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By Y. Pinchover, J. Rubenstein

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Write it in the coordinates s = x + y, t = 2x. 1 Introduction A first-order PDE for an unknown function u(x1 , x2 , . . , xn ) has the following general form: F(x1 , x2 , . . , xn , u, u x1 , u x2 , . . 1) where F is a given function of 2n + 1 variables. First-order equations appear in a variety of physical and engineering processes, such as the transport of material in a fluid flow and propagation of wavefronts in optics. Nevertheless they appear less frequently than second-order equations. For simplicity we shall limit the presentation in this chapter to functions in two variables.

Inverting the mapping (x(t, s), y(t, s)) we obtain t = y 1/3 − 1, s = x + 1 − y 1/3 . Hence the explicit solution to the PDE is u(x, y) = x + y 1/3 , which is indeed singular on the x axis. 8 Solve the equation (y + u)u x + yu y = x − y subject to the initial conditions u(x, 1) = 1 + x. This is an example of a quasilinear equation. The characteristic equations and the initial data are: (i) xt = y + u, (ii) yt = y, (iii) u t = x − y, x(0, s) = s, y(0, s) = 1, u(0, s) = 1 + s. Let us examine the transversality condition.

The variable y has the physical interpretation of time. We shall show that the solutions to this equation often develop a special singularity that is called a shock wave. In hydrodynamics the equation is called the Euler equation (cf. Chapter 1; the reader may be baffled by now by the multitude of differential equations that are called after Euler. We have to bear in mind that Euler was a highly prolific mathematician who published over 800 papers and books). 41) to a larger family of equations, and in particular, we shall apply the theory to study traffic flow.

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