http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2013 , Hindawi Publishing Corporation . All rights reserved. Tue, 18 Jun 2013 18:18:36 +0000 http://www.hindawi.com/journals/ijde/2013/874196/ We study the dynamical behavior of solutions of an n-dimensional nonlinear Schrödinger equation with potential and linear derivative terms under the presence of phenomenological damping. This equation is a general version of the dissipative Gross-Pitaevskii equation including terms with first-order derivatives in the spatial coordinates which allow for rotational contributions. We obtain conditions for the existence of a global attractor and find bounds for its dimension. Renato Colucci, Gerardo R. Chacón, and Andrés Vargas Copyright © 2013 Renato Colucci et al. All rights reserved. Sun, 16 Jun 2013 12:44:09 +0000 http://www.hindawi.com/journals/ijde/2013/740980/ We study a new kind of asymptotic behaviour near for the nonautonomous system of two linear differential equations: , , where the matrix-valued function has a kind of singularity at . It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that as and the length of the solution curve of is finite (resp., infinite) for every . It is characterized in terms of certain asymptotic behaviour of the eigenvalues of near . Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at . Yūki Naito and Mervan Pašić Copyright © 2013 Yūki Naito and Mervan Pašić. All rights reserved. Mon, 10 Jun 2013 10:13:08 +0000 http://www.hindawi.com/journals/ijde/2013/852851/ A. Guezane-Lakoud and R. Khaldi Copyright © 2013 A. Guezane-Lakoud and R. Khaldi. All rights reserved. Wed, 05 Jun 2013 10:17:17 +0000 http://www.hindawi.com/journals/ijde/2013/490673/ We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory. Zhenbin Fan and Gisèle Mophou Copyright © 2013 Zhenbin Fan and Gisèle Mophou. All rights reserved. Thu, 30 May 2013 10:36:47 +0000 http://www.hindawi.com/journals/ijde/2013/502963/ This paper studies a class of periodic n species cooperative Lotka-Volterra systems with continuous time delays and feedback controls. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin, some new sufficient conditions on the existence of positive periodic solutions are established. Tursuneli Niyaz and Ahmadjan Muhammadhaji Copyright © 2013 Tursuneli Niyaz and Ahmadjan Muhammadhaji. All rights reserved. Thu, 16 May 2013 08:22:27 +0000 http://www.hindawi.com/journals/ijde/2013/865464/ The main feature of the boundary layer flow problems of nanofluids or classical fluids is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the series methods when applied to solve such a kind of problems. In order to solve these difficulties, the authors usually resort to either Padé approximants or the commercial numerical codes. However, an intensive work is needed to perform the calculations using Padé technique. Due to the importance of the nanofluids flow as a growing field of research and the difficulties caused by using Padé approximants to solve such problems, a suggestion is proposed in this paper to map the semi-infinite domain into a finite one by the help of a transformation. Accordingly, the differential equations governing the fluid flow are transformed into singular differential equations with classical boundary conditions which can be directly solved by using the differential transformation method. The numerical results obtained by using the proposed technique are compared with the available exact solutions, where excellent accuracy is found. The main advantage of the present technique is the complete avoidance of using Padé approximants to treat the infinity boundary conditions. Abdelhalim Ebaid, Hassan A. El-Arabawy, and Nader Y. Abd Elazem Copyright © 2013 Abdelhalim Ebaid et al. All rights reserved. Sun, 12 May 2013 16:18:01 +0000 http://www.hindawi.com/journals/ijde/2013/532987/ We study the asymptotic behavior at small diffusivity of the solutions, , to a convection-diffusion equation in a rectangular domain . The diffusive equation is supplemented with a Dirichlet boundary condition, which is smooth along the edges and continuous at the corners. To resolve the discrepancy, on , between and the corresponding limit solution, , we propose asymptotic expansions of at any arbitrary, but fixed, order. In order to manage some singular effects near the four corners of , the so-called elliptic and ordinary corner correctors are added in the asymptotic expansions as well as the parabolic and classical boundary layer functions. Then, performing the energy estimates on the difference of and the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces. Gung-Min Gie, Chang-Yeol Jung, and Roger Temam Copyright © 2013 Gung-Min Gie et al. All rights reserved. Sun, 12 May 2013 07:59:18 +0000 http://www.hindawi.com/journals/ijde/2013/297085/ We study the oscillation of all solutions of a general class of forced second-order differential equations, where their second derivative is not necessarily a continuous function and the coefficients of the main equation may be discontinuous. Our main results are not included in the previously published known oscillation criteria of interval type. Many examples and consequences are presented illustrating the main results. Siniša Miličić, Mervan Pašić, and Darko Žubrinić Copyright © 2013 Siniša Miličić et al. All rights reserved. Mon, 15 Apr 2013 13:20:58 +0000 http://www.hindawi.com/journals/ijde/2013/693529/ We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed. Brahim Amaziane and Leonid Pankratov Copyright © 2013 Brahim Amaziane and Leonid Pankratov. All rights reserved. Wed, 03 Apr 2013 15:38:47 +0000 http://www.hindawi.com/journals/ijde/2013/476781/ We prove the existence and uniqueness of entropy solution for nonlinear anisotropic elliptic equations with Neumann homogeneous boundary value condition for -data. We prove first, by using minimization techniques, the existence and uniqueness of weak solution when the data is bounded, and by approximation methods, we prove a result of existence and uniqueness of entropy solution. B. K. Bonzi, S. Ouaro, and F. D. Y. Zongo Copyright © 2013 B. K. Bonzi et al. All rights reserved. Wed, 27 Mar 2013 14:05:19 +0000 http://www.hindawi.com/journals/ijde/2013/435456/ Timoshenko beam equations with external damping and internal damping terms and forcing terms are investigated, and boundary conditions (end conditions) to be considered are hinged ends (pinned ends), hinged-sliding ends, and sliding ends. Unboundedness of solutions of boundary value problems for Timoshenko beam equations is studied, and it is shown that the magnitude of the displacement of the beam grows up to ∞ as under some assumptions on the forcing term. Our approach is to reduce the multidimensional problems to one-dimensional problems for fourth-order ordinary differential inequalities. Kusuo Kobayashi and Norio Yoshida Copyright © 2013 Kusuo Kobayashi and Norio Yoshida. All rights reserved. Wed, 06 Mar 2013 17:22:22 +0000 http://www.hindawi.com/journals/ijde/2013/595819/ We consider the free boundary problem for current-vortex sheets in ideal incompressible magnetohydrodynamics. The problem of current-vortex sheets arises naturally, for instance, in geophysics and astrophysics. We prove the existence of a unique solution to the constant-coefficient linearized problem and an a priori estimate with no loss of derivatives. This is a preliminary result to the study of linearized variable-coefficient current-vortex sheets, a first step to prove the existence of solutions to the nonlinear problem. Davide Catania Copyright © 2013 Davide Catania. All rights reserved. Thu, 28 Feb 2013 10:22:34 +0000 http://www.hindawi.com/journals/ijde/2013/857410/ We derive some simple sufficient conditions on the amplitude , the phase and the instantaneous frequency such that the so-called chirp function is fractal oscillatory near a point , where and is a periodic function on . It means that oscillates near , and its graph is a fractal curve in such that its box-counting dimension equals a prescribed real number and the -dimensional upper and lower Minkowski contents of are strictly positive and finite. It numerically determines the order of concentration of oscillations of near . Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions. Mervan Pašić and Satoshi Tanaka Copyright © 2013 Mervan Pašić and Satoshi Tanaka. All rights reserved. Thu, 14 Feb 2013 07:55:11 +0000 http://www.hindawi.com/journals/ijde/2013/802324/ Fawang Liu, Om P. Agrawal, Shaher Momani, Nikolai N. Leonenko, and Wen Chen Copyright © 2013 Fawang Liu et al. All rights reserved. Wed, 13 Feb 2013 15:17:07 +0000 http://www.hindawi.com/journals/ijde/2013/704547/ We use Sadovskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one. Lakshman Mahto, Syed Abbas, and Angelo Favini Copyright © 2013 Lakshman Mahto et al. All rights reserved. Sun, 30 Dec 2012 18:16:47 +0000 http://www.hindawi.com/journals/ijde/2012/408637/ We consider a delayed SIR epidemic model in which the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. We investigate the qualitative behaviour of the model and find the conditions that guarantee the asymptotic stability of corresponding steady states. We present the conditions in the time lag in which the DDE model is stable. Hopf bifurcation analysis is also addressed. Numerical simulations are provided in order to illustrate the theoretical results and gain further insight into the behaviour of this system. Fathalla A. Rihan and M. Naim Anwar Copyright © 2012 Fathalla A. Rihan and M-. Naim Anwar. All rights reserved. Sun, 23 Dec 2012 07:51:18 +0000 http://www.hindawi.com/journals/ijde/2012/491874/ Ebrahim Momoniat, T. G. Myers, Mapundi Banda, and Jean Charpin Copyright © 2012 Ebrahim Momoniat et al. All rights reserved. Thu, 06 Dec 2012 11:47:24 +0000 http://www.hindawi.com/journals/ijde/2012/472030/ In this study, fractional Rosenau-Hynam equations is considered. We implement relatively new analytical techniques, the variational iteration method and the homotopy perturbation method, for solving this equation. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for fractional Rosenau-Hynam equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity. R. Yulita Molliq and M. S. M. Noorani Copyright © 2012 R. Yulita Molliq and M. S. M. Noorani. All rights reserved. Wed, 28 Nov 2012 09:56:53 +0000 http://www.hindawi.com/journals/ijde/2012/780619/ A difference equation is a relation between the differences of a function at one or more general values of the independent variable. These equations usually describe the evolution of certain phenomena over the course of time. The present paper deals with the existence and uniqueness of solutions of fractional difference equations. J. Jagan Mohan and G. V. S. R. Deekshitulu Copyright © 2012 J. Jagan Mohan and G. V. S. R. Deekshitulu. All rights reserved. Wed, 21 Nov 2012 11:38:03 +0000 http://www.hindawi.com/journals/ijde/2012/845945/ This paper suggests two component homotopy method to solve nonlinear fractional integrodifferential equations, namely, Volterra's population model. Padé approximation was effectively used in this method to capture the essential behavior of solutions for the mathematical model of accumulated effect of toxins on a population living in a closed system. The behavior of the solutions and the effects of different values of fractional-order are indicated graphically. The study outlines significant features of this method as well as sheds some light on advantages of the method over the other. The results show that this method is very efficient, convenient, and can be adapted to fit a larger class of problems. Najeeb Alam Khan, Amir Mahmood, Nadeem Alam Khan, and Asmat Ara Copyright © 2012 Najeeb Alam Khan et al. All rights reserved. Thu, 08 Nov 2012 10:44:30 +0000 http://www.hindawi.com/journals/ijde/2012/530268/ Khaled Boukerrioua Copyright © 2012 Khaled Boukerrioua. All rights reserved. Wed, 07 Nov 2012 16:05:57 +0000 http://www.hindawi.com/journals/ijde/2012/572723/ We have presented a numerical integration method to solve a class of singularly perturbed delay differential equations with small shift. First, we have replaced the second-order singularly perturbed delay differential equation by an asymptotically equivalent first-order delay differential equation. Then, Simpson’s rule and linear interpolation are employed to get the three-term recurrence relation which is solved easily by discrete invariant imbedding algorithm. The method is demonstrated by implementing it on several linear and nonlinear model examples by taking various values for the delay parameter and the perturbation parameter . Gemechis File and Y. N. Reddy Copyright © 2012 Gemechis File and Y. N. Reddy. All rights reserved. Wed, 31 Oct 2012 14:45:57 +0000 http://www.hindawi.com/journals/ijde/2012/975829/ A fractional version of logistic equation is solved using new iterative method proposed by Daftardar-Gejji and Jafari (2006). Convergence of the series solutions obtained is discussed. The solutions obtained are compared with Adomian decomposition method and homotopy perturbation method. Sachin Bhalekar and Varsha Daftardar-Gejji Copyright © 2012 Sachin Bhalekar and Varsha Daftardar-Gejji. All rights reserved. Tue, 16 Oct 2012 15:56:32 +0000 http://www.hindawi.com/journals/ijde/2012/720687/ I study the geometric notion of a differential system describing surfaces of a constant negative curvature and describe a family of pseudospherical surfaces for the nonlinear partial differential equations with constant Gaussian curvature . G. M. Gharib Copyright © 2012 G. M. Gharib. All rights reserved. Thu, 04 Oct 2012 13:47:31 +0000 http://www.hindawi.com/journals/ijde/2012/296591/ We investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions. William C. Troy and Edward P. Krisner Copyright © 2012 William C. Troy and Edward P. Krisner. All rights reserved. Wed, 26 Sep 2012 10:11:29 +0000 http://www.hindawi.com/journals/ijde/2012/521750/ This paper builds upon our recent paper on generalized fractional variational calculus (FVC). Here, we briefly review some of the fractional derivatives (FDs) that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives (GFDs) which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases of the proposed GFDs. We develop several parts of FVC in terms of one parameter GFDs. We point out how many other parts could be developed using the properties of the one-parameter GFDs. Subsequently, we introduce two new two- and three-parameter GFDs. We introduce some of their properties, and discuss how they can be used to develop FVC. In addition, we indicate how these formulations could be used in various fields, and how the generalizations presented here can be further extended. Om Prakash Agrawal Copyright © 2012 Om Prakash Agrawal. All rights reserved. Tue, 25 Sep 2012 11:51:23 +0000 http://www.hindawi.com/journals/ijde/2012/842813/ We develop a generalized monotone method using coupled lower and upper solutions for Caputo fractional differential equations with periodic boundary conditions of order , where . We develop results which provide natural monotone sequences or intertwined monotone sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. However, these monotone iterates are solutions of linear initial value problems which are easier to compute. J. D. Ramírez and A. S. Vatsala Copyright © 2012 J. D. Ramírez and A. S. Vatsala. All rights reserved. Tue, 11 Sep 2012 09:42:35 +0000 http://www.hindawi.com/journals/ijde/2012/412569/ A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn(u,k). If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differential equation of the polar angle, exact solutions of the Poisson Boltzmann and the sinh-Poisson equations are found in terms of the Jacobi elliptic functions. Kim Johannessen Copyright © 2012 Kim Johannessen. All rights reserved. Sun, 09 Sep 2012 14:39:36 +0000 http://www.hindawi.com/journals/ijde/2012/570283/ We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. We obtain sufficient conditions for finite speed of support propagation and for waiting time phenomena by application of a new extension of Stampacchia's lemma for a system of functional equations. Marina Chugunova and Roman M. Taranets Copyright © 2012 Marina Chugunova and Roman M. Taranets. All rights reserved. Sun, 09 Sep 2012 13:04:29 +0000 http://www.hindawi.com/journals/ijde/2012/495202/ A numerical scheme is presented for a class of time fractional differential equations with Dirichlet's and Neumann's boundary conditions. The model solution is discretized in time and space with a spectral expansion of Lagrange interpolation polynomial. Numerical results demonstrate the spectral accuracy and efficiency of the collocation spectral method. The technique not only is easy to implement but also can be easily applied to multidimensional problems. Fenghui Huang Copyright © 2012 Fenghui Huang. All rights reserved.