- Analytical method for solution:
- Homotopy perturbation method
Before solving the proposed problem, we first introduce the Homotopy
method in this section. This method was introduced by Mr. He in 1998 and
has been used to solve the nonlinear ordinary differential equation
(ODE). In 2004, he combined this method with the boundary element method
and proposed the method of general boundary element [33]. By means
of Homotopy perturbation, a series of initial guesses can be evaluated
and by a series of auxiliary parameters, a series of answers can be
obtained, which converge to the exact answer. This method utilizes
features such as freedom of action in choosing the initial function and
linear operator. By using the same freedom of action and initial
choices, a complex nonlinear problem is solved and transformed into
smaller and simpler linear problems. Another advantage of this method is
the controllability of the convergence area, which is the most
significant feature of this method in comparison with other methods.
Some of the articles about Homotopy method are in references
[34-35].
To explain the basic ideas of this manner, we consider the following
nonlinear differential equation: