Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Wavelet method is a recently developed tool in applied mathematics. Investigation of various wavelet methods, for its capability of analyzing various dynamic phenomena through waves gained more and more attention in engineering research. Starting from ‘offering good solution to differential equations’ to capturing the nonlinearity in the data distribution, wavelets are used as appropriate tools at various places to provide good mathematical model for scientific phenomena, which are usually modeled through linear or nonlinear differential equations. Review shows that the wavelet method is efficient and powerful in solving wide class of linear and nonlinear reaction–diffusion equations. This review intends to provide the great utility of wavelets to science and engineering problems which owes its origin to 1919. Also, future scope and directions involved in developing wavelet algorithm for solving reaction–diffusion equations are addressed.     

The Kostant–García quantization of the bundle of connections

We shall call quantum states of a principal bundle π : P → M with structure group a semi-simple Lie group G, the elements of certain space of sections of the adjoint bundle , associated to the G-bundle of connections . An inner product of sections of is defined for which is a Hilbert space such that the Gauge group gau(P) of the given bundle represents in a family of self-adjoint operators. This work crystallizes some heuristic considerations, on the unitary representations of Gauge algebras, of Garcia in the already a classical article (J. Differ. Geom. 12, 209–227, 1977).     

Interannual variations in the length of day and ENSO events in 1982–1983 and 1997–1998

The length of day series during the period of 1962.0–2000.0, the atmospheric angular momentum and the Southern Oscillation Index are adopted to analyze the relationships among the ENSO events that have occurred since 1960, the changes in the length of day and the atmospheric angular momentum. Attention is particularly given to the different effects of the 1982–1983 and 1997–1998 ENSO events on the variations of Earth rotation. The synthetic excitation effects of multi-scale atmospheric oscillations on the anomalous variations of the interannual rates of Earth rotation are revealed by means of the time-frequency spectrum of the wavelet transform.     

Characterization of Levi-Civita and Newton–Cartan connections in dimension 2

In dimension 2, we give a local characterization of Levi-Civita connections, improving a result by G. Thompson (1991), and of Newton–Cartan connections (connections which are limit of Levi-Civita connections).     

A numerical analysis of the Cahn–Hilliard equation with non-permeable walls

In this article we consider the numerical analysis of the Cahn–Hilliard equation in a bounded domain with non-permeable walls, endowed with dynamic-type boundary conditions. The dynamic-type boundary conditions that we consider here have been recently proposed in Ruiz Goldstein et al. (Phys D 240(8):754–766, 2011) in order to describe the interactions of a binary material with the wall. The equation is semi-discretized using a finite element method for the space variables and error estimates between the exact and the approximate solution are obtained. We also prove the stability of a fully discrete scheme based on the backward Euler scheme for the time discretization. Numerical simulations sustaining the theoretical results are presented.     

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.     

Chaotic analysis of pressure fluctuation signal in the gas–liquid–solid slurry column

The pressure signal of a slurry column is easily obtained by using a pressure sensor, and a chaotic analysis method is used to analyze these signals in order to indicate the flow pattern of the slurry column. The slopes of the correlation integral curve indicate the flow pattern of the slurry column in various operating conditions. The flow pattern is dispersed bubble regime when the superficial velocity is low and the correlation integral curve has two slopes. The flow pattern changes into transition regime with increase in the superficial velocity, the correlation integral curve has only one slope. In the case of the flow pattern becoming a slugging regime, there are several slopes to the correlation integral curve. So it is convenient to find out the flow pattern in the slurry column by solving the slopes of the correlation integral of the pressure signal. The maximum Lyapunov exponent represents the chaos in a slurry column with various solid holdups. The maximum Lyapunov exponent is nearly similar at different heights when the flow patterns are dispersed bubble regime and slugging regime, but the maximum Lyapunov exponent at the axial height is quite different when the flow pattern is transition regime.     

Nonlinear differential systems in the 3–space: A note on periodic solutions by the analysis of two examples

In this note, we address the problem of the existence and location of periodic solutions in nonlinear differential systems in the 3–space. Our main motivation is the study, via bifurcation theory, of periodic solutions (especially limit cycles). We study this problem in two simple polynomial (chaotic) systems: The first one, due to Muthuswamy and Chua, is the mathematical model to the simplest chaotic circuit, and the second is due to Sprott et al.     

A new foundation for the mean–variance analysis

In this paper, I re-examine how the mean–variance analysis is consistent with its traditional theoretical foundations, namely, stochastic dominance and the expected utility theory. Then I propose a simplified version of the coarse utility theory as a new foundation. I prove that, by assuming risk aversion and the normality of asset variables, the simplified model is well behaved; indifference curves are convex and the opportunity set is concave. Therefore, there exist global optimal portfolios in the market. Finally, I prove that decision-making in accordance with the simplified model is consistent with the mean–variance analysis.     

On the spectrum of Euler–Lagrange operator in the stability analysis of Bénard problem

In studying the stability of Bénard problem, we usually have to solve a variational problem to determine the critical Rayleigh number for linear or nonlinear stability. To solve the variational problem, one usually transforms it to an eigenvalue problem which is called Euler–Lagrange equations. An operator related to the Euler–Lagrange equations is usually referred to as Euler–Lagrange operator whose spectrum is investigated in this paper. We have shown that the operator possesses only the point spectrum consisting of real number, which forms a countable set. Moreover, it is found that the spectrum of the Euler–Lagrange operator depends on the thickness of the fluid layer.     

Bifurcation analysis of the Gierer–Meinhardt system with a saturation in the activator production

The reaction–diffusion Gierer–Meinhardt system with a saturation in the activator production is considered. Stability of the unique positive constant steady state solution is analysed, and associated Hopf bifurcations and steady state bifurcations are obtained. A global bifurcation diagram of non-trivial periodic orbits and steady state solutions with respect to key system parameters is obtained, which improves the understanding of dynamics of Gierer–Meinhardt system with a saturation in different parameter regimes.     

Paley–Wiener–Schwartz type theorem for the wavelet transform

In the present paper, we discuss about extension of the wavelet transform on distribution space of compact support and develop the Paley–Wiener–Schwartz type theorem for the wavelet transform on the same. Furthermore, Paley–Wiener–Schwartz type theorem for the wavelet transform is also established using the relation between the wavelet transform and double Fourier transform.     

A note on regularity of the pressure in the Navier–Stokes system

In this note we improve the standard regularity of the dynamic part of the pressure in the Navier–Stokes system. Using the theory of elliptic equations with \(L^1\) right-hand side we prove that, in addition to be in \(L^2\), the dynamic pressure belongs to \(W^{1,\alpha }_{loc} \) with \(1<\alpha <\frac{n}{n-1}\), in case of Dirichlet boundary condition. For pressure boundary condition the dynamic pressure is proved to be in \(W^{1,\alpha } \). As a consequence, for the force \(\mathbf{f} \in L^q (\Omega )^n \) and \(q>n /2 \) the pressure turns out to be continuous.     

A Kantorovich-type convergence analysis of the Newton–Josephy method for solving variational inequalities

We present a Kantorovich-type semilocal convergence analysis of the Newton–Josephy method for solving a certain class of variational inequalities. By using a combination of Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we provide an analysis with the following advantages over the earlier works (Wang 2009, Wang and Shen, Appl Math Mech 25:1291–1297, 2004) (under the same or less computational cost): weaker sufficient convergence conditions, larger convergence domain, finer error bounds on the distances involved, and an at least as precise information on the location of the solution.     

A number of stable matchings in models of the Gale–Shapley type

We give some upper bounds on the maximum number of stable matchings in the Gale–Shapley marriage model with n$n$ men and n$n$ women. We also characterize, with the use of some graph-theoretical notions, the exact number of such matchings, assuming that the preferences of men and women are given.     

Symmetry analysis of the bidimensional Lane–Emden systems

We carry out a complete group classification of the nonlinear Lane–Emden systems in dimension two. The Noether symmetries are found and their corresponding conservation laws are established.     

A characterisation of the Hoffman–Wohlgemuth surfaces in terms of their symmetries

For an embedded singly periodic minimal surface [(M)\tilde]{\tilde{M}} with genus r 3 4{\varrho\ge4} and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman–Wohlgemuth examples. We give a very geometrical proof of this fact, along which they come out many valuable clues for the understanding of these surfaces.