MODIFICATION OF THE PARAMETRIZATION METHOD FOR SOLVING A BOUNDARY VALUE PROBLEM FOR LOADED DIFFERENTIAL EPCAG
DOI:
https://doi.org/10.26577/JMMCS.2022.v115.i3.02Keywords:
load, piecewise-constant argument, two-point boundary value problem, parametrization method, numerical solutionAbstract
In this paper, modification of Dzhumabaev parameterization method is developed to a boundary value problem for systems of loaded differential equations with piecewise constant argument of generalized type (EPCAG). The method is based on reducing the considering problem to an equivalent multi-point boundary value problem for ordinary differential equations with parameters. An equivalent boundary value problem with parameters consists of the Cauchy problem for a system of ordinary differential equations with parameters, a two-point condition, a continuity condition, and additional conditions for a piecewise constant argument. The solution of the Cauchy problem for a system of ordinary differential equations with parameters is constructed using the fundamental matrix of the differential equation. A system of linear algebraic equations for the parameters is compiled using the values of the solution at the corresponding points and substituting them into the two-point condition, the continuity condition, and the conditions for the piecewise constant argument. A modification of Dzhumabaev parameterization method for solving the considering boundary value problem is proposed which is based on solving the constructed system and the 4th order Runge-Kutta method for solving the Cauchy problem on subintervals. The obtained results are verified by a numerical example. Numerical analysis showed high efficiency of the constructed modification of Dzhumabaev parameterization method.
References
[2] Cooke K.L., Wiener J., "Retarded differential equations with piecewise constant delays" , J Math Anal Appl, 99, (1984): 179-189.
[3] Akhmet M., Yilmaz E., "Neural Networks with Discontinuous/Impact Activations" , (Springer, New York, 2014).
[4] Akhmet M.U., "Nonlinear hybrid continuous/discrete time models" , ( Atlantis, Amsterdam-Paris, 2011).
[5] Akhmet M.U., "Almost periodic solution of differential equations with piecewise-constant argument of generalized type" , Nonlinear Analysis-Hybrid Systems, 2, (2008): 456-467.
[6] Akhmet M.U., "On the reduction principle for differential equations with piecewise-constant argument of generalized type" , J. Math. Anal. Appl., 1, (2007): 646-663.
[7] Dzhumabaev D.S., "Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation" , USSR Comput. Math. Math. Phys., 1, (1989): 34–46.
[8] Temesheva S.M., Dzhumabaev D.S., Kabdrakhova S.S. "On One Algorithm To Find a Solution to a Linear Two-Point Boundary Value Problem" , Lobachevskii J. of Math., 42, (2021): 606-612.
[9] Nakhushev A.M., "Loaded equations and their applications" , (Nauka, Moscow, 2012) (in Russian).
[10] Nakhushev A.M., "An approximation method for solving boundary value problems for differential equations with applications to the dynamics of soil moisture and groundwater" , Differential Equations, 18, (1982): 72–81.
[11] Dzhenaliev M.T. "Loaded equations with periodic boundary conditions" , Differential Equations, 37, (2001): 51-57.
[12] Abdullaev V.M., Aida-zade K.R. "Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations" , Comput. Math. Math. Phys., 54, (2014): 1096-1109.
[13] Assanova A.T., Kadirbayeva Zh.M. "Periodic problem for an impulsive system of the loaded hyperbolic equations" , Electronic Journal of Differential Equations, 72, (2018): 1-8.
[14] Assanova A.T., Imanchiyev A.E., Kadirbayeva Zh.M. "Numerical solution of systems of loaded ordinary differential equations with multipoint conditions" , Comput. Math. Math. Phys., 58, (2018): 508–516.
[15] Assanova A.T., Kadirbayeva Zh.M. "On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations" , Comp. and Applied Math., 37, (2018): 4966–4976.
[16] Kadirbayeva Zh.M., Kabdrakhova S.S., Mynbayeva S.T. "A Computational Method for Solving the Boundary Value Problem for Impulsive Systems of Essentially Loaded Differential Equations" , Lobachevskii J. of Math., 42, (2021): 3675- 3683.
[17] Dzhumabaev D.S. "On one approach to solve the linear boundary value problems for Fredholm integro-differential equations" , J. of Comp. and Applied Math., 294, (2016): 342-357
