International Journal of Differential Equations https://www.hindawi.com The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. Global Dynamics of a Periodic SEIRS Model with General Incidence Rate Thu, 09 Nov 2017 07:04:27 +0000 http://www.hindawi.com/journals/ijde/2017/5796958/ We consider a family of periodic SEIRS epidemic models with a fairly general incidence rate of the form , and it is shown that the basic reproduction number determines the global dynamics of the models and it is a threshold parameter for persistence of disease. Numerical simulations are performed using a nonlinear incidence rate to estimate the basic reproduction number and illustrate our analytical findings. Eric Ávila-Vales, Erika Rivero-Esquivel, and Gerardo Emilio García-Almeida Copyright © 2017 Eric Ávila-Vales et al. All rights reserved. The Morbidity of Multivariable Grey Model MGM Sun, 05 Nov 2017 07:24:28 +0000 http://www.hindawi.com/journals/ijde/2017/2495686/ This paper proposes the morbidity of the multivariable grey prediction MGM model. Based on the morbidity of the differential equations, properties of matrix, and Gerschgorin Panel Theorem, we analyze the factors that affect the morbidity of the multivariable grey model and give a criterion to justify the morbidity of MGM. Finally, an example is presented to illustrate the practicality of our results. Haixia Wang, Lingdi Zhao, and Mingzhao Hu Copyright © 2017 Haixia Wang et al. All rights reserved. Solving Nonlinear Fourth-Order Boundary Value Problems Using a Numerical Approach: th-Step Block Method Sun, 29 Oct 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/4925914/ Nonlinear boundary value problems (BVPs) are more tedious to solve than their linear counterparts. This is observed in the extra computation required when determining the missing conditions in transforming BVPs to initial value problems. Although a number of numerical approaches are already existent in literature to solve nonlinear BVPs, this article presents a new block method with improved accuracy to solve nonlinear BVPs. A -step block method is developed using a modified Taylor series approach to directly solve fourth-order nonlinear boundary value problems (BVPs) where is the order of the differential equation under consideration. The schemes obtained were combined to simultaneously produce solution to the fourth-order nonlinear BVPs at points iteratively. The derived block method showed improved accuracy in comparison to previously existing authors when solving the same problems. In addition, the suitability of the -step block method was displayed in the solution for magnetohydrodynamic squeezing flow in porous medium. Oluwaseun Adeyeye and Zurni Omar Copyright © 2017 Oluwaseun Adeyeye and Zurni Omar. All rights reserved. Analysis of a Predator-Prey Model with Switching and Stage-Structure for Predator Wed, 27 Sep 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/2653124/ This paper studies the behavior of a predator-prey model with switching and stage-structure for predator. Bounded positive solution, equilibria, and stabilities are determined for the system of delay differential equation. By choosing the delay as a bifurcation parameter, it is shown that the positive equilibrium can be destabilized through a Hopf bifurcation. Some numerical simulations are also given to illustrate our results. T. Suebcharoen Copyright © 2017 T. Suebcharoen. All rights reserved. Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate Sun, 27 Aug 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/8372140/ We propose a fractional order model in this paper to describe the dynamics of human immunodeficiency virus (HIV) infection. In the model, the infection transmission process is modeled by a specific functional response. First, we show that the model is mathematically and biologically well posed. Second, the local and global stabilities of the equilibria are investigated. Finally, some numerical simulations are presented in order to illustrate our theoretical results. Adnane Boukhouima, Khalid Hattaf, and Noura Yousfi Copyright © 2017 Adnane Boukhouima et al. All rights reserved. Cyclic Growth and Global Stability of Economic Dynamics of Kaldor Type in Two Dimensions Sun, 02 Jul 2017 07:36:05 +0000 http://www.hindawi.com/journals/ijde/2017/2062819/ This article proposes nonlinear economic dynamics continuous in two dimensions of Kaldor type, the saving rate and the investment rate, which are functions of ecological origin verifying the nonwasting properties of the resources and economic assumption of Kaldor. The important results of this study contain the notions of bounded solutions, the existence of an attractive set, local and global stability of equilibrium, the system permanence, and the existence of a limit cycle. Aka Fulgence Nindjin, Albin Tetchi N’Guessan, Hypolithe Okou, and Kessé Thiban Tia Copyright © 2017 Aka Fulgence Nindjin et al. All rights reserved. Identifying Initial Condition in Degenerate Parabolic Equation with Singular Potential Wed, 14 Jun 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/1467049/ A hybrid algorithm and regularization method are proposed, for the first time, to solve the one-dimensional degenerate inverse heat conduction problem to estimate the initial temperature distribution from point measurements. The evolution of the heat is given by a degenerate parabolic equation with singular potential. This problem can be formulated in a least-squares framework, an iterative procedure which minimizes the difference between the given measurements and the value at sensor locations of a reconstructed field. The mathematical model leads to a nonconvex minimization problem. To solve it, we prove the existence of at least one solution of problem and we propose two approaches: the first is based on a Tikhonov regularization, while the second approach is based on a hybrid genetic algorithm (married genetic with descent method type gradient). Some numerical experiments are given. K. Atifi, Y. Balouki, El-H. Essoufi, and B. Khouiti Copyright © 2017 K. Atifi et al. All rights reserved. Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations Tue, 13 Jun 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/2097317/ An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations. Abdulnasir Isah, Chang Phang, and Piau Phang Copyright © 2017 Abdulnasir Isah et al. All rights reserved. On a Singular Second-Order Multipoint Boundary Value Problem at Resonance Sun, 11 Jun 2017 09:20:09 +0000 http://www.hindawi.com/journals/ijde/2017/8579065/ The aim of this paper is to derive existence results for a second-order singular multipoint boundary value problem at resonance using coincidence degree arguments. S. A. Iyase and O. F. Imaga Copyright © 2017 S. A. Iyase and O. F. Imaga. All rights reserved. Existence and Uniqueness of Solution of Stochastic Dynamic Systems with Markov Switching and Concentration Points Sun, 30 Apr 2017 06:19:33 +0000 http://www.hindawi.com/journals/ijde/2017/7958398/ In this article the problem of existence and uniqueness of solutions of stochastic differential equations with jumps and concentration points are solved. The theoretical results are illustrated by one example. Taras Lukashiv and Igor Malyk Copyright © 2017 Taras Lukashiv and Igor Malyk. All rights reserved. An Analysis of the Replicator Dynamics for an Asymmetric Hawk-Dove Game Mon, 24 Apr 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/8781570/ We analyze, using a dynamical systems approach, the replicator dynamics for the asymmetric Hawk-Dove game in which there is a set of four pure strategies with arbitrary payoffs. We give a full account of the equilibrium points and their stability and derive the Nash equilibria. We also give a detailed account of the local bifurcations that the system exhibits based on choices of the typical Hawk-Dove parameters and . We also give details on the connections between the results found in this work and those of the standard two-strategy Hawk-Dove game. We conclude the paper with some examples of numerical simulations that further illustrate some global behaviours of the system. Ikjyot Singh Kohli and Michael C. Haslam Copyright © 2017 Ikjyot Singh Kohli and Michael C. Haslam. All rights reserved. Fractional Variational Iteration Method for Solving Fractional Partial Differential Equations with Proportional Delay Mon, 13 Mar 2017 07:50:09 +0000 http://www.hindawi.com/journals/ijde/2017/5206380/ This paper deals with an alternative approximate analytic solution to time fractional partial differential equations (TFPDEs) with proportional delay, obtained by using fractional variational iteration method, where the fractional derivative is taken in Caputo sense. The proposed series solutions are found to converge to exact solution rapidly. To confirm the efficiency and validity of FRDTM, the computation of three test problems of TFPDEs with proportional delay was presented. The scheme seems to be very reliable, effective, and efficient powerful technique for solving various types of physical models arising in science and engineering. Brajesh Kumar Singh and Pramod Kumar Copyright © 2017 Brajesh Kumar Singh and Pramod Kumar. All rights reserved. Critical Oscillation Constant for Euler Type Half-Linear Differential Equation Having Multi-Different Periodic Coefficients Mon, 27 Feb 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/5042421/ We compute explicitly the oscillation constant for Euler type half-linear second-order differential equation having multi-different periodic coefficients. Adil Misir and Banu Mermerkaya Copyright © 2017 Adil Misir and Banu Mermerkaya. All rights reserved. Approximate Controllability of Semilinear Control System Using Tikhonov Regularization Thu, 23 Feb 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/1684637/ For an approximately controllable semilinear system, the problem of computing control for a given target state is converted into an equivalent problem of solving operator equation which is ill-posed. We exhibit a sequence of regularized controls which steers the semilinear control system from an arbitrary initial state to an neighbourhood of the target state at time under the assumption that the nonlinear function is Lipschitz continuous. The convergence of the sequences of regularized controls and the corresponding mild solutions are shown under some assumptions on the system operators. It is also proved that the target state corresponding to the regularized control is close to the actual state to be attained. Ravinder Katta and N. Sukavanam Copyright © 2017 Ravinder Katta and N. Sukavanam. All rights reserved. An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations Wed, 08 Feb 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/7269450/ In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods. Süleyman Cengizci Copyright © 2017 Süleyman Cengizci. All rights reserved. Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation Sun, 29 Jan 2017 00:00:00 +0000 http://www.hindawi.com/journals/ijde/2017/4683581/ In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: ,  , , where , , , and . The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle. Cheng-Min Su, Jian-Ping Sun, and Ya-Hong Zhao Copyright © 2017 Cheng-Min Su et al. All rights reserved. Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case Mon, 23 Jan 2017 12:00:01 +0000 http://www.hindawi.com/journals/ijde/2017/3879017/ We consider the Cauchy problem for the Ostrovsky-Hunter equation , ,  , , where . Define . Suppose that is a pseudodifferential operator with a symbol such that , , and . For example, we can take . We prove the global in time existence and the large time asymptotic behavior of solutions. Fernando Bernal-Vílchis, Nakao Hayashi, and Pavel I. Naumkin Copyright © 2017 Fernando Bernal-Vílchis et al. All rights reserved. A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems Sun, 22 Jan 2017 13:47:16 +0000 http://www.hindawi.com/journals/ijde/2017/9293530/ In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages. F. F. Ngwane and S. N. Jator Copyright © 2017 F. F. Ngwane and S. N. Jator. All rights reserved. A Family of Boundary Value Methods for Systems of Second-Order Boundary Value Problems Sun, 15 Jan 2017 13:29:21 +0000 http://www.hindawi.com/journals/ijde/2017/2464759/ A family of boundary value methods (BVMs) with continuous coefficients is derived and used to obtain methods which are applied via the block unification approach. The methods obtained from these continuous BVMs are weighted the same and are used to simultaneously generate approximations to the exact solution of systems of second-order boundary value problems (BVPs) on the entire interval of integration. The convergence of the methods is analyzed. Numerical experiments were performed to show efficiency and accuracy advantages. T. A. Biala and S. N. Jator Copyright © 2017 T. A. Biala and S. N. Jator. All rights reserved. Modelling the Potential Role of Media Campaigns in Ebola Transmission Dynamics Thu, 12 Jan 2017 14:34:31 +0000 http://www.hindawi.com/journals/ijde/2017/3758269/ A six-compartment mathematical model is formulated to investigate the role of media campaigns in Ebola transmission dynamics. The model includes tweets or messages sent by individuals in different compartments. The media campaigns reproduction number is computed and used to discuss the stability of the disease states. The presence of a backward bifurcation as well as a forward bifurcation is shown together with the existence and local stability of the endemic equilibrium. Results show that messages sent through media have a more significant beneficial effect on the reduction of Ebola cases if they are more effective and spaced out. Sylvie Diane Djiomba Njankou and Farai Nyabadza Copyright © 2017 Sylvie Diane Djiomba Njankou and Farai Nyabadza. All rights reserved. Numerical Solution of Piecewise Constant Delay Systems Based on a Hybrid Framework Thu, 29 Dec 2016 14:07:02 +0000 http://www.hindawi.com/journals/ijde/2016/9754906/ An efficient numerical scheme for solving delay differential equations with a piecewise constant delay function is developed in this paper. The proposed approach is based on a hybrid of block-pulse functions and Taylor’s polynomials. The operational matrix of delay corresponding to the proposed hybrid functions is introduced. The sparsity of this matrix significantly reduces the computation time and memory requirement. The operational matrices of integration, delay, and product are employed to transform the problem under consideration into a system of algebraic equations. It is shown that the developed approach is also applicable to a special class of nonlinear piecewise constant delay differential equations. Several numerical experiments are examined to verify the validity and applicability of the presented technique. H. R. Marzban and S. Hajiabdolrahmani Copyright © 2016 H. R. Marzban and S. Hajiabdolrahmani. All rights reserved. Some Comparison of Solutions by Different Numerical Techniques on Mathematical Biology Problem Wed, 07 Dec 2016 11:33:01 +0000 http://www.hindawi.com/journals/ijde/2016/8921710/ We try to compare the solutions by some numerical techniques when we apply the methods on some mathematical biology problems. The Runge-Kutta-Fehlberg (RKF) method is a promising method to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for insect population, one-species Lotka-Volterra model. The technique is described and illustrated by numerical examples. We modify the population models by taking the Holling type III functional response and intraspecific competition term and hence we solve it by this numerical technique and show that RKF method gives good results. We try to compare this method with the Laplace Adomian Decomposition Method (LADM) and with the exact solutions. Susmita Paul, Sankar Prasad Mondal, Paritosh Bhattacharya, and Kripasindhu Chaudhuri Copyright © 2016 Susmita Paul et al. All rights reserved. Existence of Solutions for Fractional Impulsive Integrodifferential Equations in Banach Spaces Tue, 06 Dec 2016 08:48:47 +0000 http://www.hindawi.com/journals/ijde/2016/5648798/ We investigate the existence of solutions for a class of impulsive fractional evolution equations with nonlocal conditions in Banach space by using some fixed point theorems combined with the technique of measure of noncompactness. Our results improve and generalize some known results corresponding to those obtained by others. Finally, two applications are given to illustrate that our results are valuable. Haide Gou and Baolin Li Copyright © 2016 Haide Gou and Baolin Li. All rights reserved. The Dynamical Analysis of a Prey-Predator Model with a Refuge-Stage Structure Prey Population Thu, 24 Nov 2016 07:48:20 +0000 http://www.hindawi.com/journals/ijde/2016/2010464/ We proposed and analyzed a mathematical model dealing with two species of prey-predator system. It is assumed that the prey is a stage structure population consisting of two compartments known as immature prey and mature prey. It has a refuge capability as a defensive property against the predation. The existence, uniqueness, and boundedness of the solution of the proposed model are discussed. All the feasible equilibrium points are determined. The local and global stability analysis of them are investigated. The occurrence of local bifurcation (such as saddle node, transcritical, and pitchfork) near each of the equilibrium points is studied. Finally, numerical simulations are given to support the analytic results. Raid Kamel Naji and Salam Jasim Majeed Copyright © 2016 Raid Kamel Naji and Salam Jasim Majeed. All rights reserved. About a Problem for Loaded Parabolic-Hyperbolic Type Equation with Fractional Derivatives Wed, 23 Nov 2016 06:07:26 +0000 http://www.hindawi.com/journals/ijde/2016/9815796/ An existence and uniqueness of solution of local boundary value problem with discontinuous matching condition for the loaded parabolic-hyperbolic equation involving the Caputo fractional derivative and Riemann-Liouville integrals have been investigated. The uniqueness of solution is proved by the method of integral energy and the existence is proved by the method of integral equations. Let us note that, from this problem, the same problem follows with continuous gluing conditions (at ); thus an existence theorem and uniqueness theorem will be correct and on this case. Kishin B. Sadarangani and Obidjon Kh. Abdullaev Copyright © 2016 Kishin B. Sadarangani and Obidjon Kh. Abdullaev. All rights reserved. Piecewise Approximate Analytical Solutions of High-Order Singular Perturbation Problems with a Discontinuous Source Term Tue, 15 Nov 2016 06:04:00 +0000 http://www.hindawi.com/journals/ijde/2016/1015634/ A reliable algorithm is presented to develop piecewise approximate analytical solutions of third- and fourth-order convection diffusion singular perturbation problems with a discontinuous source term. The algorithm is based on an asymptotic expansion approximation and Differential Transform Method (DTM). First, the original problem is transformed into a weakly coupled system of ODEs and a zero-order asymptotic expansion of the solution is constructed. Then a piecewise smooth solution of the terminal value reduced system is obtained by using DTM and imposing the continuity and smoothness conditions. The error estimate of the method is presented. The results show that the method is a reliable and convenient asymptotic semianalytical numerical method for treating high-order singular perturbation problems with a discontinuous source term. Essam R. El-Zahar Copyright © 2016 Essam R. El-Zahar. All rights reserved. Positive Solutions to Periodic Boundary Value Problems of Nonlinear Fractional Differential Equations at Resonance Wed, 09 Nov 2016 09:26:56 +0000 http://www.hindawi.com/journals/ijde/2016/9260726/ By Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima, we discuss the existence of positive solutions to fractional order with periodic boundary conditions at resonance. At last, an example is presented to demonstrate the main results. Lei Hu Copyright © 2016 Lei Hu. All rights reserved. Multiplicity Results for the -Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition Mon, 07 Nov 2016 14:15:31 +0000 http://www.hindawi.com/journals/ijde/2016/3149482/ We investigate the singular Neumann problem involving the -Laplace operator:   , in  , where is a bounded domain with boundary, is a positive parameter, and , and are assumed to satisfy assumptions (H0)–(H5) in the Introduction. Using some variational techniques, we show the existence of a number such that problem has two solutions for one solution for , and no solutions for . K. Saoudi, M. Kratou, and S. Alsadhan Copyright © 2016 K. Saoudi et al. All rights reserved. Dirichlet Boundary Value Problem for the Second Order Asymptotically Linear System Thu, 03 Nov 2016 12:45:41 +0000 http://www.hindawi.com/journals/ijde/2016/5676217/ We consider the second order system with the Dirichlet boundary conditions , where the vector field is asymptotically linear and . We provide the existence and multiplicity results using the vector field rotation theory. A. Gritsans, F. Sadyrbaev, and I. Yermachenko Copyright © 2016 A. Gritsans et al. All rights reserved. Error Analysis of an Implicit Spectral Scheme Applied to the Schrödinger-Benjamin-Ono System Thu, 03 Nov 2016 06:28:50 +0000 http://www.hindawi.com/journals/ijde/2016/6930758/ We develop error estimates of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a coupled nonlinear Schrödinger-Benjamin-Ono system that describes the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. The accuracy of the numerical solver is checked using some exact travelling wave solutions of the system. Juan Carlos Muñoz Grajales Copyright © 2016 Juan Carlos Muñoz Grajales. All rights reserved.