Open AccessAPPROXIMATE SOLUTIONS OF SOME NONLINEAR PROBLEMS FOR THE MONGE-AMPERE EQUATIO
Author(s) -
Н.Б. Искакова,
А.С. Рысбек,
Н.С. Серік
Publication year - 2020
Publication title -
habaršy. fizika-matematika seriâsy
Language(s) - English
Resource type - Journals
ISSN - 1728-7901
DOI - 10.51889/2020-1.1728-7901.16
Subject(s) - mathematics , boundary value problem , nonlinear system , linearization , mathematical analysis , ordinary differential equation , ampere , differential equation , monge–ampère equation , partial differential equation , current (fluid) , physics , quantum mechanics , thermodynamics
Due to numerous applications in various fields of science, including gas dynamics, meteorology, differential geometry, and others, the Monge – ampere equation is one of the most intensively studied equations of nonlinear mathematical physics.In this report, we study a nonlinear boundary value problem for the inhomogeneous Monge-ampere equation, the right part of which contains power nonlinearities in derivatives and arbitrary nonlinearity from the desired function.Based on linearization, the studied boundary value problems are reduced to a system of ordinary first-order differential equations with initial conditions that depend on the parameter.Methods for constructing exact and approximate solutions of some boundary value problems for the Monge-ampere equation are proposed.Using the Mathcad software package, numerical implementation of methods for constructing approximate solutions of the obtained systems of ordinary differential equations with a parameter is performed.Three-dimensional graphs of exact and approximate solutions of the problems under consideration in the Grafikus service are constructed.