Basic Idea of Homotopy Perturbation Method
To illustrate the basic ideas of the new method, we consider the following nonlinear differential equation,
(1)With boundary conditions of:
(2) where
A(u) is defined as follows:
(3) where
stands for the linear and
for the nonlinear part. Homotopy perturbation structure is shown as the following equation:
(4) where,
(5)Obviously, using Eq. (4) we have:
(6) where
is an embedding parameter and u
0 is the first approximation that satisfies the boundary condition. The process of changes in p from zero to unity is that of
changing from
to
We consider
, as the following:
(7) and the best approximation for the solution is:
(8)The above convergence is discussed in Cai et. Al (2006) and Cveticanin (2005).