Quasilinearization Approach to Nonlinear Problems in Physics with Application to Nonlinear ODEs
V. B. Mandelzweig,Frank Tabakin +1 more
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In this paper, 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.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.read more
Citations
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An approximation algorithm for the solution of the nonlinear lane-emden type equations arising in astrophysics using hermite functions collocation method
TL;DR: A collocation method for solving some well-known classes of Lane–Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain based on a Hermite function collocation (HFC) method is proposed.
Journal ArticleDOI
Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method
Ahmet Yildirim,Turgut Öziş +1 more
TL;DR: In this article, a new scheme, deduced from He's homotopy perturbation method, is presented for solving Lane-Emden type singular IVPs problem, and only a few terms are required to obtain accurate computable solutions.
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Solutions of singular IVPs of Lane–Emden type by the variational iteration method
Ahmet Yildirim,Turgut Öziş +1 more
TL;DR: In this article, approximate-exact solutions of a class of Lane-Emden type singular IVPs problems, by the variational iteration method, are presented, which yields solutions in the forms of convergent series with easily calculable terms.
Journal ArticleDOI
Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type
TL;DR: A pseudospectral technique is proposed to solve the Lane-Emden type equations on a semi-infinite domain based on rational Legendre functions and Gauss-Radau integration to solve a system of nonlinear algebraic equations.
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Series approach to the Lane–Emden equation and comparison with the homotopy perturbation method
TL;DR: In this article, series solutions of the Lane-Emden equation based on either a Volterra integral equation formulation or the expansion of the dependent variable in the original ordinary differential equation are presented and compared with series solutions obtained by means of integral or differential equations based on a transformation of dependent variables.
References
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Book
Boundary layer theory
TL;DR: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part, denoted as turbulence as discussed by the authors, and the actual flow is very different from that of the Poiseuille flow.
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Nonlinear Differential Equations
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Quasilinearization and nonlinear boundary-value problems
Richard Bellman,Robert E. Kalaba +1 more
TL;DR: Quasilinearization and nonlinear boundary value problems as discussed by the authors, where the boundary value problem is formulated as a quadratic equation of the value of a boundary value.