ABSTRACT
In this chapter we use a simple nonlinear ordinary differential equation as an example to introduce the basic ideas of the homotopy analysis method.
Consider a free sphere dropping in the air from a static state. Let t˜ denote the time, U(t˜) the velocity of the sphere, m the mass, and g the acceleration of gravity. Assume that the air resistance on the sphere is a U2(t˜), where a is a constant. Then, due to Newton’s second law, it holds
m dU(t˜) dt˜
= mg − aU2(t˜), (2.1)
subject to the initial condition
U(0) = 0. (2.2)
Physically speaking, the speed of a freely dropping sphere is increased due to the gravity until a steady velocity U∞ is reached. So, even not knowing the solution U(t˜) in detail, we can gain the limit velocity U∞ directly from (2.1), i.e.,
U∞ = √
mg
a . (2.3)
Using U∞ and U∞/g as the characteristic velocity and time, respectively, and writing
t˜ = (
U∞ g
) t, U(t˜) = U∞V (t), (2.4)
we have the dimensionless equation
V˙ (t) + V 2(t) = 1, t ≥ 0, (2.5) subject to the initial condition
V (0) = 0, (2.6)
denotes the with respect to t. Obviously, as t → +∞, i.e., t˜ → ∞ and U(t˜) → U∞, we have from (2.4) that
lim t→+∞V (t) = 1, (2.7)
even without solving Equations (2.5) and (2.6). The exact solution of Equations (2.5) and (2.6) is
V (t) = tanh(t), (2.8)
useful for the comparisons of different approximations.
