ABSTRACT
It is well known that many nonlinear problems have multiple solutions. For example, let us consider again the so-called Duffing oscillator in space, governed by
v′′ + (v − v3) = 0, (7.1) subject to the boundary conditions
v(0) = v(π) = 0, (7.2)
where the prime denotes the derivation with respect to ξ. In Chapter 6, we use the homotopy analysis method to solve the same problem and correctly discover its critical condition = 1 for the simple bifurcation and express its solution by such a set of base functions
{sin[(2m + 1)ξ] | m ≥ 0} . (7.3)
Notice that there exist an infinite number of sets of base functions denoted by
{sin[(2m + 1)κ ξ] | m ≥ 0, κ ≥ 1} , (7.4) where κ ≥ 1 is a positive integer, which can be used to express a real function satisfying the boundary conditions (7.2). This implies that Equations (7.1) and (7.2) might have multiple solutions. This is indeed true. We show in this chapter that, using base functions denoted by (7.4), we can gain all multiple solutions of Equations (7.1) and (7.2) by means of the homotopy analysis method.
