Introduction to perturbation techniques
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.
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adjoint problem amplitude asymptotic development asymptotic expansion auxiliary polynomial becomes binomial theorem boundary conditions boundary layer composite expansion Consider the equation corresponding cos2 cos3 curves Determine a first-order Determine the solvability differential equation dimensionless Duffing equation Eliminating the secular equating coefficients equating the coefficients exact solution example Expanded for small exponentially expressed first-order uniform expansion follows frequency function Hence homogeneous equation homogeneous problem inhomogeneous interval leading term Lindstedt-Poincare linear linearly independent linearly independent solutions matching matching principle matrix method of averaging method of multiple method of renormalization multiple scales neighborhood nonlinear nontrivial solution numerically obtain parameter particular solution preceding section resonances rewrite rewritten roots saddle point scales and averaging second-order secular terms self-adjoint Show sin2 solvability condition solve stationary point straightforward expansion Substituting Taylor series technique tion transformation trigonometric identities two-term valid values vanish WKB approximation yields