On a problem for a delay differential equation

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 const...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2023-06, Vol.46 (9), p.11283-11297
Main Authors: Iskakova, Narkesh, Temesheva, Svetlana, Uteshova, Roza
Format: Article
Language:English
Subjects:
Algebra
algorithm
Algorithms
Boundary conditions
Boundary value problems
Cauchy problems
Delay
delay differential equation
Differential equations
Formulas (mathematics)
Initial conditions
isolated solution
Mathematical analysis
nonlinear boundary‐value problem
Parameterization
Parameters
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Summary: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.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.9181