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Asymptotic Expansion and Perturbation
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
In a chapter of this size, we can only introduce the basic ideas of this advanced technique that can provide approximate but insightful results for highly nonlinear problems, which are otherwise unsolvable. Our discussion only targets elementary and introductory levels. In particular, asymptotic expansions and their application in the regular perturbation method in solving ODEs are discussed. Singular perturbation methods are discussed briefly, and they include the method of matched asymptotic expansion (or boundary layer analysis), multiple scale analysis, and WKB approximation. The application of asymptotic expansion in evaluating integrals with a large parameter is discussed in light of the Laplace method, Riemann (or Debye) method of steepest descent (or saddle point method), and the method of stationary phase of Kelvin and Stokes. We conclude the chapter by looking at how to apply the perturbation method to convert the Navier-Stokes equation of fluid flows into a linear solvable problem. The Airy water wave and shallow water wave are considered as an example.
Analytical investigation of asymmetric forced vibration behavior of functionally graded porous plates with structural damping
Published in Mechanics Based Design of Structures and Machines, 2023
S. Karen Alavi, Majid R. Ayatollahi, Mohd Yazid Yahya, S. S. Rahimian Koloor
Various lengths or time scales in general are related to different natural processes. Failure to recognize the reliance on more than a one-time scale is a major cause of nonuniformity in perturbation expansions. For both small and large values of the independent variables, multiple-scale analysis is a combination of mathematical and physical methodologies for creating uniformly valid approximations to the solutions to perturbation problems. This is accomplished by establishing fast-scale and slow-scale variables for an independent variable, and then treating the fast and slow variables separately. As regular perturbation procedures malfunction, this scheme is employed (Salih 2014). As a result, the multiple scales methodology of the perturbation expansion is used to solve the resulted formula. Initially, the time scales of T0 = t* and T1 = εt* are represented as follows (Nayfeh 2011):
Study on fundamental damped frequency parameters of plates made of FG porous material: a close-form solution
Published in Waves in Random and Complex Media, 2022
S. Karen Alavi, Majid R. Ayatollahi, Seyed Saeid Rahimian Koloor, Michal Petru
Some natural processes, in general, are connected with many characteristic length or time scales. A major source of non-uniformity in perturbation expansions is the failure to identify dependency on more than a one-time scale. Multiple-scale analysis is a set of techniques used in mathematics and physics to develop uniformly valid approximations to the solutions of perturbation problems, for both small and large values of the independent variables. To achieve this, fast-scale and slow-scale variables are defined for an independent variable, and the fast and slow variables are then treated as independent variables. The procedure is used when regular perturbation approaches fail [62]. Hence, the multiple scales procedure of perturbation technique is employed to solve the obtained equation. Firstly, the time scales and are defied in the following [63]: