Applications in Physics
Real problems of physics are generally nonlinear and often stochastic as well. Linearity and determinism should be viewed as special cases only. The general practices of linearization, perturbation, white noise, and quasimonochromatic approximations necessarily change the problems whose solutions are desired, to be tractable by convenient mathematics. They are not then identical to the physical solutions which we seek. The alternative of using the decomposition method will be explored here as we consider examples of problems of physics. These problems are often quite difficult because nonlinearity and stochasticity are involved. Decomposition makes unnecessary procedures such as closure approximations  and perturbation and white noise processes in differential equations which involve stochastic process parameters, inputs, or initial/boundary conditions. Decomposition also avoids discretization and consequent intensive computer calculation and yields analytic expressions rather than tables of numbers. Thus quantitative solutions are obtained for dynamical systems. (When the systems are stochastic as well, the decomposition series involves stochastic terms from which statistics can be calculated.) The method applies to linear or nonlinear, ordinary or partial differential equations and is useful for many algebraic, integral, and delaydifferential equations . This chapter will outline procedures for typical applications.
KeywordsNonlinear Schrodinger Equation Decomposition Form Decomposition Series Random Boundary Condition Linear Random Operator
Unable to display preview. Download preview PDF.
- 1.G. Adomian, Analytic Solution of the Stochastic Navier-Stokes System, Found. of Physics, 21, (831–843) (July 1991).Google Scholar
- 2.G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer (1989).Google Scholar
- 3.G. Adomian, Linear Random Operator Equations in Mathematical Physics I, II, III, J. Math. Phys., 11,3, (1069–1084) (1970), 12, 9, (1971), (1944–1955).Google Scholar
- 5.I. Segal, Quantization and Dispersion for Nonlinear Relativistic Equations, in The Mathematical Theory of Elementary Particles, R. Goodman and I. Segal (eds.), MIT Press (1965).Google Scholar
- 6.D.S. Saxon, Elementary Quantum Mechanics, Holden-Day (1968).Google Scholar
- 1.A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics I-II, J. Lumley (ed. ), MIT Press (1971).Google Scholar
- 2.L.D. Landau and E.M. Lifshitz, Quantum Mechanics, J. B. Sykes and J. S. Bell (transi. ), Addison-Wesley (1958).Google Scholar
- 3.L.D. Landau and E.M. Lifshitz, Mechanics, J. B. Sykes and J. S. Bell (transi. ), Addison-Wesley (1960).Google Scholar