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This paper proposes a completely perturbed mixed ℓ 2 /ℓ p minimization to deal with a model of completely perturbed block-sparse compressed sensing. Based on the block restricted isometry property (BRIP), the paper extends the study to a complete perturbation model which considers not only noise but also perturbation, establishes a sufficient condition for efficiently recovering the block-sparse signal under the complete perturbation case, and offers eventually a superior approximation precision. The precision, in this paper, can be characterized in terms of the total noise and the best K -term approximation. The adopted mixed ℓ 2 /ℓ p minimization also gains better robustness and stability than ever that on recovering the block-sparse signal with the presence of total noise. Especially, the analysis of this study shows the condition is the best sufficient condition δ 2 K  < 1 [20] when p tends to zero and a  > 1 for the complete perturbation and block-sparse signal. The numerical experiments carried out confirm excellently the assessed performance. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the homotopy analysis method (HAM) proposed by Liao in 1992 and the homotopy perturbation method (HPM) proposed by He in 1998 are compared through an evolution equation used as the second example in a recent paper by Ganji et al. [D.D. Ganji, H. Tari, M.B. Jooybari, Variational iteration method and homotopy perturbation method for nonlinear evolution equations. Comput. Math. Appl. 54 (2007) 1018–1027]. It is found that the HPM is a special case of the HAM when . However, the HPM solution is divergent for all x and t except . It is also found that the solution given by the variational iteration method (VIM) is divergent too. On the other hand, using the HAM, one obtains convergent series solutions which agree well with the exact solution. This example illustrates that it is very important to investigate the convergence of approximation series. Otherwise, one might get useless results. [Copyright &y& Elsevier]

In this paper, the Lie approximate symmetry analysis is applied to investigate new exact solutions of the singularly perturbed Boussinesq equation. The tanh-function method, is employed to solve some of the obtained reduced ordinary differential equations.We construct new analytical solutions with small parameter which is effectively obtained by the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a perturbation method based on invariance factors and multiple scales will be presented for weakly nonlinear, regularly perturbed systems of ordinary difference equations. Asymptotic approximations of first integrals will be constructed on long iteration-scales, that is, on iteration-scales of order , where is a small parameter. It will be shown that all invariance factors have to satisfy a functional equation. To show how this perturbation method works, the method is applied to a Van der Pol equation, and a Rayleigh equation. It will be explicitly shown for the first time in the literature how these multiple scales should be introduced for systems of difference equations to obtain very accurate approximations of first integrals on long iteration-scales. [Copyright &y& Elsevier]

Abstract: In this paper, the homotopy analysis method is applied to deduce analytical approximations of limit cycles and their frequencies in general planar self-excited systems with strong nonlinearity. After changing general planar self-excited systems to the canonical forms by several linear transformations, the auxiliary linear operators and the initial guess of solutions are introduced. Hence, the homotopy analysis solving is set up. Importantly, in solving the higher-order deformation equations, the idea of a perturbation procedure of limit cycles’ approximation proposed in the setting of second-order self-excited equations is embedded. As an application, a Rosenzweig–MacArthur predator–prey model is studied in detail. By choosing the suitable convergence-control parameters, the accurately analytical approximations of the large amplitude limit cycles and their frequency of the model are obtained. The high accuracy of the analytical results are illustrated by comparing with those of numerical integrations. [Copyright &y& Elsevier]

Abstract: On the basis of the geometric singular perturbation theory and the theory of delayed Hopf bifurcation in slow–fast systems with delay, the stability of nonlinear systems with slowly and periodically varying delay is investigated in this paper. Sufficient conditions ensuring asymptotic stability of those systems are obtained. Especially, though a time-varying delay usually increases complexity in the analysis of system dynamics and it usually deteriorates system stability as well, the study indicates that under certain conditions, the stability of the systems with a time-invariant delay only can be improved by incorporating a slowly and periodically varying part into the constant delay. Two illustrative examples are given to validate the analytical results. [Copyright &y& Elsevier]

Abstract: The influence of a time-dependent gravity on the convective instability of reaction fronts in porous media is investigated in this paper. It is assumed that the time-dependent modulation is quasi-periodic with two frequencies σ 1 and σ 2 that are incommensurate with each other. The model consists of the heat equation, the equation for the depth of conversion and the equations of motion under the Darcy law. The convective threshold is approximated performing a linear stability analysis on a reduced singular perturbation problem using the matched asymptotic expansion method. The reduced interface problem is solved using numerical simulations. It is shown that if the reacting fluid is heated from below, a stabilizing effect of a reaction fronts in a porous medium can be gained for appropriate values of amplitudes and frequencies ratio of the quasi-periodic vibration. [Copyright &y& Elsevier]

Abstract: This paper derives an explicit series approximation solution for the optimal exercise boundary of an American put option by means of a new analytical method for strongly nonlinear problems, namely the homotopy analysis method (HAM). The Black–Sholes equation subject to the moving boundary conditions for an American put option is transferred into an infinite number of linear sub-problems in a fixed domain through the deformation equations. Different from perturbation/asymptotic approximations, the HAM approximation can be applicable for options with much longer expiry. Accuracy tests are made in comparison with numerical solutions. It is found that the current approximation is as accurate as many numerical methods. Considering its explicit form of expression, it can bring great convenience to the market practitioners. [Copyright &y& Elsevier]

Abstract: The stochastic perturbation of solitons due to Alfven waves in plasmas, is studied in this paper, in addition to the deterministic perturbation terms. The Langevin equations are derived and it is proved that the soliton travels through the plasma with a fixed mean velocity. [Copyright &y& Elsevier]

Abstract: We study in this paper the active control of a driven class of Van der Pol oscillator which exhibits three limit cycles. We begin by investigating the dynamics and stability analysis of the system under active control. We also analyze the effects of a time periodic perturbation included in the control process. In all these cases the domain of control gain parameters leading to a good control is obtained and verified numerically. [Copyright &y& Elsevier]

Abstract: Consider the equation −ε 2 Δu ε + q(x)u ε = f(u ε ) in , ∣u(∞)∣<∞, ε =const>0. Under what assumptions on q(x) and f(u) can one prove that the solution u ε exists and lim ε→0 u ε = u(x), where u(x) solves the limiting problem q(x)u = f(u)? These are the questions discussed in the paper. [Copyright &y& Elsevier]

Abstract: A methodology for the local singularity of non-smooth dynamical systems is systematically presented in this paper, and a periodically forced, piecewise linear system is investigated as a sample problem to demonstrate the methodology. The sliding dynamics along the separation boundary are investigated through the differential inclusion theory. For this sample problem, a perturbation method is introduced to determine the singularity of the sliding dynamics on the separation boundary. The criteria for grazing bifurcation are presented mathematically and numerically. The grazing flows are illustrated numerically. This methodology can be very easily applied to predict grazing motions in other non-smooth dynamical systems. The fragmentation of the strange attractors of chaotic motion will be presented in the second part of this work. [Copyright &y& Elsevier]

The approximate analytical solution of a classical quartic anharmonic oscillator with periodic forcing is obtained by using a complete analytical approach. The present approach starts with a trial solution of time dependent parameters. The trial solution is guessed from the knowledge of the perturbation theory. Finally, the time dependent parameters in the trial solution are found by using appropriate initial conditions. In the limiting situations, the present approach gives rise to the standard (earlier) results. The solution presented in this paper has a great deal of academic interests. The removal of secular terms and the small divisor terms are carefully discussed. It is found that the small divisor terms involving harmonic and subharmonic terms are removed automatically from the solution. However, the removal of terms involving fundamental frequency requires standard methods. [Copyright &y& Elsevier]