The generalized Cauchy problem with data on two surfaces for a quasilinear analytic system

Abstract

We consider the generalized Cauchy problem with data on two surfaces for a second-order quasilinear analytic system. The distinction of the generalized Cauchy problem from the traditional statement of the Cauchy problem is that the initial conditions for different unknown functions are given on different surfaces: for each unknown function we pose its own initial condition on its own coordinate axis. Earlier, the generalized Cauchy problem was considered in the works of C. Riquier, N. M. Gyunter, S. L. Sobolev, N. A. Lednev, V. M. Teshukov, and S. P. Bautin. In this article we construct a solution to the generalized Cauchy problem in the case when the system of partial differential equations additionally contains the values of the derivatives of the unknown functions (in particular outer derivatives) given on the coordinate axes. The last circumstance is a principal distinction of the problem in the present article from the generalized Cauchy problems studied earlier.Keywords: quasilinear system of partial differential equations, initial-boundary value problem, theorem on existence and uniqueness of an analytic solution One of the current problems of the theory of partial differential equations consists in proving some analogs and generalizations of the Kovalevskaya theorem.Generalization of the Cauchy problem (CP) to the case when the initial data for different functions are given on different surfaces was first considered in the monograph [1] of the French mathematician Riquier. Riquier's monograph is a rare book; exposition of some facts of Riquier's theory can be found in S. P. Finikov's monograph [2]. Riquier's results were further developed in the works of the Russian mathematicians N. M. Gyunter [3,4], S. L. Sobolev [5,6], and N. A. Lednev [7]. Namely, N. A. Lednev proposes the term "the generalized Cauchy problem" (GCP).S. P. Bautin [8,9] studied the GCP for a second-order quasilinear analytic system; in particular, he stated some necessary and sufficient conditions for existence of a solution in the form of formal double series and some sufficient conditions for convergence of the series. Using the methods of [9], S. P. Bautin and the author managed to prove in [10] a theorem on existence and uniqueness of a locally analytic solution to the GCP for a second-order quasilinear system with a singularity of the form u/x or x/u.Solution of problems of continuum mechanics, in particular gas dynamics, is a practical application of the theory of partial differential equations. Constructing piecewise analytic gas flows with shock waves in [11][12][13][14], V. M. Teshukov solved specific GCP's on using particular methods. Moreover, he verified the necessary and sufficient conditions for existence of a formal solution to the problems in question. Under some constraints due to the physical meaning of the problems, convergence of series was proven. In [15,16] S. P. Bautin and the author, using the GCP, studied the gas flow with shock waves in a neighborhood of a symmetry axis or symmetry center where the system of eq...

Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo gH-Type Derivatives

“…In the 20th century, French, Japanese, and Russian mathematicians published numerous publications on similar subjects (see, for example, [31,32]). Recently, in Kazakov [33,34],…”

“…(iii) While (4) and (5) are similar in form to the problems studied by Riquier [30] and Kazakov [33], they rely on gH-type derivatives. So we register that (4) and (5) are brand new and have not been reported in the literature.…”