×

Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. (English) Zbl 0991.65065

Summary: The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential equations is straightforward. The method, whose mathematical basis in physics was discussed recently by one of the present authors, approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories is not based on the existence of some kind of a small parameter.
It is shown that the quasilinearization method gives excellent results when applied to different nonlinear ordinary differential equations in physics, such as the Blasius, Duffing, Lane-Emden and Thomas-Fermi equations. The first few quasilinear iterations already provide extremely accurate and numerically stable answers.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Mandelzweig, V. B., Quasilinearization method and its verification on exactly solvable models in quantum mechanics, J. Math. Phys., 40, 6266 (1999) · Zbl 0969.81007
[2] Krivec, R.; Mandelzweig, V. B., Numerical investigation of quasilinearization method in quantum mechanics, Comput. Phys. Comm., 138, 69 (2001) · Zbl 0984.81173
[3] Kalaba, R., On nonlinear differential equations, the maximum operation and monotone convergence, J. Math. Mech., 8, 519 (1959) · Zbl 0092.07703
[4] Bellman, R. E.; Kalaba, R. E., Quasilinearization and Nonlinear Boundary-Value Problems (1965), Elsevier Publishing Company: Elsevier Publishing Company New York · Zbl 0139.10702
[5] Conte, S. D.; de Boor, C., Elementary Numerical Analysis (1981), McGraw-Hill International Editions
[6] Ralston, A.; Rabinowitz, P., A First Course in Numerical Analysis (1988), McGraw-Hill International Editions
[7] Lakshmikantham, V.; Vatsala, A. S., Generalized Quasilinearization for Nonlinear Problems. Generalized Quasilinearization for Nonlinear Problems, Mathematics and its Applications, 440 (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0997.34501
[8] Calogero, F., Variable Phase Approach to Potential Scattering (1965), Academic Press: Academic Press New York · Zbl 0193.57501
[9] Babikov, V. V., Sov. Phys. Uspekhi, 10, 271 (1967), Nauka: Nauka Moscow
[10] Adrianov, A. A.; loffe, M. I.; Cannata, F., Modern Phys. Lett., 11, 1417 (1996)
[11] Jameel, M., J. Phys. A: Math. Gen., 21, 1719 (1988)
[12] Raghunathan, K.; Vasudevan, R., J. Phys. A: Math. Gen., 20, 839 (1987)
[13] Hooshyar, M. A.; Razavy, M., Nuovo Cimento B, 75, 65 (1983)
[14] Volterra, V., Theory of Functionals (1931), Blackie and Son: Blackie and Son London
[15] Liao, S.-J., Int. J. Non-Linear Mechanics, 34, 759 (1999)
[16] Yu, L.-T.; Chen, C.-K., Math. Comput. Modelling, 28, 101 (1998)
[17] Bin, L.; Jiangong, Y., Nonlinear Anal., 33, 645 (1998)
[18] Wang, H.; Li, Y., Nonlinear Anal., 24, 961 (1995)
[19] Wazwaz, A.-M., Appl. Math. Comput.. Appl. Math. Comput., Appl. Math. Comput., 118, 324 (2001)
[20] Goenner, H.; Havas, P., J. Math. Phys., 41, 7029 (2000)
[21] Pert, G. J., J. Phys. B, 32, 249 (1999)
[22] Bender, C. M.; Milton, K. A.; Pinsky, C. C.; Simmons, L. M., J. Math. Phys., 30, 1447 (1989)
[23] Thomas, L. H., Proc. Cambridge Phil., 23, 542 (1927)
[24] Fermi, E., Z. Physik, 48, 73 (1928)
[25] Bethe, H. A.; Jackiw, R. W., Intermediate Quantum Mechanics (1968), W.A. Benjamin Inc.: W.A. Benjamin Inc. New York
[26] Schlichting, H., Boundary Layer Theory (1978), McGraw-Hill: McGraw-Hill New York
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.