Abstract
The method of solution continuation with respect to a parameter is used to solve an initial value problem for a system of ordinary differential equations with several limiting singular points. The solution is continued using an argument (called the best) measured along the integral curve of the problem. Additionally, a modified argument is introduced that is locally equivalent to the best one in the considered domain. Theoretical results are obtained concerning the conditioning of the Cauchy problem parametrized by the modified argument in a neighborhood of each point of its integral curve.
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References
V. I. Shalashilin and E. B. Kuznetsov, Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics (Editorial URSS, Moscow, 1999; Kluwer Academic, Dordrecht, 2003).
E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations. I: Nonstiff Problems (Springer, Berlin, 1987).
M. P. Galanin and S. R. Khodzhaeva, Preprint No. 98, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2013). http://library.keldysh.ru/preprint.asp?id=2013-98
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer-Verlag, Berlin, 1996; Mir, Moscow, 1999).
D. F. Davidenko, “Ob odnom novom metode chislennogo resheniya sistem nelineinykh uravnenii,” Dokl. Akad. Nauk SSSR 88 (4), 601–602 (1953).
D. F. Davidenko, “On the approximate solution of systems of nonlinear equations,” Ukr. Mat. Zh. 5 (2), 196–206 (1953).
E. I. Grigolyuk and V. I. Shalashilin, Problems of Nonlinear Deformation: The Continuation Method Applied to Nonlinear Problems in Solid Mechanics (Nauka, Moscow, 1988; Springer, 1991).
E. B. Kuznetsov and V. I. Shalashilin, “The Cauchy problem as a problem of the continuation of a solution with respect to a parameter,” Comput. Math. Math. Phys. 33 (12), 1569–1579 (1993).
E. B. Kuznetsov, “The best parameterization in curve construction,” Comput. Math. Math. Phys. 44 (9), 1462–1472 (2004).
E. B. Kuznetsov and A. Yu. Yakimovich, “Optimal parametrization in approximation of curves and surfaces,” Comput. Math. Math. Phys. 45 (5), 732–745 (2005).
E. B. Kuznetsov, “On the best parametrization,” Comput. Math. Math. Phys. 48 (12), 2162–2171 (2008).
E. B. Kuznetsov and S. S. Leonov, “Pure bending for the multimodulus material beam under creep conditions,” Vestn. Yuzhno-Ural. Gos. Univ. Ser. Mat. Model. Program. 6 (4), 26–38 (2013).
N. N. Kalitkin and I. P. Poshivaylo, “Solving the Cauchy problem with guaranteed accuracy for stiff systems by the arc length method,” Math. Models Comput. Simul. 7 (1), 24–35 (2015).
S. D. Krasnikov and E. B. Kuznetsov, “Parametrization of a solution at bifurcation points,” Differ. Equations 45 (8), 1218–1222 (2009).
S. D. Krasnikov and E. B. Kuznetsov, “Numerical continuation of solution at singular points of codimension one,” Comput. Math. Math. Phys. 55 (11), 1802–1822 (2015).
N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (BINOM, Moscow, 2006) [in Russian].
L. D. Kudryavtsev, A Course in Mathematical Analysis, Vol. 2: Series and Differential and Integral Calculus of Functions of Several Variables (Drofa, Moscow, 2004) [in Russian].
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Original Russian Text © E.B. Kuznetsov, S.S. Leonov, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 6, pp. 934–957.
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Kuznetsov, E.B., Leonov, S.S. Parametrization of the Cauchy problem for systems of ordinary differential equations with limiting singular points. Comput. Math. and Math. Phys. 57, 931–952 (2017). https://doi.org/10.1134/S0965542517060094
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DOI: https://doi.org/10.1134/S0965542517060094