A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well‐posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.
Well-posedness of a periodic boundary value problem for the system of hyperbolic equations with delayed argument The periodic boundary value problem for the system of hyperbolic equations with delayed argument is considered. By method of introduction a new functions the investigated problem reduce to an equivalent problem, consisting the family of periodic boundary value problem for a system of differential equations with delayed argument and integral relations. Relationship of periodic boundary value problem for the system of hyperbolic equations with delayed argument with the family of periodic boundary value problems for the system of ordinary differential equations with delayed argument is established. Algorithms for finding solutions of the equivalent problem are constructed and their convergence is proved. Sufficient and necessary conditions of well-posedness of periodic boundary value problem for the system of hyperbolic equations with delayed argument are obtained.
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.
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