Application of the regular perturbation method for the solution of first-order initial value problems

Several analytical and numerical methods are globally used in the solution of linear and non-linear ordinary differential equations. In recent times, an analytical method called the perturbation method was proposed by JiHuan He. This technique involves the use of the perturbation parameter, ε. The types of perturbation method which are usually used in Mathematics are the regular perturbation method and the singular perturbation method. In this research, various numerical examples of first-order IVPs are solved using the regular perturbation method. This method gives an approximate analytical solution to the first order IVPs and are discovered to be efficient and easy to adopt.


Introduction
Differential equation is one of the significant areas in mathematics. They are not just used in mathematics; various disciplines such as Physics, Astronomy, Biology, Meteorology, and Economics apply differential equations in their study. Some of these applications include: the use of differential equations: to describe various exponential decays and growths; to describe the change in investment return over time, in the medical science field to model the development of cancer or disease. The numerical methods of finding the solution of Ordinary Differential Equations (ODEs) problems have been a matter of great deal and many researchers have brought about several and different approaches and methods to solve ODE problems [1][2]. One of such methods is the Perturbation method. The perturbation approach is applied by obtaining an approximate solution to the ODE problem, starting with the exact solution of a related, clearer problem. Some applications of perturbation method include: the use of perturbation method for fractional Fornberg-Whitham equation [3]. The perturbation approach is now employed by several researchers for the application of several differential problems [4 -7]. The homotopy perturbation method (HPM) is another important method that have been introduced to solve differential equations and it is based on the perturbation method. The HPM and its modifications have been used in the solution of generalized linear and non-linear Riccati differential equation. The modified homotopy perturbation method was said to have been thoroughly tested in the solving of several implementations and it was shown that the method was efficient and accurate [8]. Meaningful interpretable explanation of Black boxes in a study done by [9]. The Least Squares Homotopy Perturbation Method (HPM) was used in the solution of nonlinear differential equations and the main feature of this method is the accelerated convergence which when compared to the regular perturbation method gives more accurate results [10]. In a study shown in [11], the HPM was well discussed as the problem of its convergence is not prominent work done by other researchers. The perturbation method was analyzed and the convergence of this method is discussed. It also showed that the homotopy perturbation approach converges to the exact solution of the ordinary or partial differential equations which are not linear [12 -17].

Methodology
In this section, the regular perturbation method would be discussed in respect to the solution of first order ordinary differential equations. It would explain how it should be applied to the first order differential equation.

Regular Perturbation Method
The regular perturbation method is implemented by getting an estimated solution to the ODE problem, beginning with the specific solution of a similar, more straightforward problem. A first order differential equation such as: where and yf are vector functions, x is an independent scalar variable and  is a small parameter, said to be regularly perturbed if In such a case, the solution of such system is found in the form of an asymptotic series which is Considering an ODE: yx is the solution of an ODE in which the ordinary differential equation and the initial conditions solely depends on a very small parameter  . We would need the Taylor Series expansion in order to compute the solution, ( , ) yx using perturbation. In a Taylor Series expansion, we suppose a function () gy is differentiable at ( yy  as: The Taylor Series expansion of ( , ) yx is given as: which is the perturbation solution.
In most cases,  is sufficiently small and then the perturbation parameter p which has the range [0, 1], is the power of  and it needs to be ignored when 2 p  . Since we are ignoring the power of  when 2 p  , we therefore get: Therefore, we would know that 01 ( , ) y x y y   .
(10) After the Taylor Series expansion, we then compute the steps involved to show an instance of how it goes. FIRST STEP: The equation and the initial conditions are written down. Then we focus on the variable which the parameter  appears. For instance: In that equation, we focus on the variable that has  in it. Also, we put down ( , ) x y  .
in which we obtain our approximate solution. If for any reason

Numerical Examples
In this section, the regular perturbation method is applied to both linear and nonlinear IVPs.
Expanding we have: In every perturbation case, if for any reason in the equation the power of  is greater than 2, we ignore it. Ignoring that in (16) we then have: Thereafter we set 0   in equations (17) and (19)

 
The exponential solution is then given as: The next thing is to obtain the value of 1 y , in order to do that, we differentiate equation (17) and (19) respectively with respect to  , doing that, we get: Recall that, if for any reason in the equation the power of  is greater than 2, we ignore it. Therefore, we get: We then set 0   in (30) and (31) order to obtain the value for Thus, the regular perturbative solution of this differential equation is: in the original equation and the initial condition would be given as: Expanding this we have: If for any reason in the equation the power of  is greater than 2, we ignore it. Therefore, we get: The initial condition is given as: Solving for the value of 0 y , we get: (2 ) mx  Substituting 2 mx , the exponential solution is given by: In order to obtain the value of 0 y , we need to get the value of A, we make use of the initial condition as given in (41).

Conclusion
The regular perturbation method was applied to three initial value problems (linear and nonlinear).
Using this method to solve differential equations of first order, it is found that the method can provide numerical approximations to ordinary differential equations of any order. From the examples solved, the regular perturbation method is reliable and efficient. The major advantage of this method is the fact that it can be used to solve both linear and non-linear equations and the major disadvantage is that several and many iterations are needed in order to solve those equations.