6.1 Method of Characteristics

A general first-order partial differential equation (PDE) in n independent variables x = (x1, … , xn) can be written as F (x, u, p) = 0. Here, u = u(x) is the unknown function to be determined and is its gradient. The gradient of a function u = u(x) is also denoted by Du = Dxu = ∇u = ∇xu, to emphasize the variables with respect to which the derivatives are taken; and, F is a given real valued Ck (k ⩾ 2) function defined on Ω × ℝ × ℝn, where Ω is a domain in ℝn. In the two-dimensional case, we also use the notation (x, y) to denote the independent variables and for the first-order derivatives. Sometimes, the variable u is replaced z, as more of a convention. We consider a Cauchy problem or an initial value problem (IVP) associated with the given equation F = 0:

where S is a smooth (n − 1) dimensional surface sitting inside Ω. If S has the parametric representation S = ﹛ℎ(s) = (ℎ1(s), … , ℎn(s)) : s ∈ V ﹜, where V is a region in ℝn−1 and ℎ1, … , ℎn are smooth real valued functions defined on V , then the initial condition is written as u(ℎ(s)) = u0(s) for s ∈ V , where u0 is a given function defined on V. In general, the function F is also required to satisfy a compatibility condition on S.

We attempt to analyse the IVP (6.1.1) through the method of characteristics. The characteristic curves x = x(t) associated with the equation F = 0 are the solutions of the first-order system of ordinary differential equation In general, since F also depends on u and p, this system alone is insufficient to solve for x(t).