Introduction to perturbation techniques

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Singularly perturbed differential equations arise in many applications, such as wave propagation and quantum mechanics. Progress can be made by studying Stokes phenomena and exponential asymptotics for the solutions in the complex plane Chapman et al Singularly perturbed differential equations can yield solutions containing regions of rapid variation rapid compared to the regular length scale for the problem. Constructing a solution to a differential equation or system involves several steps: identifying the locations of layers boundary or internal , deriving asymptotic approximations to the solution in the different regions corresponding to different distinguished limits in the equations , and ultimately, forming a uniformly valid solution over the entire domain.

Solutions obtained for the layers singular distinguished limits are usually termed inner solutions while the slowly varying solutions for the regular distinguished limits are referred to as outer solutions.

The uniformly valid solution can be constructed through asymptotic matching of the inner and outer solutions, which relies on the fundamental assumption that the different solution forms overlap at on some identifiable region see Figure 1. Procedures for matching asymptotic expansions have been examined by Kaplun, Van Dyke and others Lagerstrom , Van Dyke , Kevorkian and Cole , Eckhaus , but there are still some fundamental theoretical issues to be resolved.

Unlike WKB theory, this approach can also be applied to nonlinear equations, and this versatility allows a wide range of problems to be tackled:. Equation 9 is a weakly-nonlinear oscillator , reducing to the linear oscillator equation at leading order.

The growing cumulative error in phase and amplitude apparent in the regular expansion is a consequence of the limitations of the ansatz, particularly in deviations from the unperturbed natural frequency of the system. Various perturbation methods have been developed for dealing with such problems. These include:. Similar ideas also arise in homogenization theory , considering the averaging of spatially periodic structures of materials Bensoussan et al , Holmes We have only scratched the surface of this research area, but it is hoped that the above illustrates the power and usefulness of the methods grouped under singular perturbation theory.

As discussed above, singular perturbation theory tackles difficult problems by investigating various reduced problems and then assembling the results together in an appropriate form. These reductions could, for example, simply be to a lower order polynomial in an algebraic problem, or could be more significant, such as in the reduction of a PDE to an ODE, or a functional equation to one of algebraic form.

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The reduced problems can still be mathematically challenging, with the construction of a uniformly valid solution requiring an involved analysis. While some singular perturbation methods are based on rigorous analysis, the vast range of applications and available techniques typically restrict against such results. : Introduction to Perturbation Techniques () : Ali H. Nayfeh : Books

Consequently, the methods are often classed as formal techniques. However, this is not considered to be a significant problem: any a priori assumptions can be checked for consistency once suitable expansions have been derived; furthermore, formal results obtained by these techniques have been known to provide direction for additional rigorous theory Smith , Eckhaus In fact, some authors have seen the generality of the methods described above as representative of some more fundamental notion.

For example, Kruskal went as far as to introduce the term asymptotology in referring to the art of dealing with applied mathematical systems in limiting cases Kruskal and considered singular perturbation theory and asymptotic methods in general as a component of asymptotology. Averaging , Multiple Scale Analysis , Normal forms. Thomas Witelski and Mark Bowen , Scholarpedia, 4 4 Jump to: navigation , search. Post-publication activity Curator: Mark Bowen Contributors:. Robert O'Malley. Sponsored by: Eugene M.

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Allow this favorite library to be seen by others Keep this favorite library private. Find a copy in the library Finding libraries that hold this item Similarities, differences, advantages and limitations of perturbation techniques are pointed out concisely. The techniques are described by means of examples that consist mainly of algebraic and ordinary differential equations. Each chapter contains a number of exercises. Read more Reviews User-contributed reviews Add a review and share your thoughts with other readers. Be the first. Add a review and share your thoughts with other readers.

Equations -- Numerical solutions. Perturbation Mathematics Differential equations - Numerical solutions. Equations - Numerical solutions. Numerische Mathematik. Numerisches Verfahren. Linked Data More info about Linked Data. All rights reserved.

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Introduction to perturbation techniques Introduction to perturbation techniques
Introduction to perturbation techniques Introduction to perturbation techniques
Introduction to perturbation techniques Introduction to perturbation techniques
Introduction to perturbation techniques Introduction to perturbation techniques
Introduction to perturbation techniques Introduction to perturbation techniques
Introduction to perturbation techniques Introduction to perturbation techniques
Introduction to perturbation techniques Introduction to perturbation techniques

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