By Yehuda Pinchover and Jacob Rubinstein
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41) to a larger family of equations, and in particular, we shall apply the theory to study trafﬁc ﬂow. As a warm-up we start with the simple linear equation u y + cu x = 0. 42) the ﬂow speed is given by the positive constant c. 43) 42 First-order equations will be used for both equations. 42) we get (x, y, u) = (s + ct, t, h(s)). Eliminating s and t yields the explicit solution u = h(x − cy). The solution implies that the initial proﬁle does not change; it merely moves with speed c along the positive x axis, namely, we have a ﬁxed wave, moving with a speed c while preserving the initial shape.
3 The method of characteristics 29 Upon substituting into the initial conditions, we ﬁnd x(t, s) = t + s, y(t, s) = t, u(t, s) = 2t + s 2 . We have thus obtained a parametric representation of the integral surface. To ﬁnd an explicit representation of the surface u as a function of x and y we need to invert the transformation (x(t, s), y(t, s)), and to express it in the form (t(x, y), s(x, y)), namely, we have to solve for (t, s) as functions of (x, y). In the current example the inversion is easy to perform: t = y, s = x − y.
Recall that the implicit function theorem implies that such a transformation is invertible if the Jacobian J = ∂(x, y)/∂(t, s) = 0. But we observe that while the dependence of the characteristic curves on the variable t is derived from the PDE itself, the dependence on the variable s is derived from the initial condition. Since the equation and the initial condition do not depend upon each other, it follows that for any given equation there exist initial curves for which the Jacobian vanishes, and the implicit function theorem does not hold.