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Nonlinear stochastic systems in physics and mechanics. (English) Zbl 0623.60084
Singapore: World Scientific Publishing Co. Pte. Ltd. Distr. by John Wiley & Sons Ltd., Chichester. XIII, 244 p.; £28.40 (1987).
In considering the time evolution of real physical systems, we must consider the fact that they are generally nonlinear and generally stochastic so that linear systems or deterministic systems become special cases. Thus differential or partial differential equations can involve not only nonlinearities but stochastic parameters, inputs, and boundary conditions. The decomposition method has provided a new continuous analytical approximation scheme which not only covers differential and partial differential equations but avoids smallness assumptions, linearizations, and the discretized methods which result in such intensive computations in complex system problems.
The authors have written an excellent and readable book discussing stochastic modeling, perturbation procedures, the more powerful and widely applicable decomposition method, and an introduction to random partial differential equations and current research in stochastic continuum mechanics. The book contains interesting applications, e.g. to the dynamics of satellites and the classical Kepler-Poinsot motion, and also considerations to the existence and uniqueness of solutions.
This book as well as G. Adomian’s books ”Stochastic systems”. (1983; Zbl 0523.60056); ”Nonlinear stochastic operator equations”. (1986; Zbl 0609.60072) and ”Nonlinear stochastic systems theory and applications to physics”. Kluwer (1987; Zbl 0659.93003) should be in every applied mathematics library and constitutea valuable step forward.
Reviewer: R.Rach

MSC:
60H99 Stochastic analysis
60-02 Research exposition (monographs, survey articles) pertaining to probability theory