Goursat’s problem on the plane for a quasilinear hyperbolic equation

Abstract

A classical solution of the problem for a quasilinear hyperbolic equation in the case of two independent variables with given conditions for the desired function on the characteristic lines is obtained. The problem is reduced to a system of equations with a completely continuous operator. We constructed the unique solution by the method of successive approximations and showed the necessary and sufficient smoothness and matching conditions on given functions.

Classical Solutions of a Mixed Problem with the Zaremba Boundary Condition for a Mildly Quasilinear Wave Equation

“…It is easy to see that in this case the mixed problem (1.1) -(1.4) with conditions (2.11) and (2.12) splits into the Cauchy problem in the domain Q (1) and the Picard problems in the domains Ω 1 and Ω 2 . The first of the conditions (2.13) shows that if the comparability condition (2.3) is satisfied, then the solution of the problem (1.1) -(1.4) with the conjugation conditions (2.11) and (2.12) will be continuous, and the second of the conditions (2.13) is the necessary and sufficient comparability condition of the Picard problem (see [49] for more details).…”

Classical solution of the Cauchy problem for a quasi-linear wave equation with discontinuous initial conditions

“…Now, we use the results of the work [19] to finally prove this lemma. Let us derive an integro-differential equation for the function u (3) .…”