On the parametrization of boundary-value problems with two-point nonlinear boundary conditions
We obtain some results for the solutions of nonlinear boundary-value problems of a certain type with two-point nonlinear boundary conditions and reveal the efficiency of the procedure of reduction of the analyzed problem to a parametrized boundary-value problem with linear boundary conditions containing certain artificially introduced parameters. To study the transformed two-point problem, we propose a method based on the approximations of a special type constructed in the analytic form. It is shown that these approximations uniformly converge to a parametrized limit function and establish the relationship between this function and the exact solution. The proposed procedure leads to a certain system of algebraic equations whose solutions give numerical values of the parameters corresponding to the solution of the given two-point nonlinear boundary-value problem.
KeywordsLimit Function Determine Equation Nonlinear Boundary Condition Linear Boundary Condition Parametrized Boundary Condition
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- 2.A. M. Samoilenko and N. I. Ronto, Numerical-Analytic Methods in the Theory of Boundary-Value Problems for Ordinary Differential Equations [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
- 8.M. I. Ronto, N. M. Shchobak, and K. V. Marynets’, “On the parametrization of boundary-value problems of the Cauchy–Nicoletti type,” Nauk. Visn. Uzhhorod. Nats. Univ., Ser. Mat. Inform., No. 16, 163–173 (2008).Google Scholar
- 9.K. V. Marynets’, “On a nonlinear three-point problem of the Cauchy–Nicoletti type,” Nauk. Visn. Uzhhorod. Nats. Univ., Ser. Mat. Inform., No. 19, 53–63 (2009).Google Scholar
- 10.K. V. Marynets’, “Investigation of the solutions of three-point problems of the Cauchy–Nicoletti type and their reduction to two-point problems,” Nauk. Visn. Kyiv. Nats. Univ., Ser. Fiz.-Mat. Nauk., No. 3, 85–90 (2009).Google Scholar
- 13.M. Farkas, Periodic Motions, Applied Mathematical Sciences, Springer, London (1994).Google Scholar