International Journal of Differential Equations http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Oscillation Criteria for Certain Even Order Neutral Delay Differential Equations Wed, 16 Apr 2014 12:34:18 +0000 http://www.hindawi.com/journals/ijde/2014/437278/ We establish sufficient conditions for the oscillation of solutions of even order neutral type differential equations of the form . Ruba Al-Hamouri and Ali Zein Copyright © 2014 Ruba Al-Hamouri and Ali Zein. All rights reserved. The Research of Periodic Solutions of Time-Varying Differential Models Wed, 16 Apr 2014 06:24:39 +0000 http://www.hindawi.com/journals/ijde/2014/430951/ We have studied the periodicity of solutions of some nonlinear time-varying differential models by using the theory of reflecting functions. We have established a new relationship between the linear differential system and the Riccati equations and applied the obtained results to discuss the behavior of periodic solutions of the Riccati equations. Wenjun Liu, Yingxin Pan, and Zhengxin Zhou Copyright © 2014 Wenjun Liu et al. All rights reserved. On Inequality Applicable to Partial Dynamic Equations Tue, 15 Apr 2014 06:59:55 +0000 http://www.hindawi.com/journals/ijde/2014/949860/ The main objective of the paper is to study new integral inequality on time scales which is used for the study of some partial dynamic equations. Some applications of our results are also given. Deepak B. Pachpatte Copyright © 2014 Deepak B. Pachpatte. All rights reserved. Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions Sun, 13 Apr 2014 16:38:35 +0000 http://www.hindawi.com/journals/ijde/2014/724837/ We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under suitable conditions the solution is global in time and the solution with a negative initial energy blows up in finite time. Ülkü Dinlemez and Esra Aktaş Copyright © 2014 Ülkü Dinlemez and Esra Aktaş. All rights reserved. Linearization of a Matrix Riccati Equation Associated to an Optimal Control Problem Sun, 06 Apr 2014 13:07:15 +0000 http://www.hindawi.com/journals/ijde/2014/741390/ The matrix Riccati equation that must be solved to obtain the solution to stochastic optimal control problems known as LQG homing is linearized for a class of processes. The results generalize a theorem proved by Whittle and the one-dimensional case already considered by the authors. A particular two-dimensional problem is solved explicitly. Foued Zitouni and Mario Lefebvre Copyright © 2014 Foued Zitouni and Mario Lefebvre. All rights reserved. On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations Sun, 06 Apr 2014 13:06:41 +0000 http://www.hindawi.com/journals/ijde/2014/495357/ Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods. Tahereh Eftekhari Copyright © 2014 Tahereh Eftekhari. All rights reserved. Periodic Boundary Value Problems for First-Order Impulsive Functional Integrodifferential Equations with Integral-Jump Conditions Sun, 23 Mar 2014 07:32:01 +0000 http://www.hindawi.com/journals/ijde/2014/791240/ By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive functional integrodifferential equations with integral-jump conditions. An example is also given to illustrate our results. Chatthai Thaiprayoon, Decha Samana, and Jessada Tariboon Copyright © 2014 Chatthai Thaiprayoon et al. All rights reserved. Multiscale Splitting Method for the Boltzmann-Poisson Equation: Application to the Dynamics of Electrons Mon, 03 Mar 2014 11:40:10 +0000 http://www.hindawi.com/journals/ijde/2014/178625/ We present a model based on dynamics of electrons in a plasma using a simplified Boltzmann equation coupled with Poisson’s equation. The motivation arose from simulating active plasma resonance spectroscopy, which is used for plasma diagnostic techniques; see Braithwaite and Franklin (2009), Lapke et al. (2010), and Oberrath et al. (2011). Mathematically, we are interested in designing splitting methods for the model problem. While the full Boltzmann equation is delicate to solve, we decouple it into a transport and collision part, which are then solved in different ways. First we reduce it to a simplified transport-collision equation and start to analyse the abstract Cauchy problem using semigroup methods. Second, we pass to the coupled transport and collision model and apply the splitting ideas, resecting the different discretization schemes. The results are discussed first with numerical experiments and then we verify the underlying theoretical novelties. Jürgen Geiser Copyright © 2014 Jürgen Geiser. All rights reserved. Conjugacy of a Discrete Semidynamical System in a Neighbourhood of the Nontrivial Invariant Manifold Tue, 25 Feb 2014 11:14:08 +0000 http://www.hindawi.com/journals/ijde/2014/703868/ The conjugacy of a discrete semidynamical system and its partially decoupled discrete semidynamical system in a Banach space is proved in a neighbourhood of the nontrivial invariant manifold. Andrejs Reinfelds Copyright © 2014 Andrejs Reinfelds. All rights reserved. Existence and Regularity for Boundary Cauchy Problems with Infinite Delay Mon, 20 Jan 2014 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2014/379698/ The aim of this work is to investigate a class of boundary Cauchy problems with infinite delay. We give some sufficient conditions ensuring the uniqueness, existence, and regularity of solutions. For illustration, we apply the result to an age dependent population equation, which covers some special cases considered in some recent papers. Jung-Chan Chang Copyright © 2014 Jung-Chan Chang. All rights reserved. Qualitative Analysis of Solutions of Nonlinear Delay Dynamic Equations Mon, 30 Dec 2013 11:40:20 +0000 http://www.hindawi.com/journals/ijde/2013/764389/ We use the fixed point theory to investigate the qualitative analysis of a nonlinear delay dynamic equation on an arbitrary time scales. We illustrate our results by applying them to various kind of time scales. Mehmet Ünal and Youssef N. Raffoul Copyright © 2013 Mehmet Ünal and Youssef N. Raffoul. All rights reserved. A New Technique of Laplace Variational Iteration Method for Solving Space-Time Fractional Telegraph Equations Sun, 22 Dec 2013 13:17:26 +0000 http://www.hindawi.com/journals/ijde/2013/256593/ In this paper, the exact solutions of space-time fractional telegraph equations are given in terms of Mittage-Leffler functions via a combination of Laplace transform and variational iteration method. New techniques are used to overcome the difficulties arising in identifying the general Lagrange multiplier. As a special case, the obtained solutions reduce to the solutions of standard telegraph equations of the integer orders. Fatima A. Alawad, Eltayeb A. Yousif, and Arbab I. Arbab Copyright © 2013 Fatima A. Alawad et al. All rights reserved. Global Positive Periodic Solutions of Generalized -Species Gilpin-Ayala Delayed Competition Systems with Impulses Tue, 26 Nov 2013 14:38:30 +0000 http://www.hindawi.com/journals/ijde/2013/617824/ We consider the following generalized -species Lotka-Volterra type and Gilpin-Ayala type competition systems with multiple delays and impulses: , , ; , , By applying the Krasnoselskii fixed-point theorem in a cone of Banach space, we derive some verifiable necessary and sufficient conditions for the existence of positive periodic solutions of the previously mentioned. As applications, some special cases of the previous system are examined and some earlier results are extended and improved. Zhenguo Luo, Liping Luo, Jianhua Huang, and Binxiang Dai Copyright © 2013 Zhenguo Luo et al. All rights reserved. Behavior of the -Laplacian on Thin Domains Mon, 18 Nov 2013 17:22:54 +0000 http://www.hindawi.com/journals/ijde/2013/210270/ We give the characterization of the limiting behavior of solutions of elliptic equations driven by the -Laplacian operator with Neumann boundary conditions posed in a family of thin domains. Ricardo P. Silva Copyright © 2013 Ricardo P. Silva. All rights reserved. Embedded Zassenhaus Expansion to Splitting Schemes: Theory and Multiphysics Applications Tue, 24 Sep 2013 14:27:15 +0000 http://www.hindawi.com/journals/ijde/2013/314290/ We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications to multiphysics problems. We treat iterative splitting methods that can be improved by means of the Zassenhaus product formula, which is a sequential splitting scheme. The main idea for reducing the computation time needed by the iterative scheme is to embed fast and cheap Zassenhaus product schemes, since the computation of the commutators involved is very cheap, since we are dealing with nilpotent matrices. We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence. While the iterative splitting schemes converge slowly in their first iterative steps, we improve the initial convergence rates by embedding the Zassenhaus product formula. The applications are to multiphysics problems in fluid dynamics. We consider phase models in computational fluid dynamics and analyse how to obtain higher order operator splitting methods based on the Zassenhaus product. The computational benefits derive from the use of sparse matrices, which arise from the spatial discretisation of the underlying partial differential equations. Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators, we have accelerated the iterative splitting schemes. Jürgen Geiser Copyright © 2013 Jürgen Geiser. All rights reserved. Analysis of a Model Arising from Invasion by Precursor and Differentiated Cells Mon, 23 Sep 2013 08:58:34 +0000 http://www.hindawi.com/journals/ijde/2013/341473/ We study the wave solutions for a degenerated reaction diffusion system arising from the invasion of cells. We show that there exists a family of waves for the wave speed larger than or equal a certain number and below which there are no monotonic wave solutions. We also investigate the monotonicity, uniqueness, and asymptotics of the waves. Xiaojie Hou Copyright © 2013 Xiaojie Hou. All rights reserved. Nonlinear Extension of Multiproduct Expansion Schemes and Applications to Rigid Bodies Wed, 18 Sep 2013 11:17:24 +0000 http://www.hindawi.com/journals/ijde/2013/681575/ In this paper we discuss time integrators for nonlinear differential equations. In recent years, splitting approaches have become an important tool for reducing the computational time needed to solve differential equations. Moreover, nonlinearity is a challenge to splitting schemes, while one has to extend the exp-functions in terms of a nonlinear Magnus expansion. Here we discuss a novel extension of the so-called multiproduct expansion methods, which is used to improve the standard Strang splitting schemes as to their nonlinearity. We present an extension of linear splitting schemes and concentrate on nonlinear systems of differential equations and generalise in this respect the recent MPE method; see (Chin and Geiser, 2011). Some first numerical examples, of rigid body problems, are given as benchmarks. Jürgen Geiser Copyright © 2013 Jürgen Geiser. All rights reserved. Qualitative Analysis of Differential Equations Tue, 17 Sep 2013 14:45:00 +0000 http://www.hindawi.com/journals/ijde/2013/598956/ Ondřej Došlý, Jaroslav Jaroš, Mervan Pašić, and Norio Yoshida Copyright © 2013 Ondřej Došlý et al. All rights reserved. Periodic Solutions for Impulsive Stochastic BAM Neural Networks with Time-Varying Delays in Leakage Terms Tue, 17 Sep 2013 14:03:12 +0000 http://www.hindawi.com/journals/ijde/2013/543947/ By using an integral inequality, we establish some sufficient conditions for the existence and p-exponential stability of periodic solutions for a class of impulsive stochastic BAM neural networks with time-varying delays in leakage terms. Moreover, we present an example to illustrate the feasibility of our results. Li Yang and Yongkun Li Copyright © 2013 Li Yang and Yongkun Li. All rights reserved. Modelling and Simulation of a Packed Bed of Pulp Fibers Using Mixed Collocation Method Sat, 14 Sep 2013 13:46:28 +0000 http://www.hindawi.com/journals/ijde/2013/875298/ A convenient computational approach for solving mathematical model related to diffusion dispersion during flow through packed bed is presented. The algorithm is based on the mixed collocation method. The method is particularly useful for solving stiff system arising in chemical and process engineering. The convergence of the method is found to be of order 2 using the roots of shifted Chebyshev polynomial. Model is verified using the literature data. This method has provided a convenient check on the accuracy of the results for wide range of parameters, namely, Peclet numbers. Breakthrough curves are plotted to check the effect of Peclet number on average and exit solute concentrations. Ishfaq Ahmad Ganaie, Shelly Arora, and V. K. Kukreja Copyright © 2013 Ishfaq Ahmad Ganaie et al. All rights reserved. On ()-Dichotomies for Nonautonomous Linear Difference Equations in Banach Spaces Wed, 11 Sep 2013 15:34:36 +0000 http://www.hindawi.com/journals/ijde/2013/761680/ This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts (uniform, nonuniform, exponential, and polynomial). In the particular case of strongly invariant dichotomy projections, we present characterizations of these sequences and connections with other dichotomy concepts existent in the literature. Some illustrative examples clarify the implications between these concepts. Mihai Gabriel Babuţia, Mihail Megan, and Ioan-Lucian Popa Copyright © 2013 Mihai Gabriel Babuţia et al. All rights reserved. Existence of Positive Solutions for Higher Order -Laplacian Two-Point Boundary Value Problems Mon, 09 Sep 2013 09:33:08 +0000 http://www.hindawi.com/journals/ijde/2013/743943/ We derive sufficient conditions for the existence of positive solutions to higher order -Laplacian two-point boundary value problem, , , , , , , , ; , , , , , and , where are continuous functions from to , and . We establish the existence of at least three positive solutions for the two-point coupled system by utilizing five-functional fixed point theorem. And also, we demonstrate our result with an example. Rajendra Prasad Kapula, Penugurthi Murali, and Kona Rajendrakumar Copyright © 2013 Rajendra Prasad Kapula et al. All rights reserved. An Alternative Method for the Study of Impulsive Differential Equations of Fractional Orders in a Banach Space Mon, 12 Aug 2013 15:08:26 +0000 http://www.hindawi.com/journals/ijde/2013/191060/ This paper is concerned with the existence, uniqueness, and stability of the solution of some impulsive fractional problem in a Banach space subjected to a nonlocal condition. Meanwhile, we give a new concept of a solution to impulsive fractional equations of multiorders. The derived results are based on Banach's contraction theorem as well as Schaefer's fixed point theorem. Asma Bouzaroura and Saïd Mazouzi Copyright © 2013 Asma Bouzaroura and Saïd Mazouzi. All rights reserved. Some Properties of Solutions to Weakly Hypoelliptic Equations Wed, 31 Jul 2013 14:07:27 +0000 http://www.hindawi.com/journals/ijde/2013/526390/ A linear different operator is called weakly hypoelliptic if any local solution of is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and any -solution must vanish. Christian Bär Copyright © 2013 Christian Bär. All rights reserved. Picard Type Iterative Scheme with Initial Iterates in Reverse Order for a Class of Nonlinear Three Point BVPs Wed, 24 Jul 2013 09:58:15 +0000 http://www.hindawi.com/journals/ijde/2013/728149/ We consider the following class of three point boundary value problem , , where , , the source term is Lipschitz and continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well-order and reverse order cases. Under some sufficient conditions, we prove some new existence results. We use examples and figures to demonstrate that monotone iterative method can efficiently be used for computation of solutions of nonlinear BVPs. Mandeep Singh and Amit K. Verma Copyright © 2013 Mandeep Singh and Amit K. Verma. All rights reserved. Existence of Positive Periodic Solutions for Periodic Neutral Lotka-Volterra System with Distributed Delays and Impulses Mon, 15 Jul 2013 07:53:45 +0000 http://www.hindawi.com/journals/ijde/2013/890281/ By using a fixed-point theorem of strict-set-contraction, we investigate the existence of positive periodic solutions for a class of the following impulsive neutral Lotka-Volterra system with distributed delays: Some verifiable criteria are established easily. Zhenguo Luo and Liping Luo Copyright © 2013 Zhenguo Luo and Liping Luo. All rights reserved. On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups Wed, 10 Jul 2013 13:25:12 +0000 http://www.hindawi.com/journals/ijde/2013/268309/ We prove that a discrete semigroup of bounded linear operators acting on a complex Banach space is uniformly exponentially stable if and only if, for each , the sequence belongs to . Similar results for periodic discrete evolution families are also stated. Aftab Khan, Gul Rahmat, and Akbar Zada Copyright © 2013 Aftab Khan et al. All rights reserved. On the Derivation of a Closed-Form Expression for the Solutions of a Subclass of Generalized Abel Differential Equations Mon, 08 Jul 2013 14:59:16 +0000 http://www.hindawi.com/journals/ijde/2013/929286/ We investigate the properties of a general class of differential equations described by with a positive integer and , with , real functions of . For , these equations reduce to the class of Abel differential equations of the first kind, for which a standard solution procedure is available. However, for no general solution methodology exists, to the best of our knowledge, that can lead to their solution. We develop a general solution methodology that for odd values of connects the closed form solution of the differential equations with the existence of closed-form expressions for the roots of the polynomial that appears on the right-hand side of the differential equation. Moreover, the closed-form expression (when it exists) for the polynomial roots enables the expression of the solution of the differential equation in closed form, based on the class of Hyper-Lambert functions. However, for certain even values of , we prove that such closed form does not exist in general, and consequently there is no closed-form expression for the solution of the differential equation through this methodology. Panayotis E. Nastou, Paul Spirakis, Yannis C. Stamatiou, and Apostolos Tsiakalos Copyright © 2013 Panayotis E. Nastou et al. All rights reserved. Dynamics of a Gross-Pitaevskii Equation with Phenomenological Damping 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. Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations 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.