A method for solving a boundary value problem with a parameter for impulsive integro-differential equation involving integral constraints
DOI:
https://doi.org/10.70474/2ff0qt13Keywords:
boundary problem with a parameter, impulsive integro-differential equation, integral constraint, parametrization method, numerical algorithmAbstract
In this paper, we investigate a boundary value problem with a parameter for an impulsive integro-differential equations involving integral constraints. To solve the problem, we employ the Dzhumabaev parametrization method in combination with numerical techniques. By introducing additional parameters, the original boundary value problem is reduced to an equivalent system of linear algebraic equations. The coefficients and the right-hand side of this system are computed by solving Cauchy problems for ordinary differential equations on subintervals. Furthermore, sufficient conditions for the existence and uniqueness of the solution to the boundary value problem with a parameter are established. Based on the obtained theoretical results, we develop a constructive numerical method and propose algorithms for its implementation. The efficiency of the presented approach is emonstrated
through numerical example.
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Lakshmikantham V., Bainov D.D., Simeonov P.S. Theory of Impulsive Differential Equations. World Scientific, Singapore, 1989, 288 p. DOI: 10.1142/0906.
Bainov D.D., Simeonov P.S. Systems with Impulse Effect. Stability, Theory and Applications. Ellis Horwood Ltd./John Wiley & Sons, Chichester, 1989, 255 p.
Samoilenko A.M., Perestyuk N.A. Impulsive Differential Equations. World Scientific, Singapore, 1995, 472 p. DOI: 10.1142/2892.
Hammou B.S., Djerad A., Memou A., Latrous C. A boundary value problem with an integral condition for a certain fractional differential equation. Math. Model. Anal., Vol. 30, No. 3, 2025, P. 405--420. DOI: 10.3846/mma.2025.21481. DOI: https://doi.org/10.3846/mma.2025.21481
Abdullaev A., Mardiev A., Usmonov B., Makhmudova D. Unique solvability of boundary value problem with integral condition for model equation of second kind. AIP Conf. Proc., Vol. 3256, 2025, 040008. DOI: 10.1063/5.0266861. DOI: https://doi.org/10.1063/5.0266861
Kadirbayeva Z.M., Kadirbayeva R. Numerical solution of second-order impulsive differential equations with loadings subject to integral boundary conditions. Math. Meth. Appl. Sci., Vol. 48, No. 6, 2025, P. 6269--6277. DOI: 10.1002/mma.10670. DOI: https://doi.org/10.1002/mma.10670
Kadirbayeva Zh., Bakirova E., Tleulessova A. Solving Fredholm integro-differential equations involving integral condition: A new numerical method. Math. Slovaca, Vol. 74, No. 2, 2024, P. 403--416. DOI: 10.1515/ms-2024-0031. DOI: https://doi.org/10.1515/ms-2024-0031
Gallardo J.M. Second-order differential operators with integral boundary conditions and generation of analytic semigroups. Rocky Mountain J. Math., Vol. 30, No. 4, 2000, P. 1265--1291. DOI: 10.1216/rmjm/1021477351. DOI: https://doi.org/10.1216/rmjm/1021477351
Karakostas G.L., Tsamatos P.Ch. Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ., No. 30, 2002, P. 1--17.
Lomtatidze A., Malaguti L. On a nonlocal boundary-value problems for second order nonlinear singular differential equations. Georg. Math. J., Vol. 7, 2000, P. 133--154. DOI: 10.1515/GMJ.2000.133. DOI: https://doi.org/10.1515/GMJ.2000.133
Corduneanu C. Integral Equations and Applications. Cambridge Univ. Press, Cambridge, 1991.
Agarwal R.P., O'Regan D. Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Acad. Publ., Dordrecht, 2001.
Dzhumabayev D. Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation. U.S.S.R. Comput. Math. Math. Phys., Vol. 29, No. 1, 1989, P. 34--46. DOI: https://doi.org/10.1016/0041-5553(89)90038-4
Dzhumabaev D. A method for solving the linear boundary value problem for an integro-differential equation. Comput. Math. Math. Phys., Vol. 50, No. 7, 2010, P. 1150--1161. DOI: 10.1134/S0965542510070043. DOI: https://doi.org/10.1134/S0965542510070043
Dzhumabaev D. On one approach to solve the linear boundary value problems for Fredholm integro-differential equations. J. Comput. Appl. Math., Vol. 294, 2016, P. 342--357. DOI: 10.1016/j.cam.2015.08.023. DOI: https://doi.org/10.1016/j.cam.2015.08.023
Dzhumabaev D., Bakirova E. On the unique solvability of the boundary-value problems for Fredholm integro-differential equations with degenerate kernel. J. Math. Sci., Vol. 220, No. 4, 2017, P. 440--460. DOI: 10.1007/s10958-016-3194-2. DOI: https://doi.org/10.1007/s10958-016-3194-2
Bakirova E., Iskakova N., Assanova A. Numerical method for the solution of linear boundary-value problems for integro-differential equations based on spline approximations. Ukrainian Math. J., Vol. 71, No. 9, 2020, P. 1341--1358. DOI: 10.1007/s11253-020-01719-8. DOI: https://doi.org/10.1007/s11253-020-01719-8
Assanova A., Bakirova E., Kadirbayeva Zh. Two-point boundary value problem for Volterra--Fredholm integro-differential equations and its numerical analysis. Lobachevskii J. Math., Vol. 44, No. 3, 2023, P. 1100--1110. DOI: 10.1134/S1995080223030058. DOI: https://doi.org/10.1134/S1995080223030058
Bakirova E., Assanova A., Kadirbayeva Zh. Solving multi-point problem for Volterra--Fredholm integro-differential equations by using Dzhumabaev parametrization method. Open Math., Vol. 22, No. 1, 2024, 20240024. DOI: 10.1515/math-2024-0024. DOI: https://doi.org/10.1515/math-2024-0024
Bakirova E., Iskakova N., Kadirbayeva Zh. Numerical implementation for solving the boundary value problem for impulsive integro-differential equations with parameter. J. Math. Mech. Comput. Sci., Vol. 119, No. 3, 2023, P. 19--29. DOI: 10.26577/JMMCS2023v119i3a2. DOI: https://doi.org/10.26577/JMMCS2023v119i3a2
Mynbayeva S., Assanova A., Uteshova R. A solution to a linear boundary value problem for an impulsive integro-differential equation. Differ. Equ. Dyn. Syst., 2023. DOI: 10.1007/s12591-023-00665-8.
Mynbayeva S., Assanova A., Tankeyeva A. On the solvability of a quasilinear boundary value problem for an impulsive system. Lobachevskii J. Math., Vol. 45, No. 10, 2024, P. 5146--5155. DOI: 10.1134/S1995080224606088. DOI: https://doi.org/10.1134/S1995080224606088
Tleulessova A., Temesheva S., Orazbekova A. On a nonlinear boundary value problem with impulse action. Open Math., Vol. 23, No. 1, 2025, 20250176. DOI: 10.1515/math-2025-0176. DOI: https://doi.org/10.1515/math-2025-0176
Kadirbayeva Z.M. A numerical method for solving boundary value problem for essentially loaded differential equations. Lobachevskii J. Math., Vol. 42, No. 3, 2021, P. 551--559. DOI: 10.1134/S1995080221030112. DOI: https://doi.org/10.1134/S1995080221030112
Kadirbayeva Zh., Kabdrakhova S. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition. Open Math., Vol. 20, No. 1, 2022, P. 1173--1183. DOI: 10.1515/math-2022-0496. DOI: https://doi.org/10.1515/math-2022-0496
Iskakova N., Temesheva S., Uteshova R. On a problem for a delay differential equation. Math. Meth. Appl. Sci., Vol. 46, No. 9, 2023, P. 11283--11297. DOI: 10.1002/mma.9181. DOI: https://doi.org/10.1002/mma.9181
Iskakova N.B., Bakirova E.A., Khanzharova B.S., Kadirbayeva Zh.M. An algorithm for solving boundary value problems for delay differential equations with loadings. Lobachevskii J. Math., Vol. 45, 2024, P. 5032--5042. DOI: https://doi.org/10.1134/S1995080224606064
Kadirbayeva Zh., Iskakova N., Bakirova E. Numerical implementation of solving a boundary value problem including both delay and parameter. Math. Slovaca, Vol. 75, No. 3, 2025, P. 601--610. DOI: 10.1515/ms-2025-0043. DOI: https://doi.org/10.1515/ms-2025-0043
Assanova A.T., Imanchiyev A.E., Kadirbayeva Z.M. Solvability of nonlocal problems for systems of Sobolev-type differential equations with a multipoint condition. Russian Math., Vol. 63, No. 12, 2019, P. 1--12. DOI: 10.3103/S1066369X19120016. DOI: https://doi.org/10.3103/S1066369X19120016
Tokmagambetova T.D., Orumbayeva N.T. On one solution of a periodic boundary value problem for a hyperbolic equation. Bull. Karaganda Univ. Math. Ser., Vol. 109, No. 1, 2023, P. 141--155. DOI: 10.31489/2023M1/141-155. DOI: https://doi.org/10.31489/2023m1/141-155
Abdimanapova P.B., Temesheva S.M. On a solution of a nonlinear nonlocal boundary value problem for one class of hyperbolic equation. Lobachevskii J. Math., Vol. 44, No. 7, 2023, P. 2529--2541. DOI: 10.1134/S1995080223070028. DOI: https://doi.org/10.1134/S1995080223070028
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