scholarly journals Indignation moving singularity and analytical approximate solution of a nonlinear third-order equations

2021 ◽  
pp. 16-27
Author(s):  
Темирхан Султанович Алероев ◽  
Магомедюсуф Владимирович Гасанов
Keyword(s):  
Approximate Solution ◽  
Singular Point ◽  
Numerical Experiment ◽  
Existence And Uniqueness ◽  
Nonlinear Differential Equations ◽  
Third Order ◽  
Proved Theorem ◽  
Analytical Approximate Solution ◽  
Uniqueness Of The Solution ◽  
Posterior Estimate

В данной работе представлено исследование рассматриваемого класса нелинейных дифференциальных уравнений с подвижными особыми точками. Учитывая авторскую разработку теоремы существования и единственности решения построена структура аналитического приближенного решения, для которой, в данной работе, было установлено влияние возмущения подвижной особой точки. Представленные теоретические положения подтверждены с помощью численного эксперимента. Для оптимизации априорных оценок применялась апостериорная оценка. This paper presents a study of one class of nonlinear differential equations with movable singular points. On the basis of the previously proved theorem of existence and uniqueness of the solution, the structure of the analytical approximate solution was obtained, for which, in this work, the influence of the perturbation of a moving singular point was established. Results are tested using a numerical experiment. To optimize the prior estimates, the posterior estimate was used.

scholarly journals An approximate solution of the one class third order onlinear differential equation in the analyticity domain

2020 ◽  
pp. 45-54
Author(s):  
Виктор Николаевич Орлов ◽  
Рио-Рита Вадимовна Разакова
Keyword(s):  
Approximate Solution ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
Nonlinear Differential Equations ◽  
A Priori ◽  
Existence And Uniqueness Theorem ◽  
Third Order ◽  
Analyticity Domain ◽  
Priori Estimates ◽  
The One

В работе рассмотрен класс нелинейных дифференциальных уравнений третьего порядка с полиномиальной правой частью шестой степени. Доказана теорема существования и единственности решения в области аналитичности. Построено аналитическое приближенное решение. Предложен вариант оптимизации априорных оценок с помощью апостериорных. Проведен численный эксперимент. There is a class of third-order nonlinear differential equations with polynomial right part of the sixth degree considered in the paper. The existence and uniqueness theorem of a solution in the domain of analyticity is proved by authors. There is an analytical approximate solution which was constructed by V. Orlov and R. Razakova. A variant of optimization of a priori estimates using posterior ones is proposed by authors. A numerical experiment is carried out too.

scholarly journals Discrete Approximate Solution of Differential Inclusion with Normal Cone and Prox-Regular Set.

2022 ◽  
Author(s):  
Abdallah Beddani ◽  
Rahma Sahraoui
Keyword(s):  
Approximate Solution ◽  
Differential Inclusion ◽  
Existence And Uniqueness ◽  
Normal Cone ◽  
Approximation Property ◽  
Knowing That ◽  
Nite Element ◽  
New Variant ◽  
Uniqueness Of The Solution ◽  
Sweeping Processes

Abstract Our aim is to calculate the discrete approximate solution of di⁄erential inclusion with normal cone and prox-regular set, the question is how to calculate this solution? We use the discrete approximation property of a new variant of nonconvex sweeping processes involving normal cone and a nite element method. Knowing that The majority of mathematicians have proved only the existence and uniqueness of the solution for this type of inclusions, like: Mordukhovich, Thibault, Aubin, Messaoud,
...etc.

scholarly journals The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Huanting Li ◽  
Yunfei Peng ◽  
Kuilin Wu
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Initial State ◽  
Switching Line ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching <inline-formula><tex-math id="M1">\begin{document}$ k\in \mathbb{N} $\end{document}</tex-math></inline-formula> times on bounded time intervals.</p>

scholarly journals On the weighted boundary value problem for systems of degenerate ordinary differential equations

2010 ◽  
Vol 51 ◽  
Author(s):  
Stasys Rutkauskas ◽  
Igor Saburov
Keyword(s):  
Singular Point ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Linear Equations ◽  
System Of Equations ◽  
The Matrix ◽  
Uniqueness Of The Solution ◽  
Weighted Boundary ◽  
Well Posed

A system of ordinary second order linear equations with a singular point is considered. The aim of this work is such that the system of eigenvectors of the matrix that couples the system of equations is not complete. That implies a matter of the statement of a weighted boundary value problem for this system. The well-posed boundary value problem is proposed in the article. The existence and uniqueness of the solution is proved.

scholarly journals Exact boundaries for the analytical approximate solution of a class of first-order nonlinear differential equations in the real domain

2021 ◽  
Vol 25 (2) ◽  
pp. 381-392
Author(s):  
Victor Nikolaevich Orlov ◽  
Oleg Aleksandrovich Kovalchuk
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Analytical Approximate Solution ◽  
The Real ◽  
First Order ◽  
Real Domain

Дано решение одной из задач аналитического приближенного метода для одного класса нелинейных дифференциальных уравнений первого порядка с подвижными особыми точками в вещественной области. Рассматриваемое уравнение в общем случае не разрешимо в квадратурах и имеет подвижные особые точки алгебраического типа. Это обстоятельство требует решение ряда математических задач. Ранее авторами была решена задача влияния возмущения подвижной особой точки на аналитическое приближенное решение. Это решение основывалось на классическом подходе и, при этом, существенно уменьшилась область применения аналитического приближенного решения, по сравнению с областью, полученной в доказанной теореме существования и единственности решения. Поэтому в статье предлагается новая технология исследования, основанная на элементах дифференциального исчисления. Этот подход позволяет получить точные границы для аналитического приближенного решения в окрестности подвижной особой точки. Получены новые априорные оценки для аналитического приближенного решения рассматриваемого класса уравнений, хорошо согласующиеся с известными для общей области действия. При этом, представленные результаты дополняют ранее полученные, существенно расширена область применения аналитического приближенного решения в окрестности подвижной особой точки. Приведенные расчеты согласуются с теоретическими положениями, о чем свидетельствуют эксперименты, проведенные с нелинейным дифференциальным уравнением, обладающим точным решением. Дана технология оптимизации априорных оценок погрешности с помощью апостериорных оценок. В исследованиях применялись ряды с дробными отрицательными степенями.

NONLOCAL BOUNDARY VALUE PROBLEM FOR ONE MIXED THIRD ORDER EQUATION

2021 ◽  
pp. 18-24
Author(s):  
Alena G. Ezaova ◽  
Liana V. Kanukoeva ◽  
Gennady V. Kupovykh
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Original Problem ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
The Third ◽  
Uniqueness Of The Solution ◽  
The Given

The paper considers a nonlocal boundary value problem for a mixed hyperbolic-parabolic equation of the third order. The equation is considered in a finite simply connected domain consisting of a hyperbolic and a parabolic part. The solution to the problem posed is considered for various cases of the parameter λ, which is in the original equation. In the case when (1-2m)/2&lt; &lt;λ&lt;1, the solution of the problem is reduced to a singular integral equation, which is reduced by the well-known Carleman-Vekua method to the Fredholm integral equation of the third kind. In the case when λ=(1-2m)/2, a theorem on the existence and uniqueness of a solution to the problem posed is formulated and proved. To prove the uniqueness of the solution, the method of energy integrals is used and inequalities of the type are derived on the given functions that are in the boundary condition. It is shown that the homogeneous problem corresponding to the original problem, under the conditions of the uniqueness theorem, has only a trivial solution in the entire considered domain. From which we can conclude that the original problem has only a single solution. If the obtained conditions for the given functions are violated, the problem posed does not have a unique solution. When investigating the question of the existence of a solution to the problem posed, a system of two equations is considered, consisting of the basic functional relations between the trace of the desired function and the traces of the derivative of the desired function, brought to the line of degeneration y = 0. Eliminating from the system the function τ (x) - the trace of the desired solution on the line of degeneration, we arrive at an equation for the trace of the derivative of the desired function. Under the condition of the existence and uniqueness theorem, the problem posed is equivalently reduced to the Fredholm integral equation of the second kind, the unconditional solvability of which follows from the uniqueness of the solution to the problem posed.

Mathematical modeling of building structures and nonlinear differential equations

2020 ◽  
Vol 11 (03) ◽  
pp. 2050026
Author(s):  
Victor Orlov ◽  
Yulia Zheglova
Keyword(s):  
Mathematical Modeling ◽  
Approximate Solution ◽  
Differential Equations ◽  
Numerical Experiment ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Singular Points ◽  
Building Structures ◽  
Approximate Methods ◽  
Analytic Domain

Nonlinear differential equations with moving singular points require emergence and development of new approximate methods of solution. In this paper, we give a solution to one of the problems of the analytical approximate method for solving nonlinear differential equations with moving singular points, and study the influence of the perturbation of the initial conditions on the analytical approximate solution in the analytic domain. Theoretical material was tested using a numerical experiment confirming its reliability. The theoretical material presented in this paper allows researchers to use nonlinear differential equations with moving singular points when designing mathematical models of building structures.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

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