# Novel Features of Differential Equations, Mechanics, and Computation

## Content

The content of Differential Equations, Classical Mechanics, and Computation (which we refer to below as “our book”) consists of two main streams. The first stream is the theory of ordinary differential equations (ODE), and in particular theoretical discussion of:

• 1) The initial value problem (IVP) for first order equations of evolution,
• 2) Linear differential equations,
• 2) Second order Equations, The Calculus of Variations and the Euler-Lagrange Equations, Symmetry, Conservation Laws, and Noether's Principle,
• 3) Classical Mechanics: Newton's Equations, Lagrangian Formulation, Hamiltonian Formulation, Theory of small vibrations.

This comprises the first four chapters of our book. The second stream is the study of numerical methods for solving ODEs. It is introduced in the first chapter, and is treated in detail in the fifth chapter of our book. (For details about the content of this fifth chapter, see the paragraph called “Novel approach to numerics” below.) The two streams come together when results of the computational part of the book are applied in Chapter 5 to the Korteweg-deVries (KdV) equation. Such computations led to the discovery that this is an infinite dimensional completely integrable Hamiltonian system, an important kind of ODE discussed in the theoretical portion of the book. This observation was crucial in the development of soliton theory and resulted in numerous advances in physics, geometry, and other areas of mathematics, science, and even applied technology. The numerical method we introduce to solve the KdV equation uses both the Fast Fourier Transform (FFT) and split-stepping, two useful and exciting topics that are not usually examined in introductory books on numerical methods for ODEs.

## Novel Features

### Intended Audience

Unlike many ODE textbooks, designed for courses aimed at students of chemistry, biology, or engineering, ours is aimed at an audience of upper-level undergraduate students and beginning graduate students in pure and applied mathematics. However, we have also tried to make it suitable as an ODE text for self-study by anyone with a background in Calculus and Linear Algebra.

### Presentation

Liberal use is made of images, simple models, and appropriate metaphors to help users develop an intuition concerning the theory of initial value problem for equations of evolution, and to help motivate the statement and proof of important theorems.

Highly technical details are relegated to appendices to make it easier for readers to develop a clear understanding of the principle concepts and theorems without sacrificing rigor.

Numerous exercises are guided tours to the proofs of important theorems, i.e., after a theorem is stated, there is frequently an exercise that leads the reader through the proof by suggesting steps and giving hints.

Here is a long (35 page) excerpt from the book that should give you a good idea how these features work together. (Chap1Frst35Pages.pdf)

### Balance between theory and numerics

Most ODE texts treat either theory or numerics in detail and the other in a cursory manner. However, in modern applied mathematics, theory and rigorously analyzed computation go hand-in-hand, calling for a text that treats both in detail, and we have undertaken to provide one. Most of the theoretical material is treated in the first four shorter chapters, with an introduction to numerical methods given in the first chapter. Then a considerably longer fifth chapter (nearly as long as the other four combined) contains a thorough and rigorous treatment of numerical methods.

## Novel approach to numerics

Our discussion of numerical methods strives to be visual and comparative, based on carefully chosen examples of prototypical methods and model problems. For example, one-step and multistep methods are treated and compared side-by-side. This highlights the complementary challenges involved in understanding the behavior of both types of method when applied to particular problem; e.g., with multistep methods, the necessary conditions for a particular degree of accuracy are straightforward but the stability conditions required for convergence are subtle, while for one-step methods stability is automatic but the analysis of accuracy conditions is involved.

Figures illustrate the meaning of our example methods as well as the results of accuracy and stability studies performed with each method applied to a class of universal model problems, for example:

The stability regions of our example methods are collected in one figure for comparison, and juxtaposed with visual representations of the spectra of important difference operators, e.g., forward, backward, centered difference and second difference, for example:

When different ODE methods are used to approximate solutions of the two most important and universal initial value problems, the heat equation and the wave equation, the drastically different behavior that is observed is shown side-by-side, and explained as a consequence of the compatibility (or lack of compatibility) of the spectrum associated with the choice of differencing scheme and the absolute stability region of the various methods. Connections with the Fourier transform and the CFL condition are also treated in this context.

The book also points out an inconsistency in the definition of the region of absolute stability of a numerical method in the literature. This fact is discussed in greater detail in the web companion, with examples from eight well-known books, four of which use one definition, and four of which use a conflicting definition. For details see: Examples of Different Stability Region Definitions.

Rigorous convergence proofs are motivated by developing a strong intuition for the concepts involved. The meaning of different kinds of errors (e.g., local truncation error, residual errors, local error, global error) and also their relationships are depicted. The use of local error analysis for automatic step-size adjustment is handled with concrete examples that can be computed and understood exactly.

The conditions for a Runge-Kutta method to have a given degree of accuracy are analyzed visually using rooted trees as a tool. We show that the two expansions that must be matched to a given order correspond to building these trees in complementary ways. The terms on the Taylor expansion side are obtained from previous terms by the Leibniz rule combined with the chain rule. This corresponds to constructing new rooted trees by adding one edge and a leaf to each node of previously constructed trees. The terms on the Runge-Kutta expansion side are obtained from previous terms by performing a Taylor series expansion of a function whose argument is perturbed by a Taylor series. This corresponds to constructing new rooted trees by joining any number of previously constructed trees at a new root. This perspective, and the LISP notation we use to simplify the calculations, aims to introduce readers in a new and accessible way to the beautiful work of Butcher and others on Runge-Kutta methods.

The two streams of the book, the Theoretical and the Numerical, come together when results of the computational part of the book are applied to the Korteweg-deVries (KdV) equation. The numerical method we introduce to solve the KdV equation uses both the Fast Fourier Transform (FFT) and split-stepping.

...And finally,

our book is Smaller, Lighter, and Costs Less. (It's ORGANIC too, so you might want to buy a copy to take along on a camping trip.)