems which are already solved analytically or numerically with initial and boundary conditions.

The system of PDEs is of the form: $$A\dot{U} + BU^{'} + F(U) = 0$$ Where, $$A$$, and $$B$$ are constant matrices and $$F(U)$$ is a linear or nonlinear function of $$U$$.

$$\dot{\begin{Bmatrix} \\ \end{Bmatrix}}$$ represents the derivative w.r.t. time ($$t$$), and $$\begin{Bmatrix} \\ \end{Bmatrix}^{'}$$ represents the derivative w.r.t. spatial variable ($$s$$).

Example:

One of such problems is the transmission equation given by:

$$\begin{bmatrix}L & 0\\0 & C\end{bmatrix}\dot{\begin{Bmatrix}I \\V\end{Bmatrix}} + \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\begin{Bmatrix}I \\V\end{Bmatrix}^{'} + \begin{Bmatrix} RI \\GV \end{Bmatrix} = \begin{Bmatrix}0 \\0 \end{Bmatrix}$$

$$I$$ represents current, and $$V$$ represents voltage.

Where, $$L = 4$$, $$R = 4$$, $$C = 1$$, and $$G = 1$$

The initial conditions are given by:

$$I(s, 0) = I_0$$ $$V(s, 0) = V_0$$

An analytical solution is available for the above example problem, when $$RC = LG$$

If anyone has knowledge of similar system of equations (with 3 or more variables if possible) with initial and boundary conditions please let me know. I have searched online and could not find another example.

The scalar case $$a\dot u + b u' + c u =0$$ has analytical closed-form solutions deduced from the method of characteristics. It's much more difficult to do so in the case of systems $$A\dot{U} + B U' + F(U) = 0$$ where $$F(U)$$ is nonzero. Nevertheless, some boundary-value problems can be solved using the Fourier method in the case where $$F(U) = CU$$ is a linear function. I'd suggest to analyse the scalar case in Fourier space before moving to other systems.