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Unlike traditional programming languages such as Fortran and C, a computer algebra language such as Maple allows one to compute not only with numbers, but also with symbols, formulas, equations, and so on. Using a computer algebra system (CAS), symbolic computation can be done on the computer, replacing the traditional pen-and-paper approach with the keyboard/mouse and computer display. By entering short, simple, transparent commands on the computer keyboard (which will be referred to as the “classic” approach), or by selecting mathematical symbols from a palette with the mouse, the CAS user can quickly and accurately generate symbolic input and output on the computer screen. Mathematical operations such as differentiation, integration, and series expansion of functions can be done analytically on the computer.

Consider a system of two first-order coupled ODEs of the general structure where and are known functions of the dependent variables and , and the independent variable has been taken to be the time . In other model equations, the independent variable could be a spatial coordinate, e.g., the Cartesian coordinate . For compactness, the dot notation of (1.1) will often be used in our text discussion for time derivatives, one dot denoting , two dots standing for /, and so on. Superscripted primes on the dependent variable indicate a spatial derivative, e.g., ′ ≡ , ″ ≡ /, etc.

In the first chapter, the reader has seen examples of phase-plane portraits for two-dimensional autonomous ODE systems of the structure where and were specified real functions. Given the mathematical forms of and , the number and locations of the fixed points is easily established, either analytically or numerically. Quite generally, the topological nature of a fixed point can then be determined by examining the flow of tangent arrows in its vicinity and/or the temporal evolution of a nearby trajectory. For nonlinear systems, this was done with numerically based graphing commands. In this chapter, we will complement this approach by introducing , which involves analytically examining the nature of the trajectories at ordinary points lying near each fixed point. The method can be generalized [Hay64] to three-dimensional systems, but becomes considerably more complicated.

In the two chapters of the Appetizers, we presented only a few examples of linear ODEs, because the graphical and numerical techniques were more suitable for nonlinear systems where generally exact analytic solutions simply do not exist. Also, modern research often involves nonlinear ODE (and PDE) systems, a subject that is almost completely neglected in undergraduate mathematics training. So, one of the goals of this text is to partially fill in this “hole,” showing how a computer algebra system may be used to explore nonlinear models without getting “buried” in messy and complicated mathematical details. However, we would be remiss if we did not provide some coverage of linear differential equation systems, both ordinary and partial. The former are covered in this chapter, while linear PDEs are dealt with in Chapters 5 and 6. In between, we shall explore nonlinear ODEs further in Chapter 4.

In Chapter 1, phase-plane portraits were used to explore some simple nonlinear ODE models whose temporal evolution could not have been predicted, even qualitatively, before the portraits were numerically constructed. An example was the period-doubling route to chaos exhibited by the Duffing equation when the amplitude of the driving force was increased, the other parameters being held fixed. If the nonlinear term, , were not present, this “bizarre” period-doubling behavior would not even be possible. If we were to change the various coefficient values in (4.1), the response of the nonlinear system would in general be entirely different and not easily predicted on the basis of mathematical or physical intuition alone. To aid in the qualitative understanding of the behavior of nonlinear ODE systems such as this one, the concepts of fixed points and phase-plane analysis were discussed in Chapter 2.

Because linear partial differential equations play such an important role in the mathematical description of electromagnetic waves, heat flow, elastic vibrations, and many other scientific phenomena, there is an abundance of wonderful examples that can be solved using computer algebra. For this reason, this topic is split over two chapters. We begin with examples of checking PDE solutions, either obtained by intelligent guessing or quoted, without derivation, in some scientific reference. Diffusion and Laplace’s equation models are then presented.

In the previous chapter, the computer algebra recipes concentrated on mathematical models formulated in terms of the diffusion and Laplace equations. In this continuation of our study of linear PDE models, the wave equation and a variety of models involving semi-infinite and infinite domains are presented.

Nonlinear PDEs display a rich spectrum of solutions that in most cases must be obtained by numerical means. However, there exist special analytic solutions to some nonlinear PDEs of physical interest, the best known being solutions of nonlinear wave equations. A soliton is a stable solitary wave, which is a localized pulse solution that can propagate at some characteristic velocity without changing shape despite the “tug of war” between “competing terms” in the governing equation of motion.

In the Appetizers, the reader was introduced to the concept of phase-plane analysis of nonlinear ODE models. This involved the creation of phase-plane portraits and the location and identification of the relevant stationary points of the ODE system. This graphical approach was extended in the last chapter to finding solitary wave solutions of physically important nonlinear PDEs.