Nonlinear stochastic systems theory and applications to physics. (English) Zbl 0659.93003

Adomian, George

Mathematics and Its Applications, 46. Dordrecht (Netherlands) etc.: Kluwer Academic Publishers Group. xx, 224 p. £49.90; $ 69.00; Dfl. 140.00 (1989).

This monograph is devoted to the decomposition method developed by the author. The theoretical basis is presented in the first part: A summary of the decomposition method. The application of the method to various problems includes: Burger’s equation; nonlinear oscillations in physical systems; KdV equation; Benjamin-Ono equation; nonlinear Schrödinger equation and generalized Schrödinger equation; nonlinear plasmas; Tricomi problem; initial value problem for the wave equation; nonlinear dispersive or dissipative waves; nonlinear Klein-Gordon equation; analysis of model equations of gas dynamics; a new approach to the Effiger model for nonlinear quantum theory for gravitating particles; Navier-Stokes equations.
The book is written for the applied mathematicians and for persons interested in solving real, physical problems. Some numerical results are included, in order to compare the author’s method with other approximation methods. The author’s method appears to work well but the theoretical foundations still leave much to be explored, thus posing a fascinating challenge. It seems that the method can be automatized through symbolic programming.

Reviewer: E.Nicolau

MSC:

93-02	Research exposition (monographs, survey articles) pertaining to systems and control theory
35Q99	Partial differential equations of mathematical physics and other areas of application
93E03	Stochastic systems in control theory (general)
35Q30	Navier-Stokes equations
35R60	PDEs with randomness, stochastic partial differential equations
41A10	Approximation by polynomials
76D05	Navier-Stokes equations for incompressible viscous fluids
76N15	Gas dynamics (general theory)
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
82D10	Statistical mechanics of plasmas
93C10	Nonlinear systems in control theory

Keywords:

decomposition method; Burger’s equation; KdV equation; nonlinear Schrödinger equation; nonlinear plasmas; nonlinear Klein-Gordon equation; Navier-Stokes equations