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# Perturbation Methods

1. James Murdock

Published Online: 15 JUL 2005

DOI: 10.1002/3527600434.eap315.pub2

## Encyclopedia of Applied Physics

#### How to Cite

Murdock, J. 2005. Perturbation Methods. Encyclopedia of Applied Physics. .

#### Author Information

1. Iowa State University, Ames, USA

#### Publication History

1. Published Online: 15 JUL 2005

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### Abstract

When a solvable mathematical problem is changed slightly, or perturbed, it may become unsolvable (in closed form). Perturbation theory enables an approximate solution to be constructed; in the simplest cases, this is done by adding successive correction terms to the solution of the unperturbed problem. The roots of the method, which lie in celestial mechanics, fluid flow, and quantum mechanics are briefly described. The notions of perturbation series, uniform asymptotic validity, and regular and singular problems are defined. In the area of nonlinear oscillations, the most important methods are the Lindstedt method (or method of strained coordinates), the method of multiple scales, and the method of averaging; these are described and compared. Special results for the averaging of multifrequency Hamiltonian systems, including the Kolmogorov–Arnol'd–Moser (KAM) theorem and the Nekhoroshev theorem, are briefly described. In dynamical systems, the Melnikov integral describes the perturbation of stable and unstable manifolds. Differential equations in which a small parameter multiplies the highest derivative lead to solutions having initial or boundary layers. The multiple-scale method and methods of matched asymptotic expansions for these problems are described. The WKB method can also be used in certain cases. The matching method for partial differential equations is illustrated by the boundary layer problem for fluid flow. Finally, several approaches to the problem of finding the eigenvalues (spectra) of perturbed matrices and linear transformations are described.

### Keywords:

• perturbation;
• singular perturbation;
• nonlinear oscillations;
• boundary layer;
• WKB method