Classical Mathematical Theory
This first chapter contains the classical theory of differential equations, which we judge useful and important for a profound understanding of numerical processes and phenomena. It will also be the occasion of presenting interesting examples of different
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This first chapter contains the classical theory of differential equations, which we judge useful and important for a profound understanding of numerical processes and phenomena. It will also be the occasion of presenting interesting examples of differential equations and their properties. We first retrace in Sections I.2-I.6 the historical development of classical integration methods by series expansions, quadrature and elementary functions, from the beginning (Newton and Leibniz) to the era of Euler, Lagrange and Hamilton. The next part (Sections I.7-I.14) deals with theoretical properties of the solutions (existence, uniqueness, stability and differentiability with respect to initial values and parameters) and the corresponding flow (increase of volume, preservation of symplectic structure). This theory was initiated by Cauchy in 1824 and then brought to perfection mainly during the next 100 years. We close with a brief account of boundary value problems, periodic solutions, limit cycles and strange attractors (Sections I.15 and I.16).
I.1 Terminology
A differential equation of first order is an equation of the form y = f (x, y)
(1.1)
with a given function f (x, y) . A function y(x) is called a solution of this equation if for all x , (1.2) y (x) = f x, y(x) . It was observed very early by Newton, Leibniz and Euler that the solution usually contains a free parameter, so that it is uniquely determined only when an initial value y(x0 ) = y0 (1.3) is prescribed. Cauchy’s existence and uniqueness proof of this fact will be discussed in Section I.7. Differential equations arise in many applications. We shall see the first examples of such equations in Section I.2, and in Section I.3 how some of them can be solved explicitly. A differential equation of second order for y is of the form y = f (x, y, y ).
(1.4)
Here, the solution usually contains two parameters and is only uniquely determined by two initial values y(x0 ) = y0 ,
y (x0 ) = y0 .
(1.5)
Equations of second order can rarely be solved explicitly (see I.3). For their numerical solution, as well as for theoretical investigations, one usually sets y1 (x) := y(x) , y2 (x) := y (x) , so that equation (1.4) becomes y1 = y2
y1 (x0 ) = y0
y2
y2 (x0 ) = y0 .
= f (x, y1 , y2 )
(1.4’)
This is an example of a first order system of differential equations, of dimension n (see Sections I.6 and I.9), y1 = f1 (x, y1 , . . . , yn ) ... yn
= fn (x, y1 , . . . , yn )
y1 (x0 ) = y10 ... yn (x0 ) = yn0 .
(1.6)
I.1 Terminology
3
Most of the theory of this book is devoted to the solution of the initial value problem for the system (1.6). At the end of the 19th century (Peano 1890) it became customary to introduce the vector notation y = (y1 , . . . , yn )T ,
f = (f1 , . . . , fn )T
so that (1.6) becomes y = f (x, y) , which is again the same as (1.1), but now with y and f interpreted as vectors. Another possibility for the second order equation (1.4), instead of transforming it into a system (1.4’), is to develop methods specially adapted to
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