A problem with parameter for the integro-differential equations
The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.
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V.M. Abdullaev and K.R. Aida-zade. On the numerical solution of loaded systems of ordinary differential equations with nonseparated multipoint and integral conditions. Numer. Anal. Appl., 7(1):1–17, 2014. http://doi.org/10.1134/S1995423914010017
M.U. Akhmetov, A. Zafer and R.D. Sejilova. The control of boundary value problems for quasilinear impulsive integro-differential equations. Nonl. Anal.: Theory, Methods & Appl., 48(1):271–286, 2002.
A.T. Assanova, E.A. Bakirova and Z.M. Kadirbayeva. Numerical solution to a control problem for the integro-differential equations. Comput. Math. Math. Phys., 60(2):203–221, 2020. http://doi.org/10.1134/S0965542520020049
A.T. Assanova, A.E. Imanchiyev and Z.M. Kadirbayeva. Numerical solution of systems of loaded ordinary differential equations with multipoint conditions. Comput. Math. Math. Phys., 58(4):508–516, 2018. http://doi.org/10.1134/S096554251804005X
A.T. Assanova and Zh.M. Kadirbayeva. On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations. Comput. and Appl. Math., 37(4):4966–4976, 2018. http://doi.org/10.1007/s40314-018-0611-9
K.I. Babenko. Fundamentals of Numerical Analysis. Nauka, Moscow, 1986. (in Russian)
A.A. Boichuk and A.M. Samoilenko. Generalized inverse operators and Fredholm boundary-value problems. SP, Utrecht, Boston, 2004.
H. Brunner. Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge, 2004.
H. Cohen. Numerical approximation methods. Springer, New York, 2011.
D.S. Dzhumabaev. A method for solving the linear boundary value problem for an integro-differential equation. Comput. Math. Math. Phys., 50(7):1150–1161, 2010. http://doi.org/10.1134/S0965542510070043
D.S. Dzhumabaev. An algorithm for solving the linear boundary value problem for an integro-differential equation. Comput. Math. Math. Phys., 53(6):736–758, 2013. http://doi.org/10.1134/S0965542513060067
D.S. Dzhumabaev. Necessary and sufficient conditions for the solvability of linear boundary-value problems for the fredholm integro-differential equation. Ukr. Math. J., 66(8):1200–1219, 2015. http://doi.org/10.1007/s11253-015-1003-6
D.S. Dzhumabaev. On one approach to solve the linear boundary value problems for Fredholm integro-differential equations. J. Comput. Appl. Math., 294(2):342– 357, 2016. http://doi.org/10.1016/j.cam.2015.08.023
D.S. Dzhumabaev. Computational methods of solving the boundary value problems for the loaded differential and fredholm integro-differential equations. Math. Meth. Appl. Sci., 41(4):1439–1462, 2018. http://doi.org/10.1002/mma.4674
D.S. Dzhumabayev. Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation. USSR Comput. Math. Math. Phys., 29(1):34–46, 1989.
R. Kangro and E. Tamme. On fully discrete collocation methods for solving weakly singular integro-differential equations. Math. Model. Anal., 15(1):69–82, 2010. ttps://doi.org/10.3846/1392-6292.2010.15.69-82
A.M. Krall. The development of general differential and general differentialboundary systems. Rocky Mountain J. Math., 5(4):493–542, 1975.
A.Yu. Luchka and O.B. Nesterenko. Projection method for the solution of integro-differential equations with restrictions and control. Nonl. Oscil., 11(2):219–228, 2008. http://doi.org/10.1007/s11072-008-0025-5
A.Yu. Luchka and O.B. Nesterenko. Construction of solution of integrodifferential equations with restrictions and control by projection-iterative method. Nonl. Oscil., 12(1):85–93, 2009. http://doi.org/10.1007/s11072-009-0061-9
A.M. Nakhushev. Boundary value problems for loaded integral-differential equations of hyperbolic type and their applications to the soil moisture forecast. Differencial’nye Uravneniya, 15(1):96–105, 1979.
A.M. Nakhushev. On an approximate method of solving boundary value problems for differential equations and its applications to the dynamics of soil moisture and groundwater. Differencial’nye Uravneniya, 18(1):72–81, 1982. (in Russian)
A.M. Nakhushev. Loaded equations and their applications. Nauka, Moscow, 2012. (in Russian)
O.B. Nesterenko. Modified projection-iterative method for weakly nonlinear integro differential equations with parameters. J. Math. Sci., 198(3):328–335, 2014. http://doi.org/10.1007/s10958-014-1793-3
A. Pedas and E. Tamme. Product integration for weakly singular integro-differential equations. Math. Model. Anal., 16(1):153–172, 2011. http://doi.org/10.3846/13926292.2011.564771