Advanced Differential Equations: Asymptotics & Perturbations

TL;DR

This work surveys asymptotic and perturbation methods for differential equations, unifying numerical, analytic, and perturbative approaches to obtain meaningful approximate solutions with small parameters such as $\epsilon$. It develops a framework around perturbation theory, the Fredholm Alternative, and adjoint operators, and applies eigenfunction expansions, Green's functions, and Sturm-Liouville theory to linear and boundary-value problems. It introduces regular perturbation, the Poincaré-Lindstedt method, multiple-scale expansions, and boundary-layer theory, and demonstrates them on canonical nonlinear problems such as the pendulum and oscillator models as well as predator-prey dynamics. The text emphasizes linking qualitative phase-space analysis with rigorous asymptotics to yield tractable approximations and to illuminate how perturbations affect stability, resonance, and frequency.

Abstract

Approximation techniques have been historically important for solving differential equations, both as initial value problems and boundary value problems. The integration of numerical, analytic and perturbation methods and techniques can help produce meaningful approximate solutions for many modern problems in the engineering and physical sciences. An overview of such methods is given here, focusing on the use of perturbation techniques for revealing many key properties and behaviors exhibited in practice across diverse scientific disciplines.

Figures (37)

• Figure 1: Behavior of Node (a) and Saddle (b) which are determined by having real distinct eigenvalues which are of the same sign (a) or of opposite sign (b). Changing the signs of the eigenvalues simply changes the directions of the arrows. Thus the Node can be stable or unstable whereas the Saddle is always unstable.
• Figure 2: Proper (a) and Improper (b) Nodes corresponding to the double root case with two independent eigenvectors (a) or one eigenvector and a second generalized eigenvector (b). The nodes can be stable or unstable depending on the sign of the eigenvalue, i.e. the arrows are switched with a switch of the sign of the eigenvalue.
• Figure 3: Spiral (a) and Center (b) behavior when the eigenvalues are complex conjugates. The spiral has a nontrivial real part which determines whether the trajectories spiral in or out in (a) whereas the center has strictly periodic behavior.
• Figure 4: Schematic of pendulum oscillating from a fixed support subject to forces of gravity and damping.
• Figure 5: Behavior of solutions near each of the fixed points which are multiples of $\pi$ from the origin. Note that multiples of $2\pi$ produce centers while multiples of odd $\pi$ are saddles.