In this book the author presents the theory and techniques underlying perturbation methods in a manner that will make the book widely appealing to readers in a broad range of disciplines. Methods of algebraic equations, asymptotic expansions, integrals, PDEs, strained coordinates, and multiple scales are illustrated by copious use of examples drawn from many areas of mathematics and physics. The philosophy adopted is that there is no single or best method for such problems, but that one may exploit the small parameter given some experience and understanding of similar perturbation problems. The author does not look to perturbation methods to give quantitative answers but rather uses them to give a physical understanding of the subtle balances in a complex problem.
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amplitude asymptotic analysis asymptotic approximation asymptotic sequence boundary condition boundary layer comparing coefficients complex plane condition at infinity continuous curve contour correction term dashed curves derivative differential equations dispersion relation drift Dyke's matching rule Dyke's rule eigensolution eigenvalue eigenvector equation and comparing error error function evaluated exact solution example Exercise expansion method expansion sequence exponential integral exponentially small fixed function gives group velocity Hence higher order initial conditions integral equation integrand intermediate variable iterative method leading order approximation leading order term linear matched asymptotic expansions method of matched model problem nonlinear number of terms numerical obtain orbit ord(e ord(l oscillation outer parameter perturbation pose an expansion potential quadratic r-approximation r-region radius of convergence relaxation oscillation rescaling right hand side root saddle point satisfy the boundary secularity condition sin2 slowly varying small region solve strained co-ordinates Substituting uniformly asymptotic wavenumber WKBJ zero
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Beyond Perturbation: Introduction to the Homotopy Analysis Method
Limited preview - 2003