Perturbation theory developed by Taiwo and Osilagbum, (2011) is widely used for the simplification of complex mathematical problems. The use of perturbation theory will allow approximate solutions to determine the problems that cannot be solved by traditional analytical methods. However, in many cases, real-life situations can require much more difficult mathematical models, such as non-linear differential equations.

Perturbation theory is a powerful mathematical technique used to approximate solutions to problems that are difficult to solve exactly. It is particularly useful in physics, chemistry, engineering, and applied mathematics, where many complex systems can be represented by differential equations that are challenging to solve analytically. Perturbation theory provides a systematic way to obtain approximate solutions by breaking down the problem into simpler, more solvable parts which was investigated by Debwan and Hasan, (2020). The basic idea behind perturbation theory is to start with a known, easily solvable problem (the unperturbed problem) and then introduce small corrections or perturbations to it. These perturbations are typically represented by small parameters that quantify the deviation from the idealized, unperturbed system. By treating these perturbations as small deviations from the known solution, one can develop a series expansion in terms of these parameters, which can then be used to derive increasingly accurate approximations to the true solution. There are two main types of perturbation theory: regular perturbation theory and singular perturbation theory. Regular perturbation theory developed by Dehghan and Shakeri, (2008) in which the perturbations are assumed to be small across the entire domain of interest. This allows for the development of a systematic series expansion, such as a power series, in terms of the small parameter. Each term in the series represents a higher-order correction to the solution, and the series can often be truncated at a certain order to obtain a sufficiently accurate approximation. The other is singular perturbation theory; singular perturbation theory is used when the perturbations are not small throughout the entire domain, but rather become significant in certain regions or under certain conditions. In such cases, a straightforward series expansion may not converge, and alternative techniques, such as matched asymptotic expansions or boundary layer analysis, are employed to obtain accurate solutions near the points of interest. Once the equation is non-dimensionalized, perturbation theory requires taking advantage of a “small” parameter that appears in an equation was explained by (Shakeri and Dehghon 2008; Elrazeg et al., 2022). This parameter, usually denoted “ε” is on the order of 0 < ε << 1. 

Perturbation theory has its roots in early celestial mechanics, where the theory of epicycles was used to make small corrections to the predicted paths of planets. Perturbation methods are powerful techniques used in mathematics and physics to approximate the results to differential equations, especially when exact solutions are difficult or impossible to find. These methods are particularly useful for problems where a small parameter exists, allowing for an expansion around a known solution. Perturbation theory relies on the existence of a small parameter, often denoted as ϵ, which quantifies the deviation from a simpler problem. This parameter could represent a physical quantity like mass ratio, coupling constant or geometric scale. The central idea of perturbation theory is to approximate the solution to a complex problem by iteratively correcting a known solution to a simpler problem. This known solution is usually obtained by setting the small parameter to zero. Perturbation theory often involves analyzing the behavior of the solution as the small parameter approaches zero. Asymptotic behavior provides insights into the dominant terms contributing to the solution. The accuracy and validity of the approximate solution are validated by comparing it with exact solutions (if available), numerical simulations, or experimental data. Perturbation theory finds applications in diverse fields such as celestial mechanics, quantum mechanics, fluid dynamics, population dynamics, and many more. Advanced techniques such as multiple scales method, matched asymptotic expansions, and resummation methods are employed to improve the approximation problems (Nino et al., 2013; Gervais et al., 1975). Perturbation theory provides a systematic framework for tackling complex problems and gaining insights into the behavior of physical systems. Its versatility and applicability make it an indispensable tool in scientific and engineering research. The beginnings of perturbation theory can be traced to Isaac Newtons work on the gravitational interactions between celestial bodies in the 17th century. While Newton provided exact solutions for two-body problems, the interactions of more than two bodies posed significant challenges. In the late 18th and early 19th centuries, mathematicians such as Joseph-Louis Lagrange and Pierre-Simon Laplace made significant contributions to celestial mechanics. They developed perturbation methods by He, (2003) to study the effects of gravitational interactions among multiple celestial bodies. Aghakhani, (2015) introduced the concept of secular perturbations, which long-term effects are arising from gravitational interactions that cause slow changes in the orbits of celestial bodies over time. The development of series solutions for differential equations by mathematicians like Leonhard Euler and Joseph Fourier provided a mathematical foundation for perturbation methods. Laplace and Macgillivry, (2008) developed the methods for finding asymptotic expansions of integrals, which laid the groundwork for the asymptotic analysis used in perturbation theory. William Rowan Hamiltons reformulation of classical mechanics in terms of Hamiltonian mechanics provided a new framework for perturbation theory. Wilsen and Rallison, (2007) made significant contributions to perturbation theory in celestial mechanics, introducing the concept of canonical transformations and developing perturbation methods to study the stability of the solar system. Quantum Mechanics: Perturbation theory found widespread application in quantum mechanics, particularly in the development of quantum electrodynamics (QED) by physicists such as Paul Dirac, Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga. Feynman diagrams revolutionized perturbation theory in quantum field theory, providing a graphical representation of particle interactions and facilitating calculations of scattering amplitudes. 

Perturbation methods have been extended by Xia and Zhang, (2024) to study nonlinear dynamical systems, chaos theory, and bifurcation theory. Perturbation techniques are widely used in engineering disciplines such as fluid dynamics, structural mechanics, and control theory to analyze the effects of small disturbances on system behavior. Throughout its history, perturbation theory has evolved from its origins in celestial mechanics to become a fundamental tool in various branches of science and engineering. Its development has been driven by the need to understand and quantify the effects of small perturbations on complex systems, leading to advancements in both theory and application. Overall, the development of basic perturbation theory by Clenshaw and Norton, (1963) involves a systematic process of approximating solutions to differential equations by expanding them in powers of a small parameter and iteratively refining the solution to higher orders of accuracy. These methods provide valuable insights into complex systems and phenomena that cannot be fully understood using exact analytical techniques.

The methodology of perturbation methods involves a systematic approach to approximate results to differential equations by treating small deviations or perturbations from a known, easily solvable problem. Heres a general outline of the methodology. There are three steps and eight procedures of perturbation analysis. The three steps are given below;

• To transform the main problem into a perturbation problem by taking a small parameter δ.
• To consider an expression for the solution in the form of a perturbation series and determine the coefficient of that series.
• To regain the solution to the main problem by adding the perturbation series for the appropriate value of δ.

Step (1): There is sometimes ambiguity because there are many ways to introduce an δ. However, it is preferable to introduce δ in such a way that the zeroth-order solution i.e. the leading term in the perturbation series is obtainable as a closed-form analytic expression. 

Step (2):  By setting δ =0 in the perturbation problem, a first-order solution consists of finding the first two terms in the perturbation series, and so on. 

Step (3): Begin by identifying the differential equation that describes the problem of interest. This equation typically represents the behavior of a system under consideration, such as a physical system governed by the laws of physics or an engineering system described by mathematical models.

Now the eight procedures are as follows;

Introduce Perturbation Parameters

Identify small parameters that quantify the deviations or perturbations from the idealized, unperturbed system. These parameters may represent small variations in system parameters, initial conditions, boundary conditions, or external influences.

Decompose the Solution

Decompose the solution into two parts: the outcome of the unperturbed problem and the perturbation correction terms. The unperturbed solution represents the solution to a simplified version of the problem that is easily solvable, while the perturbation correction terms account for the effects of small deviations from this idealized solution.

Expand the Solution in a Series

Express the solution as a series expansion in terms of the small perturbation parameters. This series expansion typically takes the form of a power series, where each term represents a higher-order correction to the solution. The coefficients of the series are functions of the perturbation parameters and are determined iteratively.

Derive Equations for the Coefficients

Substitute the series expansion into the original differential equation and equate coefficients of like powers of the perturbation parameters on both sides of the equation. This yields a set of equations for the coefficients of the series expansion, which can be solved order by order to obtain increasingly accurate approximations to the solution.

Iterative Solution Procedure

Solve the equations for the coefficients iteratively, starting from the zeroth-order term corresponding to the unperturbed solution and proceeding to higher-order terms. Each successive iteration yields a more accurate approximation to the true solution by incorporating additional corrections from higher-order perturbation terms.

Truncate the Series Expansion

Truncate the series expansion at a certain order based on the desired level of accuracy or practical considerations. In many cases, only a few terms of the series are needed to obtain sufficiently accurate approximations, especially if the perturbation parameters are small.

Analyze Convergence and Validity

Analyze the convergence properties of the series expansion and assess the validity of the perturbation approach. Ensure that the perturbation parameters are indeed small and that the series converges to the true solution within the desired range of validity.

Interpretation and Application

Interpret the results into the domain and apply the approximate solution to analyze the behavior of the system, make predictions, or design engineering solutions. Understand the obstacles of the perturbation approach and validate the results through comparisons with numerical simulations or experimental data when possible. By following these steps and procedures, perturbation methods provide a systematic and powerful framework for approximating solutions to differential equations in a wide range of scientific and engineering applications, allowing researchers and engineers to tackle complex problems that are otherwise difficult to solve analytically.

Non-singular Perturbation Theory with First-order

First-order non-singular perturbation theory introduced by Nayfeh, (1981) is a technique used to approximate solutions to differential equations where perturbations become significant at certain points or under specific conditions, but are not small throughout the entire domain. This method was employed by Kumar and Parul, (2011) for the straightforward application of regular perturbation theory, which assumes small perturbations across the entire domain, is inadequate. Now, we have to solve the following differential equation,

(Salem and Thanoon, 2021; Liu and Chang, 2022) interpreted these results in the domain which can be applied the approximate solution to analyze the behavior of the system, make predictions, or design engineering solutions. Understand the limitations of the first-order non-singular perturbation approach and validate the results through comparisons with numerical simulations or experimental data when possible. By following these steps, first-order non-singular perturbation theory provides a systematic approach to approximate solutions to differential equations in situations where perturbations become significant in specific regions or under certain conditions. This method allows researchers and engineers to tackle complex problems that cannot be explained using regular perturbation theory alone.

Approximate Solution of an Initial-value Problem

The initial value problem was chosen by Fowkes, (1968) and the boundary value problem was chosen by Baltaveva and Agarwal, (2018), which gives the following,

Neumann Boundary Conditions

Neumann boundary conditions developed by Kadum and Abdul-Hassan, (2023) are commonly encountered in partial differential equations (PDEs). They specify the function at the boundary of a domain rather than specifying the function itself. Neumann boundary condition concept arise in various physical and mathematical contexts, particularly in problems involving diffusive processes, heat transfer, fluid dynamics, and electromagnetism was given by Saltzman, (1962).

The Perturbation Process employ to the outcome of an Algebraic Equation

The Perturbation Method Expanded with proposed equation 

Consider the following equation. First, the equation is introduced as seen below

The Perturbation Technique for Solving Algebraic Equation

Leading Order Solutions

The approximation process to ????²+???? ????−1= 0 is to set ???? =0

This reduces to:  

The approximation with second order,

x=±1+d (x),                           (29)

Where, δ(????) is some connection factor. 

Inserting x=±1 into x2+???? x-1=0 yields;

Now, 

For the linear ordinary differential equation can be noticed with approximations percent error in Fig. 3. 

Approximate Solutions of an IVP using the Perturbation Method 

The most important and excellent example of perturbation in this paper is example 1. It can be taken as a notable example of the perturbation solution of given equation. It shows how much good the perturbation solution is! 

Example 1

 If ???? ≪ 1, obtain the perturbed equation from ????′′ − ???????????? = 0,????(0) = 1,????′(0) = 1 

Solution: Given, 

Example 2

We want to find out its solution by different methods and show a comparison among those solutions.

Solution:

• The perfect solution given, 

• The perturbation Solution:

We change the problem into a perturbation problem by introducing the parameter ε in such a way that the unperturbed problem is solvable,

Example 3

An approximation solution of the formidable-looking non-linear two-point boundary value problem -

May be readily obtained using perturbation theory.

Since it is not possible to find the exact solution to this problem, it can also be solved numerically using nonlinear shooting with Newtons Method.

Secondly, let us assume that a perturbation expansion for y(x) of the form 

Perturbation solution

First, we introduce ε in such that the under-handed problem is solvable:

A comparison of perturbation series approximation and Numerical solution of the boundary value problem in (A) is presented in the following figure. The graphs are one-term perturbation series approximation and two-term perturbation series approximation (das-hed line) of the form in (C).

The present paper deals with the perturbation method and implies how much effective it is to solve a nonlinear differential equation involving initial as well as boundary conditions. Numerical solutions have been used to fulfill the investigation. Comparisons between numerical and perturbation solutions have been performed using Mathematica and FORTRAN programming. Perturbation methods offer a valuable approach for approximating solutions to differential equations encountered in diverse scientific and engineering contexts. By systematically treating small deviations from known solutions, perturbation methods provide insights into system behavior, facilitate analytical predictions, and aid in the design of engineering systems. Through theoretical developments, numerical techniques, and practical applications, perturbation methods continue to play a vital role in advancing our understanding of complex dynamical systems.

Firstly, I want to thank Almighty Allah. Then I like to thank my family for their support over the course of my graduate study. It is a matter of great pleasure to me expresses my heartiest grati-tude and profound respect to my supervisor Assistant Professor Bishnu Pada Ghosh, Dept. of Mathematics, Jagannath University, Dhaka. Also,  my  deepest  thanks  go  to Kbd Md. Ashraful Kabir for spending time mentoring me during  this  research .

The authors declare that there are no conflicts of interest.