chapter  3
68 Pages

Chapter 3

Miscellaneous Difference Equations

In numerical integration of a differential equation a standard approach is to replace it by a suitable difference equation whose solution can be obtained in a stable manner and without troubles from round off errors. However, often the qualitative properties of the solutions of the difference equation are quite different from the solutions of the corresponding differential equations. In this chapter we shall <:arefully <:hoose difference equation approximations of several well known ordinary and partial differential equations, and show that the solutions of these difference equations preserve most of the properties of the corresponding differential equations. We begin with Clairaut's, Euler's and Riccati's difference equations which are known for quite sometime. This is followed by Bernoulli's difference equation which can be solved in a closed form. Next we consider the Verhulst difference equation and show that its solutions correctly mimic the true solutions of the Verhulst differential equation. Then, we develop the 'best' discrete approximations of the linear differential equations with constant coefficients. Here, as an example, simple harmonic oscillator differential equation is best discretized. This is followed by Duffing's difference equation which can be solved explicitly and whose solutions have precise agreement with the solutions of Duffing's differential equation. Next we consider van der Pol's difference equation, which like van del Pol's differential equation cannot be solved, however the solutions of both the equations have same qualitative features. Then, we deal with Hill's and in particular Mathieu's difference equations, and provide conditions for basically periodic solutions of period 1r and 27r . This leads to a classification of four different types of periodic solutions, which is a well known result for the solutions of Hill's differential equation. Next we shall show that Weierstrass' elliptic differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic functions. In Section 3.12 we analyze Volterra's difference equations, their trajectories have the same closed form expression as for the Volterra's differential equations. Then, we provide several methods to

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solve linear partial difference equations with constant coefficients in two independent variables. This is followed by the best discretizations of Wave equation, FitzHugh Nagumo's equation, Korteweg de Vries' equation and Modified KdV equation. Finally, in Section 3.18 we shall formulate discrete Lagrange's equations of motion.