Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations

The functional equation f(x,ε) = 0 containing a small parameter ε and admitting regular and singular degeneracy as ε → 0 is considered. By the methods of small parameter, a function x _n ⁰(ε) satisfying this equation within a residual error of O(ε ⁿ⁺¹) is found. A modified Newton’s sequence starting from the element x _n ⁰(ε) is constructed. The existence of the limit of Newton’s sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton’s iterative sequence). The deviation of the limit of Newton’s sequence from the initial approximation x _n ⁰(ε) has the order of O(ε ⁿ⁺¹), which proves the asymptotic character of the approximation x _n ⁰(ε). The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.     

Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the Fitzhugh–Nagumo equation. The homotopy analysis method (HAM) is one of the most effective method to obtain the exact solution and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of series solution.     

Dual submanifolds in rational homology spheres

Let Σ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M_+, M_-? Σ are called dual to each other if the complement Σ-M_+ strongly homotopy retracts onto M_- or vice-versa. In this paper, we are concerned with the basic problem of which integral triples(n; m_+, m-) ∈ N~3 can appear, where n = dimΣ-1 and m_± = codim M_±-1. The problem is motivated by several fundamental aspects in differential geometry.(i) The theory of isoparametric/Dupin hypersurfaces in the unit sphere S~(n+1) initiated by′Elie Cartan, where M_± are the focal manifolds of the isoparametric/Dupin hypersurface M ? S~(n+1), and m± coincide with the multiplicities of principal curvatures of M.(ii) The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres Σ,i.e., total spaces of smooth S~3-bundles over S~4 homeomorphic but not diffeomorphic to S~7, where M_± =P_±×_(SO(4))S~3, P → S~4 the principal SO(4)-bundle of Σ and P_± the singular orbits of a cohomogeneity one SO(4) × SO(3)-action on P which are both of codimension 2.Based on the important result of Grove-Halperin, we provide a surprisingly simple answer, namely, if and only if one of the following holds true:· m_+ = m_-= n;· m_+ = m_-=1/3n ∈ {1, 2, 4, 8};· m_+ = m_-=1/4n ∈ {1, 2};· m_+ = m_-=1/6n ∈ {1, 2};·n/(m_++m_-)= 1 or 2, and for the latter case, m_+ + m_-is odd if min(m_+, m_-)≥2.In addition, if Σ is a homotopy sphere and the ratio n/(m_++m_-)= 2(for simplicity let us assume 2 m_- m_+),we observe that the work of Stolz on the multiplicities of isoparametric hypersurfaces applies almost identically to conclude that, the pair can be realized if and only if, either(m_+, m_-) =(5, 4) or m_+ + m_-+ 1 is divisible by the integer δ(m_-)(see the table on Page 1551), which is equivalent to the existence of(m_--1) linearly independent vector fields on the sphere S~(m_++m_-)by Adams' celebrated work. In contrast, infinitely many counterexamples are given if Σ is a rational homology sphere.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

Series by Haar system with monotone coefficients

Let χ = {χ _n } _n=0 ^∞ be the Haar system normalized in L ²(0, 1) and M = {M _s } _s=1 ^∞ be an arbitrary, increasing sequence of nonnegative integers. For any subsystem of χ of the form {φ _k } = χS = {χ _n } _n∈S , where S = S(M) = {n _k } _k=1 ^∞ = {n ∈ V[p]: p ∈ M}, V[0] = {1, 2} and V[p] = {2 ^p + 1, 2 ^p + 2, …, 2 ^p+1} for p = 1, 2, … a series of the form Σ _i=1 ^∞ a _i φ _i with a _i ↘ 0 is constructed, that is universal with respect to partial series in all classes L ^r (0, 1), r ∈ (0, 1), in the sense of a.e. convergence and in the metric ofL ^r (0, 1). The constructed series is universal in the class of all measurable, finite functions on [0, 1] in the sense of a.e. convergence. It is proved that there exists a series by Haar system with decreasing coefficients, which has the following property: for any ? > 0 there exists a measurable function µ(x), x ∈ [0, 1], such that 0 ≤ µ(x) ≤ 1 and |{x ∈ [0, 1], µ(x) ≠ = 1}| < ?, and the series is universal in the weighted space L _µ[0, 1] with respect to subseries, in the sense of convergence in the norm of L _µ[0, 1].     

Characterization of some finite groups by order and length of one conjugacy class

We study the possibility of characterizing S ∈ {²D_n(2), ²D_n+1(2)} by simple conditions when 2ⁿ⁺¹ > 5 is a prime. Furthermore, we will show that Thompson’s conjecture is valid under some weak condition for these groups.     

Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (^h/_2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined by plotting graphs.     

Almost everywhere convergence of sequences of two-dimensional Walsh–Fejér means of integrable functions

The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Walsh–Fourier series of bivariate integrable functions. That is, let \(a = (a_{1}, a_{2})\:\mathbb{N} \to \mathbb{N}^{2}\) such that a _j (n+1)≧δsup _k≦n a _j (n) (j=1,2, n∈?) for some δ>0 and a ₁(+∞)=a ₂(+∞)=+∞. Then for each integrable function f∈L ¹(I ²) we have the a.e. relation \(\lim_{n\to\infty}\sigma_{a_{1}(n), a_{2}(n)}f = f\). It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh–Fejér means of integrable functions which was proved earlier by the author and Weisz [3,8].     

Generalized Absolute Convergence of Series of Fourier Coefficients With Respect to Haar Type Systems

For the orthogonal systems ofHaar type, introduced by Vilenkin in 1958, we study absolute convergence of series composed from positive powers of Fourier coefficients with multiplicators from the Gogoladze–Meskhia class. The conditions for convergence of the series are given in terms of either best approximations of functions in L ^p spaces by polynomials with respect to Haar type systems or fractional modulus of continuity for functions from the Wiener spaces V _p , p > 1. We establish the sharpness of the obtained results.     

On the torus cobordant cohomology spheres

Let G be a compact Lie group. In 1960, P A Smith asked the following question: “Is it true that for any smooth action of G on a homotopy sphere with exactly two fixed points, the tangent G-modules at these two points are isomorphic?” A result due to Atiyah and Bott proves that the answer is ‘yes’ for ? _p and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on S ⁿ which are c-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in ?-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith’s question.     

A ωn-Related Family of Homotopy Elements in the Stable Homotopy of Spheres

To determine the stable homotopy groups of spheres π_*(S) is one of the central problems in homotopy theory. Let p be a prime greater than 5. The authors make use of the May spectral sequence and the Adams spectral sequence to prove the existence of a ?_n-related family of homotopy elements, β₁ω_nγ_s, in the stable homotopy groups of spheres, where n > 3, 3 ≤ s < p ? 2 and the ?_n-element was detected by X. Liu.     

On the limit cycles,period-doubling,and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

In this paper, the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol oscillator and the forced Van der Pol-Duffing oscillator are studied by combining the homotopy analysis method (HAM) with the multi-scale method analytically. Comparisons of the obtained solutions and numerical results show that this method is effective and convenient even when t is large enough, and the convergence of the approximate solutions is discussed by the so-called discrete square residual error. This method is a capable tool for solving this kind of nonlinear problems.     

The absolute convergence of the Fourier-Haar series for two-dimensional functions

It is well-known that if an one-dimensional function is continuously differentiable on [0, 1], then its Fourier-Haar series converges absolutely. On the other hand, if a function of two variables has continuous partial derivatives f _x ^′ and f _y ^′ on T ², then its Fourier series does not necessarily absolutely converge with respect to a multiple Haar system (see [1]). In this paper we state sufficient conditions for the absolute convergence of the Fourier-Haar series for two-dimensional continuously differentiable functions.     

On the homotopy analysis method for backward/forward-backward stochastic differential equations

In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forward-backward stochastic differential equations (FBSDEs), including one with high dimensionality (up to 12 dimensions). By means of the HAM, convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6-dimensional case, within less than 1 % CPU time used by a currently reported numerical method for the same case [34]. Especially, as dimensionality enlarges, the increase of computational complexity for the HAM is not as dramatic as this numerical method. All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science, engineering, and finance.     

Series Solutions of Systems of Nonlinear Fractional Differential Equations

Differential equations of fractional order appear in many applications in physics, chemistry and engineering. An effective and easy-to-use method for solving such equations is needed. In this paper, series solutions of the FDEs are presented using the homotopy analysis method (HAM). The HAM provides a convenient way of controlling the convergence region and rate of the series solution. It is confirmed that the HAM series solutions contain the Adomian decomposition method (ADM) solution as special cases.      

On the divergence of Fourier series in the spaces ϕ(<Emphasis Type="Italic">L</Emphasis>) containing <Emphasis Type="Italic">L</Emphasis>

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[?π, π], but narrower than L^p = L^p[?π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ?(L) of Kolmogorov’s theorem on the convergence of Fourier series in L^p, p ∈ (0, 1), cannot be improved.     

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X_1 and X_2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann(CR) manifolds of dimensions 2m-1 and 2n-1 in C~(m+1)and C~(n+1), respectively. We introduce the ThomSebastiani sum X = X_1 ⊕X_2which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in C~(m+n+2). Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1for all n 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X_1 ⊕ X_2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,we show that if X = X_1 ⊕ X_2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X_1 and X_2 provided that X_2 admits a transversal holomorphic S~1-action.     

Dynamic study of Schröder’s families of first and second kind

The Schröder iterative families of the first and second kind are of great importance in the theory and practice of iterative processes for solving nonlinear equations f(x) = 0. In both cases, the methods E _r (first kind) and S _r (second kind) converge locally to a zero α of f as O(|x _k ? α| ^r ). Although characteristics of these families have been studied in many papers, their dynamic and chaotic behavior has not been completely investigated. In this paper, we compare convergence properties of both iterative schemes using the two methodologies: (i) comparison by numerical examples and (ii) comparison using dynamic study of methods by basins of attraction that enable their graphic visualization. Apart from the visualization of iterative processes, basins of attraction reveal very useful features on iterations such as consumed CPU time and average number of iterations, both as functions of starting points. We demonstrate by several examples that the Schröder family of the second kind S _r possesses better convergence characteristics than the Schröder family of the first kind E _r .     

A new application of the homotopy analysis method: Solving the Sturm–Liouville problems

In this paper, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm–Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. In this paper, it can be observed that the auxiliary parameter ℏ, which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods.     

The space clcv(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>) with the Hausdorff-Bebutov metric and differential inclusions

The paper is devoted to studying the space of nonempty closed convex (but not necessarily compact) sets in ? ⁿ , a dynamical system of translations, and existence theorems for differential inclusions. We make this space complete by equipping it with the Hausdorff-Bebutov metric. The investigation of these issues is important for certain problems of optimal control of asymptotic characteristics of a control system. For example, the problem \(\dot x = A(t,u)x\), (u, x) ∈ ? ^m+n , λ _n (u(·))→ min, where λ _n (u(·)) is the largest Lyapunov exponent of the system {ie121-2} = A(t, u)x, leads to a differential inclusion with a noncompact right-hand side.