scholarly journals On differential inequalities of S.A. Chaplygin related to limit Cauchy problem for sets of ordinary differential equations of first order

2021 ◽  
pp. 95
Author(s):  
I.I. Bezvershenko
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Differential Inequalities ◽  
First Order

We prove a theorem on differential inequalities related to limit Cauchy problem for the set of ordinary differential equations$$y' = f(x,y,z),$$z' = \varphi(x,y,z)$$with boundary conditions$$\lim\limits_{x \rightarrow \infty} y(x) = y(\infty) = y_0, \; \lim\limits_{x \rightarrow \infty} z(x) = z(\infty) = z_0$$

Hodograph Method for Solving the Problem of Shallow Water Under a Solid Lid

2021 ◽  
pp. 15-24
Author(s):  
Tatiana F. Dolgikh
Keyword(s):  
Cauchy Problem ◽  
Green Function ◽  
Differential Equations ◽  
Shallow Water ◽  
Ordinary Differential Equations ◽  
Riemann Invariants ◽  
Hodograph Method ◽  
First Order ◽  
The Cauchy Problem ◽  
Derivatives Of

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

scholarly journals An error estimate for an approximate solution to ordinary differential equations obtained using the Chebyshev series

2020 ◽  
pp. 241-250
Author(s):  
О.Б. Арушанян ◽  
С.Ф. Залеткин
Keyword(s):  
Cauchy Problem ◽  
Approximate Solution ◽  
Differential Equations ◽  
Error Estimate ◽  
Ordinary Differential Equations ◽  
Chebyshev Series ◽  
Integration Interval ◽  
First Order ◽  
Partial Sum ◽  
The Cauchy Problem

Рассматривается приближенный метод решения задачи Коши для нелинейных обыкновенных дифференциальных уравнений первого порядка, основанный на применении смещенных рядов Чебышёва и квадратурной формулы Маркова. Приведены способы оценки погрешности приближенного решения, выраженного в виде частичной суммы ряда некоторого порядка. Погрешность оценивается с помощью второго приближенного решения, вычисленного специальным образом и представленного частичной суммой ряда более высокого порядка. На основе предложенных способов оценки погрешности построен алгоритм автоматического разбиения промежутка интегрирования на элементарные сегменты, делающие возможным вычисление приближенного решения с наперед заданной точностью. Работа метода проиллюстрирована примерами, в том числе примером из небесной механики. An approximate method of solving the Cauchy problem for nonlinear first-order ordinary differential equations is considered. The method is based on using the shifted Chebyshev series and a Markov quadrature formula. Some approaches are given to estimate the error of an approximate solution expressed by a partial sum of a certain order series. The error is estimated using the second approximation of the solution expressed by a partial sum of a higher order series. An algorithm of partitioning the integration interval into elementary subintervals to ensure the computation of the solution with a prescribed accuracy is discussed on the basis of the proposed approaches to error estimation.

scholarly journals A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammed AL-Smadi ◽  
Omar Abu Arqub ◽  
Ahmad El-Ajou
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Present Method ◽  
Periodic Boundary ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Periodic Boundary Conditions ◽  
First Order

The objective of this paper is to present a numerical iterative method for solving systems of first-order ordinary differential equations subject to periodic boundary conditions. This iterative technique is based on the use of the reproducing kernel Hilbert space method in which every function satisfies the periodic boundary conditions. The present method is accurate, needs less effort to achieve the results, and is especially developed for nonlinear case. Furthermore, the present method enables us to approximate the solutions and their derivatives at every point of the range of integration. Indeed, three numerical examples are provided to illustrate the effectiveness of the present method. Results obtained show that the numerical scheme is very effective and convenient for solving systems of first-order ordinary differential equations with periodic boundary conditions.

scholarly journals A numerical-analytic method for solving the Cauchy problem for ordinary differential equations

2011 ◽  
Vol 11 (4) ◽  
pp. 491-509 ◽  
Author(s):  
Volodymyr Makarov ◽  
Denys Dragunov
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Adomian Decomposition Method ◽  
Analytic Method ◽  
Discrete Method ◽  
First Order ◽  
Adomian Decomposition ◽  
The Cauchy Problem ◽  
Systems Of Odes

AbstractA functional-discrete method for solving the Cauchy problem for first-order ordinary differential equations (ODEs) is proposed. Generally speaking, this method (FD-method) is similar to the Adomian Decomposition Method (ADM). However, we have shown that for some problems the FD-method is convergent whereas the ADM is divergent. The results presented in this paper can easily be generalized to the case of systems of ODEs.

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

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