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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.

65L10Boundary value problems for ODE (numerical methods)
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
Full Text: DOI
[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) · Zbl 0139.10702
[5] Conte, S. D.; De Boor, C.: Elementary numerical analysis. (1981) · Zbl 0496.65001
[6] Ralston, A.; Rabinowitz, P.: A first course in numerical analysis. (1988) · Zbl 0976.65001
[7] Lakshmikantham, V.; Vatsala, A. S.: Generalized quasilinearization for nonlinear problems. Mathematics and its applications 440 (1998) · Zbl 0997.34501
[8] Calogero, F.: Variable phase approach to potential scattering. (1965) · Zbl 0132.43505
[9] Babikov, V. V.: Method fazovyh funkcii v kvantovoi mehanike (Variable method of phase functions in quantum mechanics) 1968 nauka Moscow sov. Phys. uspekhi. Sov. phys. Uspekhi 10, 271 (1968)
[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)
[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.. 118, 287 (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.; Jr., L. M. Simmons: 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)
[26] Schlichting, H.: Boundary layer theory. (1978) · Zbl 0096.20105