XI – Partial Differential Equations

Publisher Summary This chapter discusses the quasi-linear partial differential equations of the first order. A partial differential equation of the first order for a function u of the variables x and y is an equation of the form f(uX, uy, u, x, y) = 0, containing only the first partial derivatives of u with respect to x and y. that the chapter presents an assumption where the function f is continuous with continuous derivatives with respect to each of these arguments. For this equation, the problem to be posed is Cauchy's problem. In the (x, y) plane, there is given a curve in parametric form x = x(s), y = y(s), where x(s) and y(s) are continuously differentiable functions. Along this curve is also given u(s) as a continuously differentiable function. The integral curves are called the characteristic curves of the equations. The direction determined is the characteristic direction at the point (x, y, u).