×

Method of implicit functions in the solution of multiparameter nonlinear spectral problems. (English. Ukrainian original) Zbl 07687344

J. Math. Sci., New York 272, No. 1, 38-54 (2023); translation from Mat. Metody Fiz.-Mekh. Polya 63, No. 2, 36-50 (2020).
Summary: We propose a new numerical method for the solution of multiparameter nonlinear spectral problems of dimension \(m\) for the holomorphic operator functions defined in Banach spaces. We introduce the notion of generalized Cauchy problem, which is reduced to the solution of a system of \(m - 1\) partial differential equations of the first order with common initial condition. We also present examples of solving of two- and three-parameter spectral problems.

MSC:

47Jxx Equations and inequalities involving nonlinear operators
15Axx Basic linear algebra
34Axx General theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. I. Andriichuk, V. F. Kravchenko, P. A. Savenko, and M. D. Tkach, “Synthesis of plane radiating systems for a given power directivity pattern,” Fiz. Osnovy Priborostr.,2, No. 3, 40-55 (2013); doi:10.25210/jfop-1303-040055.
[2] G. I. Arkhipov, V. A. Sadovnichii, and V. N. Chubarikov, Lectures on Mathematical Analysis [in Russian], Vysshaya Shkola, Moscow, 1999.
[3] Ya. E. Baranetskii and P. I. Kalenyuk, “Multiparameter nonlocal spectral problems for operator-differential equations,” Mat. Met. Fiz.-Mekh. Polya, Issue 32, 26-30 (1990); English translation: J. Math. Sci.,64, No. 5, 1139-1142 (1993); doi:10.1007/BF01098835. · Zbl 1267.34105
[4] Yu. I. Bibikov, General Course of Differential Equations [in Russian], Leningrad Univ., Leningrad (1981). · Zbl 0482.34003
[5] G. M. Vainikko, Funktionalanalysis der Diskretisierungsmethoden, Teubner, Leipzig (1976). · Zbl 0343.65023
[6] G. M. Vainikko and A. M. Dement’eva, “The rate of convergence of a method of mechanical quadratures in an eigenvalue problem,” Zh. Vychisl. Mat. Mat. Fiz.,8, No. 5, 1105-1110 (1968); English translation: USSR Comput. Math. Math. Phys.,8, No. 5, 226-234 (1968); doi:10.1016/0041-5553(68)90137-7. · Zbl 0214.42302
[7] G. N. Vainikko and O. O. Karma, “The convergence rate of approximate methods in the eigenvalue problem when the parameter appears non-linearly,” Zh. Vychisl. Mat. Mat. Fiz.,14, No. 6, 1393-1408 (1974); English translation: USSR Comput. Math. Math. Phys.,14, No. 6, 23-39 (1974); doi:10.1016/0041-5553(74)90166-9. · Zbl 0352.65027
[8] Goursat, É., A Course in Mathematical Analysis (2005), Ann Arbor: Univ. of Michigan, Ann Arbor
[9] Kantorovich, LV; Akilov, GP, Functional Analysis (1982), Amsterdam: Elsevier, Amsterdam · Zbl 0484.46003
[10] Karma, O., On convergence of a difference method in nonlinear eigenvalue problems for linear differential equations, Uch. Zap. Tartu Gos. Univ., 374, 211-228 (1975) · Zbl 0395.65046
[11] Petrovsky, IG, Lectures on Partial Differential Equations (1992), New York: Dover, New York
[12] L. P. Protsakh and P. O. Savenko, “Implicit-function methods for the solution of two-parameter linear spectral problems,” Mat. Met. Fiz.-Mekh. Polya,52, No. 2, 42-49 (2009); English translation: J. Math. Sci.,170, No. 5, 612-621 (2010); doi:10.1007/s10958-010-0106-8. · Zbl 1212.15021
[13] P. A. Savenko and L. P. Protsakh, “Implicit function method in solving a two-dimensional nonlinear spectral problem,” Izv. Vyssh. Uchebn., Zaved., Mat., No. 11 (546), 41-44 (2007); English translation: Russian Math. (Izv. VUZOV).,51, No. 11, 40-43 (2007); doi:10.3103/S1066369X07110060. · Zbl 1298.47071
[14] P. Savenko, “Nonlinear two-parameter spectral problems in the theory of nonlinear integral and ordinary differential equations,” in: Abstr. of the V. Ya. Skorobogat’ko Internat.. Math. Conf., Sept. 19-23, 2011, Drogobych (2011), p. 177.
[15] P. O. Savenko, Nonlinear Problems of Synthesis of Radiating Systems with Plane Aperture [in Ukrainian], Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv (2014). · Zbl 1349.78055
[16] P. O. Savenko, “Nonlinear integral equations in the theory of synthesis of radiating systems,” in: Abstr. of the Internat. Sci. Conf. “Integral Equations - 2009”, G. Pukhov Institute of Problems of Modeling in Power Industry, National Academy of Sciences of Ukraine, Kyiv (2009), pp. 45-46.
[17] P. O. Savenko and L. P. Protsakh, “Variational approach to the solution of a nonlinear vector spectral problem in the case of selfadjoint positive semidefinite operators,” Dop. Nats. Acad. Nauk Ukr., No. 6, 26-31 (2004). · Zbl 1069.47512
[18] P. O. Savenko and Protsakh, “Numerical solution of a two-point boundary-value problem with nonlinear two-dimensional spectral parameter,” Mat. Met. Fiz.-Mekh. Polya,54, No. 1, 48-56 (2011); English translation: J. Math. Sci.,183, No. 1, 43-53 (2012); doi:10.1007/s10958-012-0796-1. · Zbl 1274.35010
[19] V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow, 1980. · Zbl 0517.46001
[20] M. Hervé, Several Complex Variables. Local Theory, Oxford Univ. Press, London, 1963. · Zbl 0113.29003
[21] M. S. Birman and N. N. Uraltseva, Nonlinear Equations and Spectral Theory, AMS Press, Providence, R.I., 2007. · Zbl 1111.35001
[22] Hendi, FA; Al-Qarni, MM, Numerical solution of nonlinear mixed integral equation with a generalized Cauchy kernel, Appl. Math., 8, 2, 209-214 (2017) · doi:10.4236/am.2017.82017
[23] G. Y. Karpushyna, P. O. Savenko, and M. D. Tkach, “Nonlinear three-parametric spectral problems in the synthesis theory of radiating systems with a flat aperture,” in: DIPED-2014: Proc. of the XIX Internat. Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory, Tbilisi, Sept. 22-25, 2014, Tbilisi, (2014), pp. 143-147.
[24] P. O. Savenko, “Application of implicit functions methods to solution of the nonlinear vector spectral problem,” Visn. Nats. Univ. “Lviv. Politekh.” Ser. Fiz.-Mat. Nauki, No. 660, 42-45 (2009). · Zbl 1289.47123
[25] Savenko, P.; Klakovych, L.; Tkach, M., Theory of Nonlinear Synthesis of Radiating Systems (2016), Saarbrücken: LAMBERT Acad. Publ, Saarbrücken
[26] Wazwaz, A-M, Linear and Nonlinear Integral Equations (2011), Springer, Berlin: Methods and Applications, Springer, Berlin · Zbl 1227.45002 · doi:10.1007/978-3-642-21449-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.