International Journal of Differential Equations
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The latest articles from Hindawi Publishing Corporation
© 2016 , Hindawi Publishing Corporation . All rights reserved.

The Dynamical Analysis of a PreyPredator Model with a RefugeStage 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 preypredator 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 ParabolicHyperbolic 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 parabolichyperbolic equation involving the Caputo fractional derivative and RiemannLiouville 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 HighOrder 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 fourthorder 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 zeroorder 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 highorder singular perturbation problems with a discontinuous source term.
Essam R. ElZahar
Copyright © 2016 Essam R. ElZahar. 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 LeggettWilliams normtype 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 SchrödingerBenjaminOno 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 FourierGalerkin numerical scheme to approximate solutions of a coupled nonlinear SchrödingerBenjaminOno system that describes the motion of two fluids with different densities under capillarygravity 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 ChunLei Tang
Copyright © 2016 Yu Duan and ChunLei 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 wellknown 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 roundoff 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 SpaceTime 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 spacetime 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.

HomogeneousLike 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 SecondOrder 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 secondorder 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 SturmLiouville 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 SturmLiouville 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 SecondOrder 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 secondorder 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 secondorder nonlinear delay differential equation and the forced secondorder 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 SpaceTime Fractional Diffusion Equation
Mon, 08 Aug 2016 09:59:49 +0000
http://www.hindawi.com/journals/ijde/2016/2371837/
We study the spacetime fractional diffusion equation with spatial RieszFeller 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 initialboundary 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 closedloop 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.

On Some Existence and Uniqueness Results for a Class of Equations of Order on Arbitrary Time Scales
Wed, 27 Jul 2016 13:43:33 +0000
http://www.hindawi.com/journals/ijde/2016/7327319/
This paper investigates the existence and uniqueness of solution for a class of nonlinear fractional differential equations of fractional order in arbitrary time scales. The results are established using extensions of KrasnoselskiiKrein, Rogers, and Kooi conditions.
Abdourazek Souahi, Assia GuezaneLakoud, and Rabah Khaldi
Copyright © 2016 Abdourazek Souahi et al. All rights reserved.

Boundary Layers and Shock Profiles for the Broadwell Model
Wed, 20 Jul 2016 05:50:57 +0000
http://www.hindawi.com/journals/ijde/2016/5801728/
We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.
Niclas Bernhoff
Copyright © 2016 Niclas Bernhoff. All rights reserved.

Improving Results on Solvability of a Class of thOrder Linear Boundary Value Problems
Mon, 18 Jul 2016 07:29:02 +0000
http://www.hindawi.com/journals/ijde/2016/3750530/
This paper presents a modification of a recursive method described in a previous paper of the authors, which yields necessary and sufficient conditions for the existence of solutions of a class of thorder linear boundary value problems, in the form of integral inequalities. Such a modification simplifies the assessment of the conditions on restricting the inequality to be verified to a single point instead of the full interval where the boundary value problem is defined. The paper also provides an error bound that needs to be considered in the integral inequalities of the previous paper when they are calculated numerically.
Pedro Almenar and Lucas Jódar
Copyright © 2016 Pedro Almenar and Lucas Jódar. All rights reserved.

Existence of Optimal Control for a NonlinearViscous Fluid Model
Mon, 27 Jun 2016 14:26:53 +0000
http://www.hindawi.com/journals/ijde/2016/9428128/
We consider the optimal control problem for a mathematical model describing steady flows of a nonlinearviscous incompressible fluid in a bounded threedimensional (or a twodimensional) domain with impermeable solid walls. The control parameter is the surface force at a given part of the flow domain boundary. For a given bounded set of admissible controls, we construct generalized (weak) solutions that minimize a given cost functional.
Evgenii S. Baranovskii and Mikhail A. Artemov
Copyright © 2016 Evgenii S. Baranovskii and Mikhail A. Artemov. All rights reserved.

Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions
Sun, 05 Jun 2016 14:00:00 +0000
http://www.hindawi.com/journals/ijde/2016/4726526/
This paper is motivated by some papers treating the fractional hybrid differential equations with nonlocal conditions and the system of coupled hybrid fractional differential equations; an existence theorem for fractional hybrid differential equations involving Caputo differential operators of order is proved under mixed Lipschitz and Carathéodory conditions. The existence and uniqueness result is elaborated for the system of coupled hybrid fractional differential equations.
Khalid Hilal and Ahmed Kajouni
Copyright © 2016 Khalid Hilal and Ahmed Kajouni. All rights reserved.

Exact Solutions Superimposed with Nonlinear Plane Waves
Thu, 02 Jun 2016 16:07:51 +0000
http://www.hindawi.com/journals/ijde/2016/1846341/
The flow of fluid in atmosphere and ocean is governed by rotating stratified Boussinesq equations. Through the literature, we found that many researchers are trying to find the solutions of rotating stratified Boussinesq equations. In this paper, we have obtained special exact solutions and nonlinear plane waves. Finally, we provide exact solutions to rotating stratified Boussinesq equations with large scale motion superimposed with the nonlinear plane waves. In support of our investigations, we provided two examples: one described the special exact solution and in second example, we have determined the special exact solution superimposed with nonlinear plane wave. Also, we depicted some integral curves that represent the flow of an incompressible fluid particle on the plane as the particular case.
B. S. Desale and Vivek Sharma
Copyright © 2016 B. S. Desale and Vivek Sharma. All rights reserved.

Static Consensus in Passifiable Linear Networks
Wed, 18 May 2016 07:39:54 +0000
http://www.hindawi.com/journals/ijde/2016/9192127/
Sufficient conditions of consensus (synchronization) in networks described by digraphs and consisting of identical deterministic SIMO systems are derived. Identical and nonidentical control gains (positive arc weights) are considered. Connection between admissible digraphs and nonsmooth hypersurfaces (sufficient gain boundary) is established. Necessary and sufficient conditions for static consensus by output feedback in networks consisting of certain class of double integrators are rediscovered. Scalability for circle digraph in terms of gain magnitudes is studied. Examples and results of numerical simulations are presented.
Ibragim A. Junussov
Copyright © 2016 Ibragim A. Junussov. All rights reserved.

Numerical Solution of FirstOrder Linear Differential Equations in Fuzzy Environment by RungeKuttaFehlberg Method and Its Application
Thu, 12 May 2016 07:20:59 +0000
http://www.hindawi.com/journals/ijde/2016/8150497/
The numerical algorithm for solving “firstorder linear differential equation in fuzzy environment” is discussed. A scheme, namely, “RungeKuttaFehlberg method,” is described in detail for solving the said differential equation. The numerical solutions are compared with (i)gH and (ii)gH differential (exact solutions concepts) system. The method is also followed by complete error analysis. The method is illustrated by solving an example and an application.
Sankar Prasad Mondal, Susmita Roy, and Biswajit Das
Copyright © 2016 Sankar Prasad Mondal et al. All rights reserved.

Multiplicity of Positive Solutions for Fractional Differential Equation with Laplacian Boundary Value Problems
Thu, 12 May 2016 06:06:30 +0000
http://www.hindawi.com/journals/ijde/2016/6906049/
We investigate the existence of multiple positive solutions of fractional differential equations with Laplacian operator , , , , where , , , , , is a fixed integer, and , by applying Leggett–Williams fixed point theorems and fixed point index theory.
Sabbavarapu Nageswara Rao
Copyright © 2016 Sabbavarapu Nageswara Rao. All rights reserved.