# 5 Classification Of First Order Systems

In addition to the second order equations of the type discussed in sections 2--3, systems of first order equations are also frequently encountered in computational science.

The current and voltage at position x and time t in a transmission line satisfy the first order equations

where R, L, C and G denote, respectively, resistance, inductance, capacitance and leakage conductance per unit length of transmission line.

(b) The first order system

governs the one dimensional flow of an ideal gas with velocity , density and pressure . is a physical constant determined by the specific heat of the gas.

Problems such as these present computational scientists with systems of first order partial differential equations. The general quasilinear system of n first order partial differential equations in two independent variables has the form

where , and may depend on . If each and is independent of , the system (15) is called almost linear. If, in addition, each depends linearly on ,

with and S functions of at most x and t, the system is said to be linear. If for , the system is called homogeneous. If C, G, R and L depend at most on x and t the transmission line equations are linear. The ideal gas equations are quasilinear.

In terms of the matrices and the column vectors , the system of equations (15) can be written as