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The integrodifferential equations of mathematical physics are objects of research, and construction of interpolating polynomials to obtain approximate solutions of such equations is the subject of investigation. This paper lays out a technique for constructing approximate expressions for functionals on solutions of integrodifferential equations which are an analog of the Hermite polynomial used to interpolate functions. In the example of the diffusion equation, it is shown that the use of such basis solutions allows a substantial increase in the accuracy of the approximate representation of the functionals in comparison to the first approximation of perturbation theory with practically the same computational costs. [ABSTRACT FROM AUTHOR]

We present molecular applications of a spin free size-extensive state-specific multireference perturbation theory (SS-MRPT), which is valid for model functions of arbitrary spin and generality. In addition to the singlet states, this method is equally capable to handle nonsinglet states. The formulation based on Rayleigh–Schrödinger approach works with a complete active space and treats each of the model space functions democratically. The method is capable of handling varying degrees of quasidegeneracy and of ensuring size consistency as a consequence of size extensivity. In this paper, we illustrate the effectiveness of the Mo\ller–Plesset (MP) partitioning based spin free SS-MRPT [termed as SS-MRPT(MP)] in computations of energetics of the nonsinglet states of several chemically interesting and demanding molecular examples such as LiH, NH2, and CH3. The spectroscopic constants of 3Σ- state of NH and OH+ molecular systems and the ground 1Σg+ as well as excited 3Σu+ states of N2 have been investigated and comparison with experimental and full configuration interaction values (wherever available) has also been provided. We have been able to demonstrate here that the SS-MRPT(MP) method is an intrinsically consistent and promising approach to compute reliable energies of nonsinglet states over different geometries. [ABSTRACT FROM AUTHOR]

This paper investigates perturbation operators for variable neighborhood search (VNS) approaches for two related problems, namely the pickup and delivery traveling salesman problem with LIFO loading (TSPPDL) and FIFO loading (TSPPDF). Our study is motivated by the fact that previously published results on VNS approaches on the TSPPDL suggest that the perturbation operation has the most significant effect on solution quality. We propose a new perturbation operator for the TSPPDL that achieves better results on average than the existing best approach. We also devise new perturbation operators for the TSPPDF that combine request removal and request insertion operations, and investigate which combination of request removal and request insertion operations produces the best results. Our resultant VNS that employs our best perturbation operator outperforms the best existing TSPPDF approach on benchmark test data. [ABSTRACT FROM AUTHOR]

In this paper, the condition number of singular value is discussed. The expressions of the zero-structured and patterned conditioning are also studied. The numerical tests are reported to provide interesting information about the singular value sensitivity when the perturbations of the matrix have arbitrarily assigned zero-structured and patterned case. [ABSTRACT FROM AUTHOR]

This paper tests the hypothesis that optimal perturbations in quasigeostrophic turbulence are excited sufficiently strongly and frequently to account for the energy-containing eddies. Optimal perturbations are defined here as singular vectors of the propagator, for the energy norm, corresponding to the equations of motion linearized about the time-mean flow. The initial conditions are drawn from a numerical solution of the nonlinear equations associated with the linear propagator. Experiments confirm that energy is concentrated in the leading evolved singular vectors, and that the average energy in the initial singular vectors is within an order of magnitude of that required to explain the average energy in the evolved singular vectors. Furthermore, only a small number of evolved singular vectors (4 out of 4000) are needed to explain the dominant eddy structure when total energy exceeds a predefined threshold. The initial singular vectors explain only 10% of such events, but this discrepancy was similar to that of the full propagator, suggesting that it arises primarily due to errors in the propagator. In the limit of short lead times, energy conservation can be expressed in terms of suitable singular vectors to constrain the energy distribution of the singular vectors in statistically steady equilibrium. This and other connections between linear optimals and nonlinear dynamics suggests that the positive results found here should carry over to other systems, provided the propagator and initial states are chosen consistently with respect to the nonlinear system. [ABSTRACT FROM AUTHOR]

This paper is concerned with asymptotic behaviour of solutions of perturbed dynamic systems on time scales. A time scale version of the Hartman-Wintner theorem is established for a class of time scales. [ABSTRACT FROM AUTHOR]

In this paper, we consider a singularly perturbed control system involving differential inclusions in Banach spaces with slow and fast solutions. Using the averaging approach, we obtain sufficient conditions for the Hausdorff convergence of the set of slow solutions in the supremum norm. We present applications of the theorem to prove convergence of the fast solutions in terms of invariant measures and convergence of equi-Lipschitz solutions. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of integrable Hamiltonian systems under small perturbations, proving a weak form of the KAM/Nekhoroshev theory for viscosity solutions of Hamilton Jacobi equations. The main advantage of our approach is that, only a finite number of terms in an asymptotic expansion are needed in order to obtain uniform control. Therefore there are no convergence issues involved. An application of these results is to show thai Diophantine invariant tori and Aubry-Mather sets are stable under small perturbations. [ABSTRACT FROM AUTHOR]

In this paper we consider a singularly perturbed Markov decision process with the limiting average cost criterion. We assume that the underlying process is composed of n separate irreducible processes, and that the small perturbation is such that it ‘unites’ these processes into a single irreducible process. We formulate the underlying control problem for the singularly perturbed MDP, and call it the ‘limit Markov control problem’ (limit MCP). We prove the validity of the ‘the limit control principle’ which states that an optimal solution to the perturbed MDP can be approximated by an optimal solution of the limit MCP for any sufficiently small perturbation. We also demonstrate that the limit Markov control problem is equivalent to a suitably constructed nonlinear program in the space of long-run state-action frequencies. This approach combines the solutions of the original separated irreducible MDPs with the stationary distribution of a certain ‘aggregated MDP’ and creates a framework for future algorithmic approaches. [ABSTRACT FROM AUTHOR]

In this paper, an implicit net with perturbations for solving the mixed equilibrium problems and fixed point problems has been constructed and it is shown that the proposed net converges strongly to a common solution of the mixed equilibrium problems and fixed point problems. Also, as applications, some corollaries for solving the minimum-norm problems are also included. [ABSTRACT FROM AUTHOR]

The property (gR), introduced in [Aiena, P., Guillen, J. and Peña, P., Property (gR) and perturbations, to appear in Acta Sci. Math. (Szeged), 2012], is an extension to the context of B-Fredholm theory, of property (R), introduced in [Aiena, P., Guillen, J. and Peña, P., Property (R) for bounded linear operators, Mediterr. J. Math. 8 (4), 491-508, 2011]. In this paper we continue the study of property (gR) and we consider its preservation under perturbations by finite rank and nilpotent operators. We also prove that if T is left polaroid (resp. right polaroid) and N is a nilpotent operator which commutes with T then T +N is also left polaroid (resp. right polaroid). [ABSTRACT FROM AUTHOR]

After introducing the context and prehistory, the paper shows the threefold rationale for proposing an extension of conventional time, able to be used in modelling living systems: failure of atem- poral modelling in ecology; unsuitability of Newtonian time for transdisciplinary research; requirements of service-oriented soft- ware engineering (SOSE). The approach is boundedly rational: the start vector contains few premises (e.g., the extension must start from and be reducible to conventional time) and flexible criteria (e.g., the extension should be mathematically "conve- nient" and exploit its roots - mainly Euler and Laplace). On this groundwork, Wienerian time is defined as complex-valued extension of physical time, its components are evaluated, and first consequences for SOSE are inferred. After abridging the "Proof-of-Concept" appliance (research toolkit aimed at "What- if" scenarios for exploring homeostasis in benthic communities) the paper focuses on the different temporal dimensions (required by the system and by any perturbation triggering its evolution toward homeostasis) and on the temporal accessibility relations between the two Kripke worlds. Instead of premature conclusions, the paper ends with assumptions inferred from the "Proof-of-Concept" application and suggestions to exploit other roots too. [ABSTRACT FROM AUTHOR]

In this paper, a semilinear elliptic equation with a nonlinear boundary condition and a perturbation in the reaction term is studied. The existence of a positive solution and another non-zero solution to the problem is proved when |λ| is small enough without any specific assumptions on the perturbation term. Moreover, it is shown that the non-zero solution becomes a positive one for small λ>0 under suitable assumptions on the perturbation term. [ABSTRACT FROM AUTHOR]

In this paper, we study the bifurcation of limit cycles of a class of planar quadratic reversible system ẋ = y +4x², ẏ = -x+2xy under quadratic perturbations. It is proved that the cyclicity of the period annulus is equal to two. [ABSTRACT FROM AUTHOR]

In this paper, we study a reversible and non-Hamitonian system with a period annulus bounded by a hemicycle in the Poincaré disk. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. This verifies some results of the conjecture given by Gautier et al. [ABSTRACT FROM AUTHOR]

In this paper a new approach combining the features of the homotopy concept with variational approach is proposed to find accurate analytical solutions for nonlinear oscillators with and without a fractional power restoring force. Since the first-order approximation leads to very accurate results, comparisons with other results are presented to show the effectiveness of this method. The validity of the method is independent of whether or not there exist small or large parameters in the considered nonlinear equations; the obtained results prove the validity and efficiency of the method, which can be easily extended to other strongly nonlinear problems. At the end we compare our procedure with the optimal homotopy perturbation method. [ABSTRACT FROM AUTHOR]

Structural studies are largely performed without taking into account vibrational effects or with incorrectly taking them into account. The paper presents a first-order perturbation theory analysis of the problem. It is shown that vibrational effects introduce errors on the order of 0.02 Å or larger (sometimes, up to 0.1-0.2 Å) into the results of diffraction measurements. Methods for calculating the mean rotational constants, mean-square vibrational amplitudes, vibrational corrections to internuclear distances, and asymmetry parameters are described. Problems related to low-frequency motions, including torsional motions that transform into free rotation at low excitation levels, are discussed. The algorithms described are implemented in the program available from the author (free). [ABSTRACT FROM AUTHOR]

The dynamics of super-sech solitons in dispersion-managed optical fibers is obtained in this paper. The dynamical system of soliton parameters is obtained for such pulses for dispersion-managed fibers, in presence of various perturbation terms. The perturbation terms studied are Hamiltonian, as well as non-Hamiltonian along with non-local types. [ABSTRACT FROM AUTHOR]

Abstract  In this paper we consider positive semigroups on Lp(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L∞(Ω) is real and satisfies b ≥ δ for some δ > 0. [ABSTRACT FROM AUTHOR]

This paper was motivated by the need to establish some new characterizations of the metric regularity of set-valued mappings. Through these new characterizations it was possible to investigate the global/local perturbation stability of the metric regularity and to extend a result by Ioffe [Set-Valued Anal., 9 (2001), pp. 101-109] on the perturbation stability of the global metric regularity when the image space is not necessarily complete. It was also possible to give a characterization of the local metric regularity and to derive a local version of the perturbation stability of the metric regularity. In this work we also describe an application of this perturbation stability and give a simple proof of a result on the error bound of 2-regular mappings established by Izmailov and Solodov [Math. Program., 89 (2001), pp. 413-435] and generalized by He and Sun [Math. Oper. Res., 30 (2005), pp. 701-717]. [ABSTRACT FROM AUTHOR]

In this paper we present a rigorous derivation of the effective model for enhanced diffusion through a narrow and long 2D pore. The analysis uses a singular perturbation technique. The starting point is a local pore scale model describing the transport by convection and diffusion of a reactive solute. The solute particles undergo a first-order reaction at the pore surface. The transport and reaction parameters are such that we have large, dominant Peclet and Damkohler numbers with respect to the ratio of characteristic transversal and longitudinal lengths (the small parameter ε). We give a rigorous mathematical justification of the effective behavior for small ε. Error estimates are presented in the energy norm as well as in L∞ and L¹ norms of the space variable. They guarantee the validity of the upscaled model. As a special case, we recover the well-known Taylor dispersion formula. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the age-structured SEIR epidemic model which is described by partial differential equations. Using theory in function analysis and small perturbation, we obtain the existence and stability of steady states under some conditions. [ABSTRACT FROM AUTHOR]

Due to the lack of regularity of the solutions to the hydrostatic approximation of Navier–Stokes equations, an energy identity cannot be deduced. By including certain nonlinear perturbations to the hydrostatic approximation equations, the solutions to the perturbed problem are smooth enough so that they satisfy the corresponding energy identity. The perturbations considered in this paper are of the monotone class. Three kinds of problems are then studied. To do that, we introduce a functional setting and show in every case that the set of smooth functions with compact support is dense in the space where we search for solutions. When the perturbations are small enough in a certain sense, the solutions of the perturbed problem are close to those of the original one. As a result, this gives a new proof of the existence of solutions to the hydrostatic approximation of Navier–Stokes equations. Finally, this regularization technique has been applied to the analysis of a one-equation hydrostatic turbulence model. [ABSTRACT FROM AUTHOR]

Spiking and bursting observed in nerve membranes seem to be important when we investigate information representation model in the brain. Many topologically different bursting responses are observed in the mathematical models and their related bifurcation mechanisms have been clarified. In this paper, we propose a design method to generate bursting responses in FitzHugh–Nagumo model with a simple periodic external force based on bifurcation analysis. Some effective parameter perturbations for the amplitude of the external input are given from the two-parameter bifurcation diagram. [ABSTRACT FROM AUTHOR]

A simple, straightforward and efficient method is proposed for the calculation of the period of anharmonic oscillators as an energy series. The approach is based on perturbation theory and the method of Lindstedt— Poincaré. [ABSTRACT FROM AUTHOR]

This paper investigates both the convexity and elasticity of the growth rate of size-classsified population models. For an irreducible population projection matrix, we discuss the convexity properties of its Perron eigenvalue under perturbation of the vital rates, extending work of Kirkland and Neumann on Leslie matrices. We also provide nonnegative attainable lower bounds on the derivatives of the elasticity of the Perron eigenvalue under perturbation of the vital rates, sharpening, in the context of population projection matrices, the main result of Kirkland, Neumann, Ormes, and Xu. [ABSTRACT FROM AUTHOR]

In this paper, the following inverse heat conduction problem: ut = uxx, x≥0, t≥0, u(x,0) = 0, x≥0 u(1,t) = g(t), t≥0, u&z.sfnc;x→∞ is bounded, is considered again. This problem is severely ill-posed: its solution (if it exists) does not depend continuously on the data; a small perturbation in the data may cause a dramatically large error in the solution for 0 < x < 1. In this paper, a new wavelet regularization method for this problem is given. Moreover, we can easily find the regularization parameter J such that some sharp stable estimates between the exact solution and the approximate one in Hr(R)-norm meaning is given. [Copyright &y& Elsevier]

Flow of a two component fluid in a capillary is studied analytically and numerically to obtain an understanding of the basic parameters and their effect on the advection-diffusion flow process. One fluid (the solvent) is Newtonian, and the other is a dilute polymer plug (the solution) introduced into the flowing solvent. The polymer plug is modeled as a Maxwell fluid. The ultimate goal is characterization of industrially manufactured polymers in real time, at-line. Experimentally, the evolution of a dilute plug introduced into a fully developed capillary solvent flow is tracked by observing the pressure drop downstream. The volumetric flow rate is held constant. While the pressure plateau has been identified with the intrinsic viscosity of the polymer, in this work, we relate the leading edge behavior of the plug/plateau to the elasticity and diffusivity of the polymer. The combination of these three parameters should unambiguously characterize the polymer chains in solution. In this paper, asymptotic expansions of the equations governing the flow and advection-diffusion are carried out in terms of a dilution parameter ∊(&Le; 1). The resulting initial-boundary value problems are solved numerically in order to characterize the effect of polymer properties on the observed pressure profile. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

We study singular perturbation of impulsive system with a proportional-integral-derivative controller (PID controller) and solve an optimal control problem. The perturbation system comprises two important variables, a fast variable and a slow variable. Because of the complexity of the system, it is difficult to find its exact solution. This paper presents an approximation method for solving it. The aim of the approximation method is to reduce the complexity of the system by eliminating the fast variable. The solution of the method is expressed in an integral form, and it is called an approximated mild solution of the perturbed system. An example is provided to illustrate our result. [ABSTRACT FROM AUTHOR]

A novel computational method, namely the unified perturbation mathematical programming (UPMP) approach, for hybrid uncertainty analysis of engineering structures is proposed in this paper. The presented study considers a mixture of random and interval system parameters which are frequently encountered in engineering applications. Within the UPMP approach, matrix perturbation theory is adopted in combination with the mathematical programming approach. The proposed computational method provides a non-simulative hybrid uncertainty analysis framework, which is competent to offer the extreme bounds of the statistical characteristics (i.e., mean and variance) of any concerned structural responses in computationally tractable fashion. In order to thoroughly explore various intricate aspects of the engineering system involving hybrid uncertainties, systematic numerical experiments have also been conducted. Diverse statistical analyses are implemented to identify the bounded probability profile of the uncertain structural responses. Both academic and practical engineering structures are investigated to justify the applicability, accuracy and efficiency of the proposed UPMP approach. [ABSTRACT FROM AUTHOR]

A second-order oscillator is considered having a random perturbation in its stiffness. This is given by a colored Gaussian or non-Gaussian process. In this way, the oscillator may be stochastically stable or unstable according to the intensity of the excitation. The almost sure (sample) stochastic stability and the stability in the first three response statistical moments are compared for different excitation processes: process with exponential autocorrelation, second-order Gaussian process, bounded noise process. Notable differences in the stability boundaries are found either according to the stability criteria or to the type of excitation. These comparisons are lacking in literature. [ABSTRACT FROM AUTHOR]

A benchmark set of relevant geometries of a model protein, the N-acetylphenylalanylamide, is presented to assess the validity of the approximate second-order coupled cluster (CC2) method in studying low-lying excited states of such bio-relevant systems. The studies comprise investigations of basis-set dependence as well as comparison with two multireference methods, the multistate complete active space 2nd order perturbation theory (MS-CASPT2) and the multireference difference dedicated configuration interaction (DDCI) methods. First of all, the applicability and the accuracy of the quasi-linear multireference difference dedicated configuration interaction method have been demonstrated on bio-relevant systems by comparison with the results obtained by the standard MS-CASPT2. Second, both the nature and excitation energy of the first low-lying excited state obtained at the CC2 level are very close to the Davidson corrected CAS+DDCI ones, the mean absolute deviation on the excitation energy being equal to 0.1 eV with a maximum of less than 0.2 eV. Finally, for the following low-lying excited states, if the nature is always well reproduced at the CC2 level, the differences on excitation energies become more important and can depend on the geometry. [ABSTRACT FROM AUTHOR]

In this paper we show numerically that the main characteristics of a dust devil-like vortex: vertical vorticity generation, eye formation, and tilting of the eye/axis of rotation, can be explained by thermoconvective mechanisms. By considering a cylinder non-homogeneously heated from below we prove that an intense localized heating on the ground generates a convective stationary axisymmetric flow that begins to spiral up around a central axis when perturbation vertical vorticity is permitted and a critical vertical temperature gradient is exceeded, thus forming an axisymmetric vortex. If the intense heating on the ground is not too localized and the temperature gradient continues increasing, central downdrafts appear in the vortex and an eye is formed. We show that the axisymmetric vortex loses stability towards a new state for which the axisymmetry is broken, the axis of rotation or proper eye displaces from the center and tilts. The vortical states found are comparable to dust devils. These findings establish the relevance of thermoconvection on the formation and evolution of these atmospheric phenomena. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we provide a family of bivariate copulas based on a perturbation of a given copula by a factor term. The new class generalizes well-known families of copulas and allows to describe a wide range of possible dependencies. [Copyright &y& Elsevier]

We experimentally investigate the impact of a liquid jet on a soap film. We observe that the jet never breaks the film and that two qualitatively different steady regimes may occur. The first one is a refractionlike behavior obtained at small incidence angles when the jet crosses the film and is deflected by the film-jet interaction. For larger incidence angles, the jet is absorbed by the film, giving rise to a new class of flows in which the jet undulates along the film with a characteristic wavelength. Besides its fundamental interest, this paper presents a different way to guide a micrometric flow of liquid in the inertial regime and to probe foam stability submitted to violent perturbations at the soap film scale. [ABSTRACT FROM AUTHOR]

This paper is devoted to the perturbation analysis for a projected generalized continuous-time Sylvester equation. Perturbation bounds of the solution based on the Euclidean norm are presented. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper consists in the finding of the solution for a stationary transport equation using the techniques of homotopy perturbation method (HPM). The results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method. [Copyright &y& Elsevier]

Abstract: This paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener–Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge–Kutta second-order stochastic scheme as an illustrative example. [Copyright &y& Elsevier]

The article discusses the criterion for the basis property of perturbed exponential systems in Lebesgue function spaces with variable exponent. It obtains the criteria for the basis property on equiconvergence and compact sets with trigonometric series in exponential systems. It points out that the range of systems covered by these criteria is larger than the range of exponential systems. It discusses the hardy classes with variable exponent, its theorem, equations, and solutions. The Reimann problem is the classes is also discussed.

Abstract: In this paper, two methods for solving a nonlinear differential equation known as He’s variational iteration and homotopy perturbation methods are applied to derive the approximate kink-type soliton solutions for a new (2+1)-dimensional simplified generalized Broer–Kaup system. Furthermore, the solutions obtained are compared with the corresponding exact solution to show the applicability, accuracy and, finally, efficiency of the present methods in solving a large class of nonlinear physics and engineering problems without substantial noisy sensitivity to the nonlinear terms or any restrictive assumptions or transformations that may change the physical behavior of the problems. [Copyright &y& Elsevier]

Abstract: This paper deals with initial value problems for the sine–Gordon equation by using the modified homotopy perturbation method. The advantage of this method that is its ability to provide the analytical or approximate solutions to linear and nonlinear equations makes it reliable for solving the sine–Gordon equation. The numerical results are presented to show the efficiency of this method. [Copyright &y& Elsevier]

Abstract: In this paper, nonlinear oscillators under mixed quadratic and cubic nonlinearities with stochastic inputs are considered. Different methods are used to obtain second order approximations, namely; the Wiener–Hermite and perturbation (WHEP) technique and the homotopy perturbation method (HPM). Some statistical moments are computed for the different methods using mathematica 5. Comparisons are illustrated through figures for different case-studies. [Copyright &y& Elsevier]

We apply the well-known Kovacic algorithm to find closed form, i.e., Liouvillian solutions, to the differential equations governing perturbations of black holes. Our analysis includes the full gravitational perturbations of Schwarzschild and Kerr, the full gravitational and electromagnetic perturbations of Reissner-Nordstrom, and specialized perturbations of the Kerr-Newman geometry. We also include the extreme geometries. We find all frequencies ω, in terms of black hole parameters and an integer n, which allow Liouvillian perturbations. We display many classes of black hole parameter values and their corresponding Liouvillian perturbations, including new closed-form perturbations of Kerr and Reissner-Nordstrom. We also prove that the only type 1 Liouvillian perturbations of Schwarzschild are the known algebraically special ones and that type 2 Liouvillian solutions do not exist for extreme geometries. In cases where we do not prove the existence or nonexistence of Liouvillian perturbations we obtain sequences of Diophantine equations on which decidability rests. [ABSTRACT FROM AUTHOR]

Consideration was given to stability of the solution of the system of differential equations to the function of a locally bounded variation added to the right-hand side of the derivative (understood in the general sense). Simple attributes of stability and asymptotic stability were established for different classes of these equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the solution of the generalized Huxley equation is obtained by means of homotopy perturbation method and Adomian decomposition method. The comparison reveals that the former method is more effective than the later. In this method, a homotopy is constructed for the equation.The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Moreover, we will show that He’s homotopy perturbation method overcome the difficulties arising in calculating Adomian polynomials. It is predicted that the HPM can be found wide application in engineering problems. [Copyright &y& Elsevier]