Criteria for the Existence of an Isolated Solution of a Nonlinear Boundary-Value Problem

Dzhumabaev, D. S.; Temesheva, S. M.

doi:10.1007/s11253-018-1507-y

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…The main feature of the parameterization method for studying and solving boundary value problems for differential equations, in particular nonlinear two‐point boundary value problems for ordinary differential equations, is that this method allows to establish their solvability criteria [18, 19]. In order to strengthen the continuity of the research of this paper in this aspect, we will further try to prove that the obtained sufficient conditions of solvability of the problem ()–() are also necessary.…”

Section: Discussionmentioning

confidence: 99%

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On a problem for a delay differential equation

Iskakova

Temesheva

Uteshova

2023

Math Methods in App Sciences

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The article considers a nonlinear boundary value problem for a linear delay differential equation. To solve the problem, the idea of parametrization method, namely, the interval at which the problem is being considered, is divided into subintervals whose lengths do not exceed the values of the constant delay; constant parameters are introduced at the left ends of these intervals; a new unknown function is introduced at each subinterval. Thus, the problem under consideration is reduced to an equivalent multipoint boundary value problem for differential equations with delay containing parameters. Auxiliary Cauchy problems without delay with zero initial conditions at the left ends of the subintervals are consistently considered on each of the subintervals. Using an analog of the Cauchy formula to represent the solution of a system of linear differential equations on each of the subintervals and given nonlinear boundary conditions, an algebraic system with respect to unknown parameters is obtained. The article proposes an algorithm for finding a solution to a multipoint boundary value problem for differential equations with a delay containing parameters. At each step of the algorithm, a system of nonlinear algebraic equations is solved to determine the values of the parameters, and an analog of the Cauchy formula is used to obtain solutions to auxiliary Cauchy problems. The results obtained are demonstrated on a test problem.

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Section: Discussionmentioning

confidence: 99%

“…Let us find the k-th approximation 𝜆 (k) of the parameter 𝜆 from the equation Q Δ 𝓁 ( 𝜆, u (k−1) ) = 0. Similarly to (19), using (25), (26) and the equality…”

Section: 𝓁N+1mentioning

confidence: 99%

On a problem for a delay differential equation

Iskakova

Temesheva

Uteshova

2023

Math Methods in App Sciences

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New General Solutions of Ordinary Differential Equations and The Methods for The Solution of Boundary-Value Problems

Dzhumabaev

2019

Ukr Math J

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Solution of Second- and Higher-Order Nonlinear Two-Point Boundary-Value Problems Using Double Decomposition Method

et al. 2022

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The present paper makes use of the efficient double decomposition method to propose a method for solving two-point boundary-value problems, featuring second- and higher-order nonlinear ordinary differential equations. The efficacy of the proposed method is demonstrated on numerous test problems. In the end, a high level of exactitude between the obtained approximate solution and the available exact solution is achieved.

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Criteria for the Existence of an Isolated Solution of a Nonlinear Boundary-Value Problem

Cited by 3 publications

References 8 publications

On a problem for a delay differential equation

On a problem for a delay differential equation

New General Solutions of Ordinary Differential Equations and The Methods for The Solution of Boundary-Value Problems

Solution of Second- and Higher-Order Nonlinear Two-Point Boundary-Value Problems Using Double Decomposition Method

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