Showing papers in "Nonlinear Analysis-real World Applications in 2009"

Variational approach to impulsive differential equations

Abstract: Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work we present a new approach via variational methods and critical point theory to obtain the existence of solutions to impulsive problems. We consider a linear Dirichlet problem and the solutions are found as critical points of a functional. We also study the nonlinear Dirichlet impulsive problem.

Analysis of viscous flow due to a stretching sheet with surface slip and suction

Abstract: The viscous flow due to a stretching sheet with slip and suction is studied. The Navier–Stokes equations admit exact similarity solutions. For two-dimensional stretching a closed-form solution is found and uniqueness is proved. For axisymmetric stretching both existence and uniqueness are shown. The boundary value problem is then integrated numerically.

Particle swarm optimization approach to portfolio optimization

Abstract: The survey of the relevant literature showed that there have been many studies for portfolio optimization problem and that the number of studies which have investigated the optimum portfolio using heuristic techniques is quite high. But almost none of these studies deals with particle swarm optimization (PSO) approach. This study presents a heuristic approach to portfolio optimization problem using PSO technique. The test data set is the weekly prices from March 1992 to September 1997 from the following indices: Hang Seng in Hong Kong, DAX 100 in Germany, FTSE 100 in UK, S&P 100 in USA and Nikkei in Japan. This study uses the cardinality constrained mean-variance model. Thus, the portfolio optimization model is a mixed quadratic and integer programming problem for which efficient algorithms do not exist. The results of this study are compared with those of the genetic algorithms, simulated annealing and tabu search approaches. The purpose of this paper is to apply PSO technique to the portfolio optimization problem. The results show that particle swarm optimization approach is successful in portfolio optimization.

Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature

Abstract: The unsteady laminar boundary layer flow over a continuously stretching permeable surface is investigated. The unsteadiness in the flow and temperature fields is caused by the time-dependence of the stretching velocity and the surface temperature. Effects of the unsteadiness parameter, suction/injection parameter and Prandtl number on the heat transfer characteristics are thoroughly examined.

Strong solution to the compressible magnetohydrodynamic equations with vacuum

Abstract: We consider the compressible magnetohydrodynamic (MHD) equations with nonnegative thermal conductivity or infinite electric conductivity. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set.

Chaotic dynamics of a discrete prey-predator model with Holling type II

Abstract: A discrete-time prey–predator model with Holling type II is investigated. For this model, the existence and stability of three fixed points are analyzed. The bifurcation diagrams, phase portraits and Lyapunov exponents are obtained for different parameters of the model. The fractal dimension of a strange attractor of the model was also calculated. Numerical simulations show that the discrete model exhibits rich dynamics compared with the continuous model, which means that the present model is a chaotic, and complex one.

Variational iteration method for fractional heat- and wave-like equations

Abstract: This paper applies the variational iteration method to obtaining analytical solutions of fractional heat- and wave-like equations with variable coefficients. Comparison with the Adomian decomposition method shows that the VIM is a powerful method for the solution of linear and nonlinear fractional differential equations.

MHD flow of a micropolar fluid near a stagnation-point towards a non-linear stretching surface

Abstract: The two-dimensional magnetohydrodynamic (MHD) stagnation-point flow of an incompressible micropolar fluid over a non-linear stretching surface is studied. The resulting non-linear system of equations is solved analytically using homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly discussed and given in the form of recurrence formulas. The influence of various pertinent parameters on the velocity, microrotation and skin-friction are shown in the tables and graphs. Comparison is also made with the corresponding numerical results of viscous ( K = 0 ) [R. Cortell, Viscous flow and heat transfer over a nonlinearly stretching sheet, Appl. Math. Comput. 184 (2007) 864–873] and hydrodynamic micropolar fluid ( M = 0 ) [R. Nazar, N. Amin, D. Filip, I. Pop, Stagnation point flow of a micropolar fluid towards a stretching sheet, Internat. J. Non-Linear Mech. 39 (2004) 1227–1235] for linear and non-linear stretching sheet. An excellent agreement is found.

The homotopy analysis method to solve the Burgers–Huxley equation

Abstract: In this paper, an analytical technique, namely the homotopy analysis method (HAM) is applied to obtain an approximate analytical solution of the Burgers–Huxley equation. This paper introduces the two theorems which provide us with a simple and convenient way to apply the HAM to the nonlinear PDEs with the power-law nonlinearity. The homotopy analysis method contains the auxiliary parameter ħ , which provides us with a simple way to adjust and control the convergence region of solution series.

Global stability of a SIR epidemic model with nonlinear incidence rate and time delay

Abstract: In this paper, a SIR epidemic model with nonlinear incidence rate and time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease free equilibrium is discussed. It is proved that if the basic reproductive number R 0 > 1 , the system is permanent. By comparison arguments, it is shown that if R 0 1 , the disease free equilibrium is globally asymptotically stable. If R 0 > 1 , by means of an iteration technique and Lyapunov functional technique, respectively, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium.

Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

Abstract: We study the well-posedness of the bidomain model that is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model as a system of coupled parabolic and elliptic PDEs for two potentials and ODEs representing the ionic activity. We first reformulate the parabolic and elliptic PDEs into a single parabolic PDE by the introduction of a bidomain operator. We properly define and analyze this operator, basically a non-differential and non-local operator. We then present a proof of existence, uniqueness and regularity of a local solution in time through a semigroup approach, but that applies to fairly general ionic models. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution in time is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh–Nagumo, Aliev–Panfilov and MacCulloch models.

Convergence of the homotopy perturbation method for partial differential equations

Abstract: In this paper, we introduce a homotopy perturbation method to obtain exact solutions to some linear and nonlinear partial differential equations. This method is a powerful device for solving a wide variety of problems. Using the homotopy perturbation method, it is possible to find the exact solution or an approximate solution of the problem. Convergence of the method is proved. Some examples such as Burgers’, Schrodinger and fourth order parabolic partial differential equations are presented, to verify convergence hypothesis, and illustrating the efficiency and simplicity of the method.

Application of He’s homotopy perturbation method for non-linear system of second-order boundary value problems

Abstract: A homotopy perturbation method (HPM) is proposed to solve non-linear systems of second-order boundary value problems. HPM yields solutions in convergent series forms with easily computable terms, and in some cases, yields exact solutions in one iteration. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The results reveal that the method is very effective, straightforward and simple.

On a Leslie―Gower predator―prey model incorporating a prey refuge

Abstract: We propose a Leslie–Gower predator–prey model incorporating a prey refuge. By constructing a suitable Lyapunov function, we show that the unique positive equilibrium of the system is globally stable, which means that for this ecosystem, prey refuge has no influence on the persistent property of the system. Mathematic analysis shows that increasing the amount of refuge can increase prey densities. As far as the predator species is concerned, when the assumption a 1 r 2 ≤ a 2 b 1 holds, increasing the amount of prey refuge can decrease the predator densities; when the assumption a 1 r 2 > a 2 b 1 holds, there exists a threshold m ∗ , such that for the prey refuge smaller than this threshold, increasing the amount of prey refuge can increase the predator densities and if the prey refuge is larger than the threshold, increasing the amount of prey refuge can decrease the predator densities.

Solutions of Emden-Fowler equations by homotopy-perturbation method

Abstract: In this paper, approximate and/or exact analytical solutions of the generalized Emden–Fowler type equations in the second-order ordinary differential equations (ODEs) are obtained by homotopy-perturbation method (HPM). The homotopy-perturbation method (HPM) is a coupling of the perturbation method and the homotopy method. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. In this work, HPM yields solutions in convergent series forms with easily computable terms, and in some cases, only one iteration leads to the high accuracy of the solutions. Comparisons with the exact solutions and the solutions obtained by the Adomian decomposition method (ADM) show the efficiency of HPM in solving equations with singularity.

The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet

Abstract: In this work, the magnetohydrodynamic (MHD) boundary layer flow is investigated by employing the modified Adomian decomposition method and the Pade approximation. The series solution of the governing non-linear problem is developed. Comparison of the present solution is made with the existing solution and excellent agreement is noted.

Multistability of competitive neural networks with time-varying and distributed delays

Abstract: In this paper, with two classes of general activation functions, we investigate the multistability of competitive neural networks with time-varying and distributed delays. By formulating parameter conditions and using inequality technique, several novel delay-independent sufficient conditions ensuring the existence of 3 N equilibria and exponential stability of 2 N equilibria are derived. In addition, estimations of positively invariant sets and basins of attraction for these stable equilibria are obtained. Two examples are given to show the effectiveness of our theory.

Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus

Abstract: This paper deals with the unsteady axial Couette flow of fractional second grade fluid (FSGF) and fractional Maxwell fluid (FMF) between two infinitely long concentric circular cylinders. With the help of integral transforms (Laplace transform and Weber transform), generalized Mittag–Leffler function and H-Fox function, we get the analytical solutions of the models. Then we discuss the exact solutions and find some results which have been known as special cases of our solutions. Finally, we analyze the effects of the fractional derivative on the models by using the numerical results and find that the oscillation exists in the velocity field of FMF.

Dynamical analysis of toxin producing Phytoplankton–Zooplankton interactions

Abstract: In the present paper we consider a toxin producing phytoplankton–zooplankton model in which the toxin liberation by phytoplankton species follows a discrete time variation Firstly we consider the elementary dynamical properties of the toxic-phytoplankton–zooplankton interacting model system in absence of time delay Then we establish the existence of local Hopf-bifurcation as the time delay crosses a threshold value and also prove the existence of stability switching phenomena Explicit results are derived for stability and direction of the bifurcating periodic orbit by using normal form theory and center manifold arguments Global existence of periodic orbits is also established by using a global Hopf-bifurcation theorem Finally, the basic outcomes are mentioned along with numerical results to provide some support to the analytical findings

Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect

Abstract: In this work, a bidimensional continuous-time differential equations system is analyzed which is derived from Leslie type predator–prey schemes by considering a nonmonotonic functional response and Allee effect on population prey. For ecological reason, we describe the bifurcation diagram of limit cycles that appear only at the first quadrant in the system obtained. We also show that under certain conditions over the parameters, the system allows the existence of a stable limit cycle surrounding an unstable limit cycle generated by Hopf bifurcation. Furthermore, we give conditions over the parameters such that the model allows long-term extinction or survival of both populations.

Reaction–diffusion systems for the macroscopic bidomain model of the cardiac electric field

Abstract: The paper deals with a mathematical model for the electric activity of the heart at the macroscopic level. The membrane model used to describe the ionic currents is a generalization of the phase-I Luo–Rudy, a model widely used in 2-D and 3-D simulations of the propagation of the action potential. From the mathematical viewpoint the model is made up of a degenerate parabolic reaction diffusion system coupled with an ODE system. We derive existence, uniqueness and some regularity results.

Finite-time synchronization of uncertain unified chaotic systems based on CLF☆

Abstract: This paper studies the problem of finite-time synchronization for the unified chaotic systems. We prove that global finite-time synchronization can be achieved for unified chaotic systems which have uncertain parameters. Simulation results for Lorenz, Lu and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.

A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients

Abstract: We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution . In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus Leray [J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math. 63 (1934) 193–248] and Caffarelli, Kohn and Nirenberg [L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771–831] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.

Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method

Abstract: The similarity transform for the steady three-dimensional problem of a condensation film on an inclined rotating disk gives a system of nonlinear ordinary differential equations which are analytically solved by applying a newly developed method namely the homotopy analysis method (HAM). The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of the Prandtl number on the heat transfer and the Nusselt number is discussed in detail. The validity of our results is verified by numerical results.

Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment

Abstract: An SIS epidemic model with treatment is proposed. The incidence rate of the model, which can include the bilinear incidence rate and the standard incidence rate, is a general nonlinear incidence rate. The global dynamics of the model are studied and then we can understand the effect of the capacity for treatment. It is found that a backward bifurcation occurs and there exist bistable endemic equilibria if the capacity is low. Mathematical results suggest that decreasing the basic reproduction number is insufficient for disease eradication and improving the efficiency and capacity of treatment is important for this end. c 2007 Elsevier Ltd. All rights reserved.

Solving systems of nonlinear equations with continuous GRASP

Abstract: A method for finding all roots of a system of nonlinear equations is described. Our method makes use of C-GRASP, a recently proposed continuous global optimization heuristic. Given a nonlinear system, we solve a corresponding adaptively modified global optimization problem multiple times, each time using C-GRASP, with areas of repulsion around roots that have already been found. The heuristic makes no use of derivative information. We illustrate the approach using systems found in the literature.

On accelerated flows of a viscoelastic fluid with the fractional Burgers’ model

Abstract: In this work, we consider the accelerated flows for a viscoelastic fluid governed by the fractional Burgers’ model The velocity field of the flow is described by a fractional partial differential equation By using the Fourier sine transform and the fractional Laplace transform, the exact solutions for the velocity distribution are obtained for the following two problems: (i) flow induced by constantly accelerating plate, and (ii) flow induced by variable accelerated plate These solutions, presented under integral and series forms in terms of the generalized Mittag–Leffler function, are presented as the sum of two terms The first terms represent the velocity field corresponding to a Newtonian fluid performing the same motion, and the second terms give the non-Newtonian contributions to the general solutions The similar solutions for second grade, Maxwell and Oldroyd-B fluids with fractional derivatives as well as those for the ordinary models, are obtained as the limiting cases of our solutions Moreover, in the special case when α = β = 1 , as it was to be expected, our solutions tend to the similar solutions for an ordinary Burgers’ fluid

Unsteady helical flows of a generalized oldroyd-b fluid with fractional derivative

Abstract: This paper deals with the unsteady helical flows of a generalized Oldroyd-B fluid between two infinite coaxial cylinders and within an infinite cylinder. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions are obtained with the help of integral transforms (Laplace transform, Weber transform and finite Hankel transform). The corresponding solutions for generalized second grade and Maxwell fluids as well as those for the Newtonian and ordinary Oldroyd-B fluids are also given in limiting cases. Finally, the influence of model parameters on the velocity field is also analyzed by graphical illustrations.

Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities

Abstract: This work was supported by the “Ministerio de Educacion y Ciencia”, Spain, under project FIS2005-05881-C02-02 and the “Generalitat Valenciana”, Spain, under project ACOMP/ 2007/020.

Global exponential stability of BAM neural networks with distributed delays and impulses

Abstract: Convergence dynamics of bi-directional associative memory (BAM) neural networks with continuously distributed delays and impulses are discussed. Without assuming the differentiability and the monotonicity of the activation functions and symmetry of synaptic interconnection weights, sufficient conditions to guarantee the existence and global exponential stability of a unique equilibrium are given.