# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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Danby, J. M. A., Computer Applications to Differential Equations (Englewood Cliffs, NJ: Prentice Hall, 1985).

Strogatz, S. H., Nonlinear Dynamics and Chaos (Reading, MA: Addison-Wesley, 1994).

A good reference on Liapunov’s second method is:

LaSalle, J., and Lefschetz, S., Stability by Liapunov’s Direct Method with Applications (New York: Academic Press, 1961).

Among the large number of more comprehensive books on differential equations are:

Arnol’d, V. I., Ordinary Differential Equations (New York/Berlin: Springer-Verlag, 1992). Translation of the third Russian edition by Roger Cooke.

Brauer, F.,andNohel, J., Qualitative Theory of Ordinary Differential Equations (New York: Benjamin, 1969; New York: Dover, 1989).

Guckenheimer, J. C., and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York/Berlin: Springer-Verlag, 1983).

A classic reference on ecology is:

Odum, E. P., Fundamentals of Ecology (3rd ed.) (Philadelphia: Saunders, 1971).

Two books dealing with ecology and population dynamics on a more mathematical level are:

May, R. M., Stability and Complexity in Model Ecosystems (Princeton, NJ: Princeton University Press, 1973).

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Chapter 9. Nonlinear Differential Equations and Stability

Pielou, E. C., Mathematical Ecology (New York: Wiley, 1977).

The original paper on the Lorenz equations is:

Lorenz, E. N., “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences 20 (1963), pp. 130-141.

A very detailed treatment of the Lorenz equations is:

Sparrow, C., The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (New York/Berlin: Springer-Verlag, 1982).

CHAPTER

10

Partial Differential Equations and Fourier Series

In many important physical problems there are two or more independent variables, so that the corresponding mathematical models involve partial, rather than ordinary, differential equations. This chapter treats one important method for solving partial differential equations, a method known as separation of variables. Its essential feature is the replacement of the partial differential equation by a set of ordinary differential equations, which must be solved subject to given initial or boundary conditions. The first section of this chapter deals with some basic properties of boundary value problems for ordinary differential equations. The desired solution of the partial differential equation is then expressed as a sum, usually an infinite series, formed from solutions of the ordinary differential equations. In many cases we ultimately need to deal with a series of sines and/or cosines, so part of the chapter is devoted to a discussion of such series, which are known as Fourier series. With the necessary mathematical background in place, we then illustrate the use of separation of variables on a variety of problems arising from heat conduction, wave propagation, and potential theory.

10.1 Two-Point Boundary Value Problems

Up to this point in the book we have dealt with initial value problems, consisting of a differential equation together with suitable initial conditions at a given point. A typical example, which was discussed at length in Chapter 3, is the differential equation

/ + p(t) Ó+ q (t )y = g(t), (1)

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Chapter 10. Partial Differential Equations and Fourier Series

with the initial conditions

y(t0) = Óî, y' = y'o ¦ (2)

Physical applications often lead to another type of problem, one in which the value of the dependent variable y or its derivative is specified at two different points. Such conditions are called boundary conditions to distinguish them from initial conditions that specify the value of y and y' at the same point. A differential equation together with suitable boundary conditions form a two-point boundary value problem. A typical example is the differential equation

y" + P(x) y'+ q (x) y = g(x) (3)

with the boundary conditions

y(a) = y0’ y(e) = Óã (4)

The natural occurrence of boundary value problems usually involves a space coordinate as the independent variable so we have used x rather than t in Eqs. (3) and (4). To solve the boundary value problem (3), (4) we need to find a function y = ô(õ) that satisfies the differential equation (3) in the interval a < x < â and that takes on the specified values y0 and y1 at the endpoints of the interval. Usually, we seek first the general solution of the differential equation and then use the boundary conditions to determine the values of the arbitrary constants.

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