scholarly journals Approximate solution of the Cauchy problem for ordinary differential equations by the method of Chebyshev series

2016 ◽  
pp. 121-131
Author(s):  
О.Б. Арушанян ◽  
С.Ф. Залеткин
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Partial Sums ◽  
Quadrature Formulas ◽  
Chebyshev Series ◽  
Adams Methods ◽  
The Cauchy Problem ◽  
Discretization Step ◽  
Derivatives Of

Рассмотрен численно-аналитический метод решения систем обыкновенных дифференциальных уравнений, разрешенных относительно производных от искомых функций. Метод основан на приближенном представлении решения и его производной в виде частичных сумм смещенных рядов Чебышёва. Коэффициенты рядов определяются с помощью итераций с применением квадратурной формулы Маркова. Метод может быть использован для интегрирования обыкновенных дифференциальных уравнений с более высокой точностью и с более крупным шагом дискретизации по сравнению с традиционными численными методами типа Рунге-Кутта и Адамса. An approximate analytical method of solving the systems of ordinary differential equations resolved with respect to the derivatives of unknown functions is considered. This method is based on the approximation of the solution to the Cauchy problem and its derivatives by partial sums of shifted Chebyshev series. The coefficients of the series are determined by an iterative process with the use of Markov's quadrature formulas. This approach can be used to solve ordinary differential equations with a higher accuracy and with a larger discretization step compared to the known Runge-Kutta and Adams methods.

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 An implementation of the Chebyshev series method for the approximate analytical solution of second-order ordinary differential equations

2019 ◽  
pp. 97-103
Author(s):  
О.Б. Арушанян ◽  
С.Ф. Залеткин
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Canonical System ◽  
Second Order ◽  
Partial Sums ◽  
Problem Solution ◽  
Chebyshev Series ◽  
Canonical Systems ◽  
The Cauchy Problem ◽  
The Right

Описан один метод по применению рядов Чебышёва для интегрирования канонических систем обыкновенных дифференциальных уравнений второго порядка. Этот метод основан на аппроксимации решения задачи Коши, его первой и второй производных частичными суммами смещенных рядов Чебышёва. Коэффициенты рядов вычисляются итерационным способом с применением соотношений, связывающих коэффициенты Чебышёва решения задачи Коши, а также коэффициенты Чебышёва первой производной решения с коэффициентами Чебышёва правой части системы. Неотъемлемым элементом вычислительной схемы является использование формулы численного интегрирования Маркова для вычисления коэффициентов Чебышёва правой части системы. В статье не только сообщаются результаты, полученные численными расчетами, но и делается упор на высокоточном аналитическом представлении решения в виде частичной суммы ряда на промежутке интегрирования. A method used to apply the Chebyshev series for solving canonical systems of second order ordinary differential equations is described. This method is based on the approximation of the Cauchy problem solution and its first and second derivatives by partial sums of shifted Chebyshev series. The coefficients of these series are determined iteratively using the relations relating the Chebyshev coefficients of the solution and its first derivative with the Chebyshev coefficients found for the right-hand side of the canonical system by application of Markov's quadrature formula. The obtained numerical results are discussed and the high-precision analytical representations of the solution are proposed in the form of partial sums of Chebyshev series on a given integration segment.

scholarly journals On an approximate analytical method of solving ordinary differential equations

2015 ◽  
pp. 235-241
Author(s):  
О.Б. Арушанян ◽  
Н.И. Волченскова ◽  
С.Ф. Залеткин
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Method ◽  
Partial Sums ◽  
Quadrature Formulas ◽  
Normal System ◽  
Cauchy Problems ◽  
Chebyshev Series ◽  
Approximation Properties ◽  
Approximate Analytical Method

Описано использование смещенных рядов Чебышёва для интегрирования обыкновенных дифференциальных уравнений. Данный подход основан на аппроксимации решения задачи Коши для нормальной системы обыкновенных дифференциальных уравнений и его производной частичными суммами ряда Фурье по смещенным многочленам Чебышёва первого рода. Коэффициенты рядов вычисляются посредством итерационного процесса с применением квадратурной формулы Маркова. Подчеркнуто, что благодаря своим аппроксимационным свойствам частичные суммы рядов Чебышёва стали основой для построения приближенного аналитического метода интегрирования дифференциальных уравнений. Наряду с общими вопросами рассмотрен ряд примеров по применению частичных сумм ряда Чебышёва для приближенного представления решения задачи Коши на заданном отрезке для обыкновенных дифференциальных уравнений. The application of shifted Chebyshev series for solving ordinary differential equations is described. This approach is based on the approximation of the solution to the Cauchy problem for a normal system of ordinary differential equations and its derivatives by partial sums of Fourier series in the Chebyshev polynomials of the first kind. The coefficients of the series are determined by an iterative process with the use of Markov's quadrature formulas. The approximation properties of shifted Chebyshev series allow us to propose an approximate analytical method for ordinary differential equations. A number of examples are considered to illustrate the application of partial sums of Chebyshev series for approximate representations of the solutions to the Cauchy problems for ordinary differential equations.

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

scholarly journals The behavior of solutions of non-linear differential equations in Hilbert space. I

1988 ◽  
Vol 11 (1) ◽  
pp. 143-165 ◽  
Author(s):  
Vladimir Schuchman
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Differential Equations ◽  
Linear Case ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Linear Differential ◽  
Degenerate Linear

This paper deals with the behavior of solutions of ordinary differential equations in a Hilbert Space. Under certain conditions, we obtain lower estimates or upper estimates (or both) for the norm of solutions of two kinds of equations. We also obtain results about the uniqueness and the quasi-uniqueness of the Cauchy problems of these equations. A method similar to that of Agmon-Nirenberg is used to study the uniqueness of the Cauchy problem for the non-degenerate linear case.

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