About one approach for solving bounary-value problems of differential equation of pantograph type
55 52
Keywords:
Pantograph equation, convergence, boundary value problem, parameterization method, solvability, Cauchy problem.Abstract
Pantograph equations have been studied for a long time. In 1940, K. Mahler introduced functional differential equations of this type into number theory. In 1971, J. Ockendon used a functional differential equation with a transformed argument, , to describe the dynamics of the pantograph of an electric locomotive. Subsequently, Pantograph-type equations were studied in the works of many authors.
This article considers a boundary value problem for a differential equation of the pantograph type. To solve the posed boundary value problem, the parameterization method proposed by Professor D. Dzhumabaev is used. To do this, we denote the value of the function at the initial point of the segment under consideration by a parameter and perform a change of variable Then the solvability of the original boundary value problem is reduced to studying the solvability of the resulting Cauchy problem for the original equation and to a linear algebraic equation to determine the introduced parameter. Next, using the method of successive approximations, we find solutions to the Cauchy problem for the Pantograph type equation. The convergence of the resulting sequence and the convergence of its solution to the solution of the Cauchy problem of Pantograph type are proved. By requiring continuity of the free term, we establish its unique solvability. Substituting the resulting solution into a linear algebraic equation to
determine the entered parameter, we calculate the values of the entered parameter using the original data. Substituting the resulting expressions into we find the solution to the original problem . And assuming the unique solvability of a linear algebraic equation, we establish the solvability of the boundary value problem for an equation of Pantograph type.
References
K. Mahler. On a special functional equation. J. London Math. Soc. – 1940. -1(2). - P. 115-123.
L. Fox, D. F. Mayers, J. R. Ockendon and A. B. Tayler. On a functional differantical equation. IMA Journal of Applied Mathematics. – 1971. - 8(3). -P. 271-307.
G R Morris, A Feldstein, and E W Bowen, The Phragmen Lindelof principle and a class of functional differential equations. Ordinary Differential Equations// Academic Press, New York. – 1972. – P. 513-540.
Быков Я.В., Быкова Л.Я., Шевцов Е.И. Достаточные условия осцилляторности решений нелинейных дифференциальных уравнений с отклоняющимся аргументом // Дифференц. уравнения. - 1973. - Т. 9, - № 9. - С. 1555–1560.
Родионов В.И. Аналог функции Коши для обобщенного уравнения с несколькими отклонениями аргумента // Дифференц. уравнения. - 2013. - Т. 49, - № 6. - С. 690–706.
A.Iserles, Yunkang Liu. Integro-differential equations and generalized hypergeometric functions. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, 1995
Dzhumabayev D.S. Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation// Computational Mathematics and Mathematical Physics. -1989. -Vol.29, - No. 1. - P. 34-46.
Джумабаев Д.С. Об одном методе решения линейной краевой задачи для интегродифференциального уравнения // Журнал вычисл. матем. и матем. физ. - 2010. - Т. 50, - № 7. - С. 1209-1221.
Dulat Dzhumabaev, “Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro-differential equations”, Mathematical Methods in the Applied Sciences. – 2018. - 41(4). - P. 1439-1462.
Nazarova K.Zh., Usmanov K.I. Unique solvability of the boundary value problem for integro-differential equations with involution // AIP Conference Proceedings. – 2021. – 2365(070012).
K. Nazarova, K. Usmanov., Оn a boundary value problem for systems of
integro-differential equations with involution // International Journal of Applied Mathematics. - 2021. –Vol.34. - P. 225-235.
Арнольд В. И. Обыкновенные дифференциальные уравнения. М.: МЦНМО, 2018. - 344 с.
K Mahler. On a special functional equation. J. London Math. Soc. – 1940. -1(2). - P. 115-123.
L Fox, D F Mayers, J R Ockendon and A B Tayler. On a functional differantical equation. IMA Journal of Applied Mathematics. – 1971. - 8(3). -P. 271-307.
G R Morris, A Feldstein, and E W Bowen, The Phragmen Lindelof principle and a class of functional differential equations. Ordinary Differential Equations// Academic Press, New York. – 1972. – P. 513-540.
Bykov Ya.V., Bykova L.Ya., Shevtsov E.I. Sufficient conditions for the oscillatory nature of solutions of nonlinear differential equations with deviating argument // Differents. Equations. - 1973. - T. 9, - No. 9. - P. 1555–1560.
Rodionov V.I. Analogue of the Cauchy function for a generalized equation with several deviations of the argument // Differents. Equations. - 2013. - T. 49, - No. 6. P. 690–706.
A.Iserles, Yunkang Liu. Integro-differential equations and generalized hypergeometric functions. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, 1995
Dzhumabayev D.S. Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation// Computational Mathematics and Mathematical Physics. -1989. -Vol.29, - No. 1. - P. 34-46.
D. S. Dzhumabaev, “On a method for solving a linear boundary value problem for an integrodifferential equation”, Comput. Math. Math. Phys., - 2010. - Т. 50, - № 7. - P. 1209-1221.
Dulat Dzhumabaev, “Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro-differential equations”, Mathematical Methods in the Applied Sciences. – 2018. - 41(4). - P. 1439-1462.
Nazarova K.Zh., Usmanov K.I. Unique solvability of the boundary value problem for integro-differential equations with involution // AIP Conference Proceedings. – 2021. – 2365(070012).
K. Nazarova, K. Usmanov., Оn a boundary value problem for systems of
integro-differential equations with involution // International Journal of Applied Mathematics. - 2021. –Vol.34. - P. 225-235.
Arnold V.I. Ordinary differential equations. M.: MTsNMO, 2018. - 344 p.
