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Computer Intensive Statistical Methods (J. S. Urban Hjorth)

in the numerical integration ofa differential equation, a standard technique is to replace it by a suitable difference equation whose solution can be obtained in a stable manner and without trouble from round-off errors. However, the qualitative properties of the solutions of difference equations are sometimes quite different from those of corresponding differential equations. The purpose of this book is to show that for several elementary differential equations, one can exploit a wider range of approximations than is customary to obtain difference equations that preserve most of the qualitative properties of the solutions. This subject is of theoretical importance. In fact, discretizations of many well-known differential equations, such as Clairaut, Euler and Riccati, have been known for quite some time, whereas Bernoulli, Verhulst, harmonic oscillator, Duffing, van der Pol, Hill, Mathieu, Weierstrass, Volterra, FitzHugh-Nagumo, Korteweg-de Vries, and Lagrange have been studied by Potts and others; see Chapter 3 in ]. In the present monograph, the author offers a set of rules to construct nonstandard difference equation approximations. However, the author’s approach is not really new. Difference equation approximations are obtained either by using the known forms of the solutions of the differential equations or by ad hoc experimentation. A more effective method is still lacking. The main contribution of the present monograph is to reinforce the practical importance and the problems of finding difference equation approximations.