Applications of nonstandard analysis to boundary value problems in singular perturbation theory

1982 ◽  
pp. 113-135 ◽  
Author(s):  
Robert Lutz ◽  
Tewfik Sari
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Boundary Value Problems ◽  
Nonstandard Analysis ◽  
Boundary Value ◽  
Singular Perturbation Theory

On the Application of Geometric Singular Perturbation Theory to Some Classical Two Point Boundary Value Problems

1998 ◽  
Vol 08 (02) ◽  
pp. 189-209 ◽  
Author(s):  
Michael Hayes ◽  
Tasso J. Kaper ◽  
Nancy Kopell ◽  
Kinya Ono
Keyword(s):  
Perturbation Theory ◽  
Dynamical Systems ◽  
Singular Perturbation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation

In this tutorial, we illustrate how geometric singular perturbation theory provides a complementary dynamical systems-based approach to the method of matched asymptotic expansions for some classical singularly-perturbed boundary value problems. The central theme is that the criterion of matching corresponds to the criterion of transverse intersection of manifolds of solutions. This theme is studied in three classes of problems, linear: ∊y″+αy′+βy=0, semilinear: ∊y″+αy′+f(y)=0, and quasilinear: ∊y″+g(y) y′+f(y)=0, on the interval [0,1], where t∈[0,1], ′=d/dt, 0<∊≪1, and general boundary conditions y(0)=A, y(1)=B hold. Chosen for their relatively simple structure, these problems provide a useful introduction to the methods of geometric singular perturbation theory that are now widely used in dynamical systems, from reaction-diffusion equations with traveling waves to perturbed N-degree-of-freedom Hamiltonian systems, and in applications to a variety of fields.

scholarly journals CEV MODEL WITH STOCHASTIC VOLATILITY

2019 ◽  
Vol 6 (3-4) ◽  
pp. 22-28
Author(s):  
IVAN BURTNYAK ◽  
ANNA MALYTSKA
Keyword(s):  
Perturbation Theory ◽  
Boundary Value Problems ◽  
Stochastic Volatility ◽  
Boundary Value ◽  
Singular Perturbation Theory ◽  
Barrier Options ◽  
Regular Perturbation ◽  
Explicit Formulas ◽  
Wide Range ◽  
Adjoint Operators

This paper develops a systematic method for calculating approximate prices for a wide range of securities implying the tools of spectral analysis, singular and regular perturbation theory. Price options depend on stochastic volatility, which may be multiscale, in the sense that it may be driven by one fast-varying and one slow-varying factor. The found the approximate price of two-barrier options with multifactor volatility as a schedule for own functions. The theorem of estimation of accuracy of approximation of option prices is established. Explicit formulas have been found for finding the value of derivatives based on the development of eigenfunctions and eigenvalues of self-adjoint operators using boundary-value problems for singular and regular perturbations. This article develops a general method of obtaining a guide price for a broad class of securities. A general theory of derivative valuation of options generated by diffusion processes is developed. The algorithm of calculating the approximate price is given. The accuracy of the estimates is established. The theory developed is applied to a diffusion operator, which is decomposed by eigenfunctions and eigenvalues. The purpose of the article is to develop an algorithm for finding the approximate price of two-barrier options and to find explicit formulas for finding the value of derivatives based on the development of self-functions and eigenvalues of self-adjoint operators using boundary-value problems for singular and regular perturbations. Price finding is reduced to the problem solving of eigenvalues and eigenfunctions of a certain equation. The main advantage of our pricing methodology is that, by combining methods in spectral theory, regular perturbation theory, and singular perturbation theory, we reduce everything to equations to find eigenfunctions and eigenvalues.

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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