1. Analytical method for solution:
  2. 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: