A perturbation method applied to the buckling of a compressed elastica

We consider the problem of carrying out an asymptotic analysis for the phenomenon of bifurcation which occurs at critical values of an axial force applied to an elastic column. In the present setting a discontinuous coefficient precludes the possibility of carrying out the usual asymptotic analysis. The problem is overcome via a nonlinear change of independent variables.     

Numerical study of homotopy‐perturbation method applied to Burgers equation in fluid

In this article, the problem of Burgers equation is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. Comparison is made between the HPM and Exact solutions. The obtained solutions, in comparison with the exact solutions, admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Multiparameter perturbation theory of Fredholm operators applied to bloch functions

In the present paper, a family of linear Fredholm operators depending on several parameters is considered. We implement a general approach, which allows us to reduce the problem of finding the set Λ of parameters t = (t ₁, ..., t _n ) for which the equation A(t)u = 0 has a nonzero solution to a finite-dimensional case. This allows us to obtain perturbation theory formulas for simple and conic points of the set Λ by using the ordinary implicit function theorems. These formulas are applied to the existence problem for the conic points of the eigenvalue set E(k) in the space of Bloch functions of the two-dimensional Schrödinger operator with a periodic potential with respect to a hexagonal lattice.     

The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation

We obtain new exact solutions to generalized Sawada-Kotera equation. Using the variational iteration method combined with the improved generalized tanh-coth method, we construct new traveling wave solutions for the standard Sawada-Kotera equation and, by means of scaling, we obtain new solutions to general Sawada-Kotera equation. Periodic and soliton solutions are formally derived for both models.     

The first-integral method applied to the Eckhaus equation

The first-integral method is a direct algebraic method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. This method is based on the theory of commutative algebra. In this work, we apply the first-integral method to study the exact solutions of the Eckhaus equation.     

高维欧式期权定价模型的奇摄动解

讨论了一类欧式期权定价问题的随机波动率模型,其随机波动率采用快速均值回归的随机波动率模型.通过采用奇摄动方法,得到了多风险资产欧式期权价格的形式渐近展开式,得到该合成展开式的一致有效误差估计.     

The tau method with perturbation term depending on the differential operator

The tau method approximates the solution of a differential equation with a polynomial, which is the exact solution of a differential equation obtained by adding to the right hand side of the given equation a perturbation term, consisting of a suitably chosen linear combination of polynomials.Until now the Chebyshev and Legendre polynomials have been used to this purpose, but the determination of the best perturbation term is, still, an open problem.In this paper a perturbation term depending on the differential equation is chosen. For this formulation of the tau method, the existence of an infinite sequence of tau approximants and the convergence of the error to zero is proved. An estimate of the local truncation error is also given, and the stability properties are discussed.Numerical results are also reported.     

The multi-resolution method applied to the sideways heat equation

We consider the sideways heat equation uxx(x,t)=ut(x,t), 0?x<1, t?0. The solution u(x,t) on the boundary x=1 is a known function g(t). This is an ill-posed problem, since the solution—if it exists—does not depend continuously on the boundary, i.e., small changes on the boundary may result in big changes in the solution. In this paper, we shall use the multi-resolution method based on the Shannon MRA to obtain a well-posed approximating problem and obtain an estimate for the difference between the exact solution and the solution of the approximating problem defined in Vj.     

Splitting method applied to hyperbolic problem with source term

This work deals with the study of an operator splitting applied to approximate scalar conservation laws with source term by use of homogeneous laws. A convergence result towards the entropy weak solution is given. The reasoning is based on fine properties of BV ∩ L^∞ functions and Kruskov techniques.     

Il'yushin approximation method applied to media with unstable properties

A method is proposed for solving the viscous-elastic integral equation with a singular kernel when the parameters depend on the temperature or some other physical factor. No assumption is made as to the existence of a temperature-time analogy. The results are used to solve boundary problems of heredity theory by the Il'yushin approximation method for time-varying properties of the material.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 3, pp. 411–419, May–June, 1970.     

多尺度高维亚式期权定价的奇摄动解

讨论了一类含有快慢变换尺度的高维亚式期权定价随机波动率模型.根据Girsanov定理和Radon-Nikodym导数实现了期望回报率与无风险利率之间的转化;定义路径依赖型的新算术平均算法,借助Feynman-Kac公式,得到了风险资产期权价格所满足的相应的Black-Scholes方程,运用奇摄动渐近展开方法,得到了期权定价方程的渐近解,并得到其一致有效估计.     

The two-scale expansion method for weakly irregular ocean waveguides with current perturbation

The two-scale expansion method is generalized to the case of weakly irregular surface and depth waveguides with current perturbation. This permits estimating the effect of currents on the acoustic field under relevant propagation conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 68–78, 1985.     

The projection method of optimization applied to engineering plasticity problems

In engineering plasticity, the behavior of a structure (e.g., a frame or truss) under a variety of loading conditions is studied. Two primary types of analysis are generally conducted. Limit analysis determines the rigid plastic collapse load for a structure and can be formulated as a linear program (LP). Deformation analysis at plastic collapse can be formulated as a quadratic program (QP). The constraints of the two optimization problems are closely related. This paper presents a specialization of the projection method for linear programming for the limit-load analysis problem. The algorithm takes advantage of the relationship between the LP constraints and QP constraints to provide advantageous starting data for the projection method applied to the QP problem. An important feature of the method is that it avoids problems of apparent infeasibility due to roundoff errors. Experimental results are given for two medium-sized problems.This work was supported by the National Research Council of Canada under Research Grant No. A8189.     

Potential method applied to Boussinesq equation

In this paper a solution of the nonlinear Boussinesq equation is presented using the potential similarity transformation method. The equation is first written in a conserved form, a potential function is then assumed reducing it to a system of equations which is further solved through the group transformation method. New transformations are found.     

The wavelet transform method applied to a coupled chaotic system with asymmetric coupling schemes

The wavelet transform method originated by Wei et al. (2002) [19] is an effective tool for enhancing the transverse stability of the synchronous manifold of a coupled chaotic system. Much of the theoretical study on this matter is centered on networks that are symmetrically coupled. However, in real applications, the coupling topology of a network is often asymmetric; see Belykh et al. (2006)  [23], [24], Chavez et al. (2005)  [25], Hwang et al. (2005)  [26], Juang et al. (2007)  [17], and Wu (2003)  [13]. In this work, a certain type of asymmetric sparse connection topology for networks of coupled chaotic systems is presented. Moreover, our work here represents the first step in understanding how to actually control the stability of global synchronization from dynamical chaos for asymmetrically connected networks of coupled chaotic systems via the wavelet transform method. In particular, we obtain the following results. First, it is shown that the lower bound for achieving synchrony of the coupled chaotic system with the wavelet transform method is independent of the number of nodes. Second, we demonstrate that the wavelet transform method as applied to networks of coupled chaotic systems is even more effective and controllable for asymmetric coupling schemes as compared to the symmetric cases.     

The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem

Summary The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL ²,H ¹ andL , provided that certain relationships hold between the frequency, mesh size and outer radius.     

The reductive perturbation method and the Korteweg-de Vries hierarchy

By using the reductive perturbation method of Taniuti with the introduction of an infinite sequence of slow time variables ₁, ₃, ₅, ..., we study the propagation of long surface-waves in a shallow inviscid fluid. The Korteweg-de Vries (KdV) equation appears as the lowest order amplitude equation in slow variables. In this context, we show that, if the lowest order wave amplitude ₀ satisfies the KdV equation in the time ₃, it must satisfy the (2n+1)th order equation of the KdV hierarchy in the time _n+1, withn=2,3,4, ... As a consequence of this fact, we show with an explicit example that the secularities of the evolution equations for the higher-order terms ( ₁, ₂, ...) of the amplitude can be eliminated when ₀ is a solitonic solution to the KdV equation. By reversing this argument, we can say that the requirement of a secular-free perturbation theory implies that the amplitude ₀ satisfies the (2n+ 1)th order equation of the KdV hierarchy in the time _2n+1. This essentially means that the equations of the KdV hierarchy do play a role in perturbation theory. Thereafter, by considering a solitary-wave solution, we show, again with an explicit, example that the elimination of secularities through the use of the higher order KdV hierarchy equations corresponds, in the laboratory coordinates, to a renormalization of the solitary-wave velocity. Then, we conclude that this procedure of eliminating secularities is closely related to the renormalization technique developed by Kodama and Taniuti.