A parametrization method for solving nonlinear two-point boundary value problems

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

doi:10.1134/s096554250701006x

Cited by 16 publications

(8 citation statements)

References 1 publication

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“…The operator Q Δ 𝓁 (𝜆, u (1) ) in S(𝜆 (1) , 𝜌 1 ) satisfies all conditions of Theorem 1 [16]. Therefore, the iterative process 𝜆 (2,0) = 𝜆 (1) , (2,m) , u (1) ) , m = 0, 1, 2, … .…”

Section: Sufficient Conditions For the Existence Of An Isolated Solut...mentioning

confidence: 90%

“…The operator Q Δ 𝓁 (𝜆, u (0) ) satisfies all assumptions of Theorem 1 [16] in S(𝜆 (0) , 𝜌 𝜆 ). Choosing a number 𝜀 0 > 0, satisfying the inequalities…”

Section: Sufficient Conditions For the Existence Of An Isolated Solut...mentioning

confidence: 98%

“…By Theorem 1 [16], the iterative process (17) converges to 𝜆 (1) ∈ S ( 𝜆 (0) , 𝜌 𝜆 ) , an isolated solution to the equation…”

Section: Sufficient Conditions For the Existence Of An Isolated Solut...mentioning

confidence: 99%

“…The authors of this paper successfully applied the parameterization method to study and solve periodic boundary value problems for linear delayed differential equations [15] and nonlinear two-point boundary value problems for ordinary differential equations [16]. The obtained results motivated the authors to study the (1)-(3) problem.…”

Section: Introduction and Statement Of The Problemmentioning

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: Sufficient Conditions For the Existence Of An Isolated Solut...mentioning

confidence: 90%

“…The operator Q Δ 𝓁 (𝜆, u (0) ) satisfies all assumptions of Theorem 1 [16] in S(𝜆 (0) , 𝜌 𝜆 ). Choosing a number 𝜀 0 > 0, satisfying the inequalities…”

Section: Sufficient Conditions For the Existence Of An Isolated Solut...mentioning

confidence: 98%

“…By Theorem 1 [16], the iterative process (17) converges to 𝜆 (1) ∈ S ( 𝜆 (0) , 𝜌 𝜆 ) , an isolated solution to the equation…”

Section: Sufficient Conditions For the Existence Of An Isolated Solut...mentioning

confidence: 99%

Section: Introduction and Statement Of The Problemmentioning

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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“…A solution to the problem ( 26)-( 29) is a pair (26), boundary conditions (27), initial conditions on splitting lines (28), and continuity condition (29).…”

Section: A Family Of Boundary Value Problems For a Differential Equation With Multiple Loadsmentioning

confidence: 99%

A Family of Two-Point Boundary Value Problems for Loaded Differential Equations

Assanova

Zholamankyzy

2021

Russ Math.

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A family of two-point boundary value problems for loaded differential equations is considered on a rectangular domain. By introducing new additional functions, the considered problem is reduced to an equivalent family of problems for differential equations with parameters. Sufficient existence conditions of the unique solution to the family of twopoint boundary value problems for loaded differential equations are established in terms of input data.

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A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

Dzhumabaev

Бакирова

Mynbayeva

2019

Math Methods in App Sciences

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A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed. KEYWORDS loaded differential equation with parameter, method of solving, nonlinear boundary value problem MSC CLASSIFICATION

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A parametrization method for solving nonlinear two-point boundary value problems

Cited by 16 publications

References 1 publication

On a problem for a delay differential equation

On a problem for a delay differential equation

A Family of Two-Point Boundary Value Problems for Loaded Differential Equations

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

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