scholarly journals A Recent Development of Computer Methods for Solving Singularly Perturbed Boundary Value Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-32 ◽  
Author(s):  
Manoj Kumar ◽  
Parul
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Physical Significance ◽  
Singularly Perturbed ◽  
Research Articles ◽  
Singular Perturbation Theory ◽  
Important Research ◽  
Computer Methods ◽  
Perturbation Problems ◽  
Work Done

This paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of ideas and methods of singular perturbation theory. The work done in this area during the periods 1984–2000 and 2000–2005 has already been surveyed in 2002 and 2007 but our main objective is to produce a collection of important research articles of physical significance. In this paper, the crux of research articles published by numerous researchers during 2006–2010 in referred journals has been presented, and this leads to conclusions and recommendations about what methods to use on singular perturbation problems.

Optimizing a Class of Nonlinear Singularly Perturbed Systems Using SDRE Technique

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Seyed Mostafa Ghadami ◽  
Roya Amjadifard ◽  
Hamid Khaloozadeh
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Riccati Equation ◽  
The State ◽  
Singularly Perturbed ◽  
Singular Perturbation Theory ◽  
Singularly Perturbed Systems ◽  
Perturbed State ◽  
State Dependent ◽  
Perturbed Systems

In this paper, we address the finite-horizon optimization of a class of nonlinear singularly perturbed systems based on the state-dependent Riccati equation (SDRE) technique and singular perturbation theory. In such systems, both slow and fast variables are nonlinear. Moreover, the performance index for the system states is nonlinearly quadratic. In this study, unlike conventional methods, linearization does not occur around the equilibrium point, and it provides a description of the system as state-dependent coefficients (SDCs) in the form f(x) = A(x)x. One of the advantages of the state-dependent Riccati equation method is that no information about the Jacobian of the nonlinear system, just like the Hamilton–Jacobi–Belman (HJB) equation, is required. Thus, the state-dependent Riccati equation has simplicity of the linear quadratic method. On the other hand, one of the advantages of the singular perturbation theory is that it reduces high-order systems into two lower order subsystems due to the interaction between slow and fast variables. In the proposed method, the singularly perturbed state-dependent Riccati equations are first derived for the system under study. Using the singular perturbation theory, the singularly perturbed state and state-dependent Riccati equations are separated into outer layer, initial, and final layer correction equations. These equations are then solved to obtain the optimal control law. Simulation results in comparison with the previous methods indicate the desirable performance and efficiency of the proposed method. However, it should be noted that due to the dependence of the proposed method on the choice of state-dependent matrices and the presence of a nonlinear optimal control problem, the results are generally suboptimal.

NAIVE SINGULAR PERTURBATION THEORY

2001 ◽  
Vol 11 (01) ◽  
pp. 119-131 ◽  
Author(s):  
ROBERT E. O'MALLEY
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Initial Value Problems ◽  
Bessel Functions ◽  
Special Functions ◽  
Singularly Perturbed ◽  
Singular Perturbation Theory ◽  
Initial Value ◽  
First Order ◽  
Initial Layers

The paper demonstrates, via extremely simple examples, the shocks, spikes, and initial layers that arise in solving certain singularly perturbed initial value problems for first-order ordinary differential equations. As examples from stability theory, they are basic to many asymptotic techniques. First, we note that limiting solutions of linear homogeneous equations [Formula: see text] on t≥0 are specified by the zeros of [Formula: see text], rather than by the turning points where a(t) becomes zero. Furthermore, solutions to the solvable equations [Formula: see text] for k=1, 2 or 3 can feature canards, where the trivial limit continues to apply after it becomes repulsive. Limiting solutions of the separable equation [Formula: see text] may likewise involve switchings between the zeros of c(x) located immediately above and below x(0), if they exist, at zeros of A(t). Finally, limiting solutions of many other problems follow by using asymptotic expansions for appropriate special functions. For example, solutions of [Formula: see text] can be given in terms of the Bessel functions Kj(t4/4ε) and Ij(t4/4ε) for j=3/8 and -5/8.

scholarly journals Renormalization group and fractional calculus methods in a complex world: A review

2021 ◽  
Vol 24 (1) ◽  
pp. 5-53
Author(s):  
Lihong Guo ◽  
YangQuan Chen ◽  
Shaoyun Shi ◽  
Bruce J. West
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Fractional Calculus ◽  
Singular Perturbation ◽  
World View ◽  
Singular Perturbation Theory ◽  
Asymptotic Behavior Of Solutions ◽  
Quantum Field ◽  
Self Similarity ◽  
Way Of Thinking

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

On voltage stability from the viewpoint of singular perturbation theory

1994 ◽  
Vol 16 (6) ◽  
pp. 409-417 ◽  
Author(s):  
N. Yorino ◽  
H. Sasaki ◽  
Y. Masuda ◽  
Y. Tamura ◽  
M. Kitagawa ◽  
...  
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Voltage Stability ◽  
Singular Perturbation Theory
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