Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs

By V. Mandelzweig, F. Physics, Hebrew University, Jérusalem, Israel, D. Physics, Astronomy, U. Pittsburgh, Pittsburgh, Pa and 1 more. 2001

308 citations recorded by the source corpus.

Abstract 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 straight forward. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), 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.

Subfields: Modeling and Simulation, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/dd5dc2a48e97cf1ffd915d73eb42b206db166f71

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In Modeling and Simulation

1. Ahmed Alsaedi
2. Bashir Ahmad
3. John P. Boyd
4. Sotiris K. Ntouyas
5. J. J. Nieto
6. Yong Zhou
7. Jinrong Wang

In Numerical Analysis

1. Ahmed Alsaedi
2. Bashir Ahmad
3. John P. Boyd
4. Sotiris K. Ntouyas
5. J. J. Nieto
6. Jinrong Wang

In Statistical and Nonlinear Physics

1. Bashir Ahmad
2. John P. Boyd
3. Ahmed Alsaedi

Citations in the corpus

• Chebyshev Spectral Methods and the Lane-Emden Problem (2011, background): “Mandelzweig & Tabakin [16] review of quasi-linearization; LE, reduced to form without 1st deriv.”
• Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions (2009, methodology): “For the details of this method, see [1,3–6,8,13,18–23] and the references therein.”