Poincaré’s Perturbation Method
This chapter, like the previous one, is devoted to the research of approximate solutions of a Cauchy problem for Equation (1.3). However, the approach presented in this chapter, which is called Poincaré’s method, follows a completely different logic (see, e.g., , , ). In fact, the power series method supplies a polynomial of a fixed degree r, which represents an approximate solution of (1.3) at least in a neighborhood of the initial value t 0 of the independent variable t. A better approximation is obtained by increasing the value r, that is by considering more terms of the power expansion, or values of the independent variable t closer to the initial value to. Poincaré’s method tries to give an approximate solution in an extended interval of the variable t, possibly for any t. In this logic one is ready to accept a less accurate solution, provided that this is uniform with respect to t. In effect, Poincaré’s approach does not always give a solution with this characteristic: More frequently, it gives an approximate solution whose degree of accuracy is not uniform with respect to time. This is due, as we shall see in this chapter, to the presence in Poincaré’s expansion of the so-called secular terms that go to infinity with t. However, by applying to the aforesaid expansion another procedure (Lindstedt, Poincaré), a uniform expansion with respect to t is derived. This approach will be analyzed in Chapter 8.
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