**1. Introduction**

The descriptions of the behaviors of the real world phenomena and systems through the use of mathematical models often involve developments of nonlinear equations which are difficult to solve exactly and analytically. Consequently, recourse is always made to numerical methods as alternative methods in solving the nonlinear equations. However, the developments of analytical solutions are obviously still very important. Analytical solutions for specified problems are also essential and required to show the direct relationship between the models parameters. When analytical solutions are available, they provide good insights into the significance of various system parameters affecting the phenomena. Such solutions provide continuous physical insights than pure numerical or computation methods. Indisputably, analytical solutions are convenient for parametric studies, accounting for the physics of the problem and appear more appealing than the numerical

solutions. Also, they help in reducing the computation and simulation costs as well as the task involved in the analysis of real-life problems.

Although, there is no general exact analytical method to solve all nonlinear problems, over the years, the nonlinear problems have been solved using different approximate analytical methods such as regular perturbation, singular perturbation method, homotopy perturbation method, homotopy analysis method, methods of weighted residual, variational iterative method, differential transformation method, variation parameter method, Adomian decomposition method, etc. The non-perturbative approximate analytic methods present explicit approximate analytical solutions which often involve complex mathematical analysis leading to analytic expressions involving large number terms. Furthermore, the methods are inherently with high computational cost and time accompanied with the requirement of high skills in mathematics. Moreover, in practice, analytical solutions with large number of terms and conditional statements for the solutions are not convenient for use by designers and engineers. Also, in these methods, there are always search for particular value(s) that will satisfy the end boundary condition(s). This always necessitates the use of software and such could result in additional computational cost in the generation of solution to the problem. Also, the quests involve applications of numerical schemes to determine the required value(s) that will satisfy the end boundary condition(s). This fact renders most of the approximate analytical methods to be taken as more of semi-analytical methods than total approximate analytical methods. Moreover, these methods have their own operational restrictions that severely narrow their functioning domain and when they are routinely implemented, they can sometimes lead to erroneous results. Specifically, the transformation of the nonlinear equations and the development of equivalent recurrence equations for the nonlinear equations using differential transformation method proved somehow difficult in some nonlinear system such as in rational Duffing oscillator, irrational nonlinear Duffing oscillator, finite extensibility nonlinear oscillator. There is difficulty in the determination of Adomian polynomials for the application of Adomian decomposition method for nonlinear problems. There are lack of rigorous theories or proper guidance for choosing initial approximation, auxiliary linear operators, auxiliary functions, and auxiliary parameters in the use of homotopy analysis method. Therefore, the need for comparatively simple, flexible, generic and high accurate total approximate analytical solutions is well established. One of the techniques that can be applied for such quest is the perturbation method. Perturbation method, although comparably old, as a pioneer method for finding approximate analytical solutions to nonlinear problems, it offers an alternative approach to solving certain types of nonlinear problems. In the limit of small parameter, perturbation method is widely used for solving many heat transfer, vibration, fluid mechanics and solid mechanics problems. It is capable of solving nonlinear, inhomogeneous and multidimensional problems with reasonable high level of accuracy. The most significant efforts and applications of the method were focused on celestial mechanics, fluid mechanics, and aerodynamics. Although, the solutions reported for other sophisticated methods to difference problems have good accuracy, they are more complicated for applications than perturbation method. Therefore, over the years, the relative simplicity and high accuracy especially in the limit of small parameter have made perturbation method an interesting tool among the most frequently used approximate analytical methods. Although, the perturbation method provides in general, better results for small perturbation parameters, besides having a handy mathematical formulation, it has been shown to have a good accuracy, even for relatively large values of the perturbation parameter [1–5].

*Perturbation Methods to Analysis of Thermal, Fluid Flow and Dynamics Behaviors of… DOI: http://dx.doi.org/10.5772/intechopen.96059*
