First-Order Partial Differential Equations

Abstract

In this chapter we study the solution of first-order partial differential equations of the form

$$F({{x}_{1}},{{x}_{2}},...,{{x}_{n}},u,{{u}_{{{x}_{1}}}},{{u}_{{{x}_{2}}}},...,{{u}_{{{x}_{n}}}})=0$$

(8.0-1)

Equation (8.0-1), with appropriate initial conditions, governs the solution of the dependent variable u as a function of the n independent variables x _i . It is a first-order equation, since only the first partial derivatives of u with respect to x _i occur. In general, F will be a nonlinear function of the u _xi . If, however, F is linear in the u _xi ,eq. (8.1-1) is said to be quasilinear, and has the most general form

$$\sum\limits_{i=1}^{n}{{{a}_{i}}}({{x}_{1}},{{x}_{2}},...,{{x}_{n}},u){{u}_{{{x}_{i}}}}=a({{x}_{1}},...,{{x}_{n}},u)$$

(8.0-2)

where the coefficients a _i and a do not depend on the u _xi but may involve u. In the special case where these coefficients do not involve u, eq. (8.0-2) is said to be linear and, finally, the simplest case results if the coefficients are constants.