Comments on ‘A finite extensibility nonlinear oscillator’

The aim of this comment is to provide more information about the study of the dynamics of a finite extensibility nonlinear oscillator conducted by Febbo [M. Febbo, A finite extensibility nonlinear oscillator, Applied Mathematics and Computation 217 (2011) 6464-6475]. We show that the linearized harmonic balance method is not sufficiently adequate for this oscillator and that the harmonic balance method (HBM) without linearization provides better results. We also discuss what happens when the oscillation amplitude approaches 1 and why the harmonic balance method does not give optimum results. For these values of the oscillation amplitude the periodic solution becomes markedly anharmonic and is almost straight between x = +A and x = −A (with negative slope) and between x = −A and x = +A (with positive slope). Finally, a ‘heuristic’ solution is proposed which is adequate for the whole amplitude range 0 < A < 1, which is consistent with the approximate solution obtained previously for A < 0.9 using the HBM.     

Analyzing the longitudinal effect of hypersonic flow past a conical cone via the perturbation method

To analyze the hypersonic flow past a conical cone, the variations of gasdynamic properties subjected to the longitudinal curvature effect by using the perturbation method. An outer perturbation expansion has been carried out by recent researchers, but a problem occurred, the outer expansion solutions are not uniformly valid in the shock layer, however, the outcome near the conical body surface called vortical layer remains deflective. This study intends to discover uniformly valid analytical solutions in the shock layer by applying the inner perturbation expansions matching with the out expansions to analyze the characteristics in the whole region including shock layer and vortical layer. Starting from the zero-order approximate solutions for hypersonic conical flow is then applied as the basic solutions for the outer perturbation expansions of a flow field. The governing equations and boundary conditions are also expanded via outer perturbations. Using an approximate analytical scheme in the derivation process, first-order perturbation equations can be simplified and the approximate closed-form solutions are obtained; furthermore, the various flow field quantities, including the normal force coefficient on the cone surface, have been calculated. According to the variations of gasdynamic properties, the longitudinal curvature effect for the hypersonic flow past a conical cone can be determined. Thicknesses of shock layer and vortical layer can be predicted as well. The physical phenomena inside both layers can be investigated carefully, the conditions for an elliptic cone with longitudinal curvature, m = 1 and n = 2 and other conditions of parameters; the perturbation parameter, ε_m2 = 0.1, semi-vertex angle of the unperturbed cone, δ = 10°, and hypersonic similarity parameter, K_δ = M_∞δ = 1.0, the thickness of vortical layer, η_VL, can be calculated at the position angle of conical cone body, ? = 30° was demonstrated here. Results show how very thin the vortical layer is approximately only 10% of the shock layer close to the body, the pressure in the whole shock layer is verified to be uniformly valid which agrees with previous studies. Large gradient changes in entropy and density are found when the flow approaches the cone surface, the most important is, this method provides a benchmark solution to the hypersonic flow past a conical cone and to assist the grids and numerics for numerical computation should be fashioned to accommodate the whole flow field region including the vortical layer of rapid adjustment, and let the analysis become more effective and low cost.     

An approximate analytical solution of time-fractional telegraph equation

In this article, the powerful, easy-to-use and effective approximate analytical mathematical tool like homotopy analysis method (HAM) is used to solve the telegraph equation with fractional time derivative α (1 < α ? 2). By using initial values, the explicit solutions of telegraph equation for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The method performs extremely well in terms of efficiency and simplicity to solve this historical model.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

Limit periodic motions in some systems with aftereffect under resonance condition

Volterra-type integrodifferential equations and their solutions are considered which, when the time increases without limit, exponentially tend to periodic modes. In the critical case of stability, when the characteristic equation has a pair of pure imaginary roots and the remaining roots have negative real parts, the problem of the existence of limit periodic solutions with resonance, caused by coincidence between the periodic part of the limit external periodic perturbation and the natural frequency of the linearized system, is solved. It is shown that, if the right-hand side of the equation is an analytic function and the existence of limit periodic solutions is determined by terms of the (2m + 1)-th order, these solutions are represented by power series in the arbitrary initial values of the non-critical variables and the parameter μ^1/(2m+1), where μ is a small parameter, characterizing the magnitude of the maximum external periodic perturbation. The amplitude equations are presented.     

The spectral approximation of multiplication operators via asymptotic (structured) linear algebra

A multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Multiplication operators with nonzero symbols, defined on L² spaces of functions, are never compact and then such approximations cannot converge in the norm topology. Instead, we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol (i.e. the multiplier). We discuss the case of real orthogonal polynomial bases and the relations with the classical Fourier basis whose choice leads to the well studied Toeplitz case. Indeed, the asymptotic approximation of the spectrum by the spectra of the associated Toeplitz sections is possible only under precise geometric assumptions on the range of the symbol. Conversely, the use of circulant approximations leads to constructive algorithms, with O(N log(N)) complexity (N = number of sections), working in general and generalizable to the separable multivariate and matrix-valued cases as well.     

Stability of travelling wave solutions of the active Josephson junction transmission line

The behavior of the Josephson line, which is a type of active pulse transmission line, is governed by a partial differential equation which is similar to the sine-Gordon equation. This equation has two solitary travelling wave solutions with different propagation speeds c₁ and c₂, 0 < c₁ < c₂, and a one-parameter family of spatially periodic travelling wave solutions whose propagation speeds range over the intervals (0, c₁) and (c₂, + ∞). First we prove the existence of these solutions. Second we consider the stability of these solutions by linearized stability analysis. It is shown that the slow solitary solution is stable in the sense of linearized stability and that the fast solitary solution is unstable. It is shown also that the periodic solution with the speed c, 0 < c < c₁, is stable in the sense of linearized stability and that the periodic solution with the speed c, c₂ < c < c₄, is unstable, where c₄ is a certain point in (c₂, + ∞).     

The stability in a strict non-linear sense of a trivial relative equilibrium position in the classical and generalized versions of Sitnikov's problem

Stability, in a strict non-linear sense, of a trivial relative equilibrium position is investigated in the classical and generalized versions of Sitnikov's problem in the case of small eccentricities of the orbits of bodies of finite dimensions. In the classical version (n = 2) of the problem, it is proved that there are no second-, third- and fourth-order resonances and a degenerate case. In the generalized version (2 < n ≤ 5 · 10⁵), it is proved that there are no second- and third-order resonances and a degenerate case. A fourth-order resonance occurs in versions of the problem in which the number of finite size bodies satisfies the inequality 45000 ≤ n ≤ 62597 and the orbital eccentricities e < 0.25. Use of the Arnold–Moser and Markeyev theorems enables one to establish the Lyapunov stability of the trivial positions of relative equilibrium in the above-mentioned versions of Sitnikov's problem.     

On the complex oscillation of meromorphic solutions of second order linear differential equations in the unit disc

The main purpose of this paper is to investigate the oscillation theory of meromorphic solutions of the second order linear differential equation f^″+A(z)f=0 for the case where A is meromorphic in the unit disc D={z:|z|<1}.     

Sharp distortion theorems for some subclass of close-to-convex functions

For real A, B such that −1 ? B < A ? 1, we denote by R(A,B) the class of functions, such that f′ ? (1 + Az)/(1 + Bz). Sharp distortion results for functions from R_A,B are obtained.     

On two perturbation estimates of the extreme solutions to the equations X ± A*XA = Q

Two perturbation estimates for maximal positive definite solutions of equations X + A*X⁻¹A = Q and X − A*X⁻¹A = Q are considered. These estimates are proved in [Hasanov et al., Improved perturbation Estimates for the Matrix Equations X ± A*X⁻¹A = Q, Linear Algebra Appl. 379 (2004) 113-135]. We derive new perturbation estimates under weaker restrictions on coefficient matrices of the equations. The theoretical results are illustrated by numerical examples.     

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x₀ of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x₀ as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.     

On p-quasi-hyponormal operators

A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p ? 1, A ∈ p − QH, if A^∗(∣A∣^2p − ∣A^∗∣^2p)A ? 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A⁻¹(0) ⊆ A^∗-1(0), A ∈ p_∗ − QH, a necessary and sufficient condition for the adjoint of a pure p_∗ − QH operator to be supercyclic is proved. Operators in p_∗ − QH satisfy Bishop’s property (β). Each A ∈ p_∗ − QH has the finite ascent property and the quasi-nilpotent part H₀(A − λI) of A equals (A − λI)^-1(0) for all complex numbers λ; hence f(A) satisfies Weyl’s theorem, and f(A^∗) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ(A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p_∗ − QH.     

Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation

The work presents a mathematical model describing the time fractional anomalous-diffusion process of a generalized Stefan problem which is a limit case of a shoreline problem. In this model, the governing equations include a fractional time derivative of order 0 < α ? 1 and variable latent heat. The approximate solution of the problem is obtained by homotopy perturbation method. The results thus obtained are compared graphically with the exact solutions. A brief sensitivity study is also performed.     

Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

Schur product of matrices and numerical radius (range) preserving maps

Let F(A) be the numerical range or the numerical radius of a square matrix A. Denote by A ° B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F(A ° B) = F(?(A) ° ?(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices.     

Exact and inexact breakdowns in the block GMRES method

This paper addresses the issue of breakdowns in the block GMRES method for solving linear systems with multiple right-hand sides of the form AX = B. An exact (inexact) breakdown occurs at iteration j of this method when the block Krylov matrix (B, AB, … , A^j−1B) is singular (almost singular). Exact breakdowns are the sign that a part of the exact solution is in the range of the Krylov matrix. They are primarily of theoretical interest. From a computational point of view, inexact breakdowns are most likely to occur. In such cases, the underlying block Arnoldi process that is used to build the block Krylov space should not be continued as usual. A natural way to continue the process is the use of deflation. However, as shown by Langou [J. Langou, Iterative Methods for Solving Linear Systems with Multiple Right-Hand Sides, Ph.D. dissertation TH/PA/03/24, CERFACS, France, 2003], deflation in block GMRES may lead to a loss of information that slows down the convergence. In this paper, instead of deflating the directions associated with almost converged solutions, these are kept and reintroduced in next iterations if necessary. Two criteria to detect inexact breakdowns are presented. One is based on the numerical rank of the generated block Krylov basis, the second on the numerical rank of the residual associated to approximate solutions. These criteria are analyzed and compared. Implementation details are discussed. Numerical results are reported.