International Journal of Differential Equations https://www.hindawi.com The latest articles from Hindawi © 2017 , Hindawi Limited . 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. A Numerical Computation of a System of Linear and Nonlinear Time Dependent Partial Differential Equations Using Reduced Differential Transform Method Thu, 27 Oct 2016 12:49:12 +0000 http://www.hindawi.com/journals/ijde/2016/4275389/ This paper deals with an analytical solution of an initial value system of time dependent linear and nonlinear partial differential equations by implementing reduced differential transform (RDT) method. The effectiveness and the convergence of RDT method are tested by means of five test problems, which indicates the validity and great potential of the reduced differential transform method for solving system of partial differential equations. Brajesh Kumar Singh and Mahendra Copyright © 2016 Brajesh Kumar Singh and Mahendra. All rights reserved. Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations on Thu, 13 Oct 2016 12:44:41 +0000 http://www.hindawi.com/journals/ijde/2016/9503073/ The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameter on is proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity and is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions. Yu Duan and Chun-Lei Tang Copyright © 2016 Yu Duan and Chun-Lei Tang. All rights reserved. On Accuracy and Stability Analysis of the Reproducing Kernel Space Method for the Forced Duffing Equation Tue, 11 Oct 2016 07:32:49 +0000 http://www.hindawi.com/journals/ijde/2016/3520815/ It is attempted to provide the stability and convergence analysis of the reproducing kernel space method for solving the Duffing equation with with boundary integral conditions. We will prove that the reproducing space method is stable. Moreover, after introducing the method, it is shown that it has convergence order two. Bahram Asadi and Taher Lotfi Copyright © 2016 Bahram Asadi and Taher Lotfi. All rights reserved. The Maximal Strichartz Family of Gaussian Distributions: Fisher Information, Index of Dispersion, and Stochastic Ordering Thu, 29 Sep 2016 09:49:30 +0000 http://www.hindawi.com/journals/ijde/2016/2343975/ We define and study several properties of what we call Maximal Strichartz Family of Gaussian Distributions. This is a subfamily of the family of Gaussian Distributions that arises naturally in the context of the Linear Schrödinger Equation and Harmonic Analysis, as the set of maximizers of certain norms introduced by Strichartz. From a statistical perspective, this family carries with itself some extrastructure with respect to the general family of Gaussian Distributions. In this paper, we analyse this extrastructure in several ways. We first compute the Fisher Information Matrix of the family, then introduce some measures of statistical dispersion, and, finally, introduce a Partial Stochastic Order on the family. Moreover, we indicate how these tools can be used to distinguish between distributions which belong to the family and distributions which do not. We show also that all our results are in accordance with the dispersive PDE nature of the family. Alessandro Selvitella Copyright © 2016 Alessandro Selvitella. All rights reserved. A Hybrid Natural Transform Homotopy Perturbation Method for Solving Fractional Partial Differential Equations Tue, 27 Sep 2016 15:42:46 +0000 http://www.hindawi.com/journals/ijde/2016/9207869/ A hybrid analytical method for solving linear and nonlinear fractional partial differential equations is presented. The proposed analytical approach is an elegant combination of the Natural Transform Method (NTM) and a well-known method, Homotopy Perturbation Method (HPM). In this analytical method, the fractional derivative is computed in Caputo sense and the nonlinear term is calculated using He’s polynomial. The proposed analytical method reduces the computational size and avoids round-off errors. Exact solution of linear and nonlinear fractional partial differential equations is successfully obtained using the analytical method. Shehu Maitama Copyright © 2016 Shehu Maitama. All rights reserved. Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation Sun, 25 Sep 2016 12:24:40 +0000 http://www.hindawi.com/journals/ijde/2016/4270724/ We investigate the symmetry properties of a variable coefficient space-time fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method. Manoj Gaur and K. Singh Copyright © 2016 Manoj Gaur and K. Singh. All rights reserved. Homogeneous-Like Generalized Cubic Systems Mon, 05 Sep 2016 11:26:55 +0000 http://www.hindawi.com/journals/ijde/2016/7640340/ We consider properties and center conditions for plane polynomial systems of the forms , where , and , are polynomials of degrees and , respectively, for integers . We restrict our attention to those systems for which . In this case the system can be transformed to a trigonometric Abel equation which is similar in form to the one obtained for homogeneous systems . From this we show that any center condition of a homogeneous system for a given can be transformed to a center condition of the corresponding generalized cubic system and we use a similar idea to obtain center conditions for several other related systems. As in the case of the homogeneous system, these systems can also be transformed to Abel equations having rational coefficients and we briefly discuss an application of this to a particular Abel equation. G. R. Nicklason Copyright © 2016 G. R. Nicklason. All rights reserved. On Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Damped Neutral Differential Equation Mon, 29 Aug 2016 16:25:02 +0000 http://www.hindawi.com/journals/ijde/2016/3746368/ This paper discusses oscillatory and asymptotic properties of solutions of a class of second-order nonlinear damped neutral differential equations. Some new sufficient conditions for any solution of the equation to be oscillatory or to converge to zero are given. The results obtained extend and improve some of the related results reported in the literature. The results are illustrated with examples. Ercan Tunç and Said R. Grace Copyright © 2016 Ercan Tunç and Said R. Grace. All rights reserved. Estimates on the Lower Bound of the Eigenvalue of the Smallest Modulus Associated with a General Weighted Sturm-Liouville Problem Mon, 29 Aug 2016 09:28:25 +0000 http://www.hindawi.com/journals/ijde/2016/7396951/ We obtain a lower bound on the eigenvalue of smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. The main motivation for this study is the result obtained by Mingarelli (1988). Mervis Kikonko and Angelo Bernado Mingarelli Copyright © 2016 Mervis Kikonko and Angelo Bernado Mingarelli. All rights reserved. Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay Dynamic Equations with Damping and Oscillatory Potential on Time Scales Wed, 24 Aug 2016 17:53:57 +0000 http://www.hindawi.com/journals/ijde/2016/3298289/ We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the form , on a time scale . We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results. Hassan A. Agwa, Ahmed M. M. Khodier, and Heba A. Hassan Copyright © 2016 Hassan A. Agwa et al. All rights reserved. Semianalytic Solution of Space-Time Fractional Diffusion Equation Mon, 08 Aug 2016 09:59:49 +0000 http://www.hindawi.com/journals/ijde/2016/2371837/ We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior. A. Elsaid, S. Shamseldeen, and S. Madkour Copyright © 2016 A. Elsaid et al. All rights reserved. A Comparison Principle for the Mean Curvature Flow Equation with Discontinuous Coefficients Thu, 04 Aug 2016 14:17:44 +0000 http://www.hindawi.com/journals/ijde/2016/3627896/ We study the level set equation in a bounded domain when the velocity of the interface is given by the mean curvature plus a discontinuous velocity. We prove a comparison principle for the initial-boundary value problem whose consequence is uniqueness of continuous solutions and well- posedness of the level set method. Cecilia De Zan and Pierpaolo Soravia Copyright © 2016 Cecilia De Zan and Pierpaolo Soravia. All rights reserved. Positive Stabilization of Linear Differential Algebraic Equation System Wed, 03 Aug 2016 07:06:10 +0000 http://www.hindawi.com/journals/ijde/2016/6346780/ We study in this paper the existence of a feedback for linear differential algebraic equation system such that the closed-loop system is positive and stable. A necessary and sufficient condition for such existence has been established. This result can be used to detect the existence of a state feedback law that makes the linear differential algebraic equation system in closed loop positive and stable. Muhafzan Copyright © 2016 Muhafzan. All rights reserved. Periodicity, Stability, and Boundedness of Solutions to Certain Second Order Delay Differential Equations Mon, 01 Aug 2016 12:56:35 +0000 http://www.hindawi.com/journals/ijde/2016/2843709/ The behaviour of solutions to certain second order nonlinear delay differential equations with variable deviating arguments is discussed. The main procedure lies in the properties of a complete Lyapunov functional which is used to obtain suitable criteria to guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results are new and also complement related ones that have appeared in the literature. Moreover, examples are given to illustrate the feasibility and correctness of the main results. A. T. Ademola, B. S. Ogundare, M. O. Ogundiran, and O. A. Adesina Copyright © 2016 A. T. Ademola et al. All rights reserved.