International Journal of Differential Equations The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations on Thu, 13 Oct 2016 12:44:41 +0000 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 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 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 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 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 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 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 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 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 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 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 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 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 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 Krasnoselskii-Krein, Rogers, and Kooi conditions. Abdourazek Souahi, Assia Guezane-Lakoud, 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 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 th-Order Linear Boundary Value Problems Mon, 18 Jul 2016 07:29:02 +0000 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 th-order 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 Nonlinear-Viscous Fluid Model Mon, 27 Jun 2016 14:26:53 +0000 We consider the optimal control problem for a mathematical model describing steady flows of a nonlinear-viscous incompressible fluid in a bounded three-dimensional (or a two-dimensional) 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 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 CarathĂ©odory 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 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 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 First-Order Linear Differential Equations in Fuzzy Environment by Runge-Kutta-Fehlberg Method and Its Application Thu, 12 May 2016 07:20:59 +0000 The numerical algorithm for solving “first-order linear differential equation in fuzzy environment” is discussed. A scheme, namely, “Runge-Kutta-Fehlberg 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 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. Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films Thu, 05 May 2016 13:00:12 +0000 For the thin-film model of a viscous flow which originates from lubrication approximation and has a full nonlinear curvature term, we prove existence of nonnegative weak solutions. Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipation of the solution arc length that is a well known property for a Hele-Shaw flow. For the classical thin-film model with linearized curvature term, under some restrictions on parameter and gradient values, we also prove analytically the arc length dissipation property for positive solutions. We compare the numerical solutions for both models, with nonlinear and with linearized curvature terms. In regimes when solutions develop finite time singularities, we explain the difference in qualitative behaviour of solutions. Matthew Michal, Marina Chugunova, and Roman Taranets Copyright © 2016 Matthew Michal et al. All rights reserved. On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations Tue, 16 Feb 2016 11:57:17 +0000 We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability. ElHassan ElJaoui and Said Melliani Copyright © 2016 ElHassan ElJaoui and Said Melliani. All rights reserved. Equivariant Hopf Bifurcation in a Time-Delayed Ring of Antigenic Variants Sun, 14 Feb 2016 12:49:05 +0000 We consider an intrahost malaria model allowing for antigenic variation within a single species. The host’s immune response is compartmentalised into reactions to major and minor epitopes. We investigate the dynamics of the model, paying particular attention to bifurcation and stability of the uniform nonzero endemic equilibrium. We establish conditions for the existence of an equivariant Hopf bifurcation in a ring of antigenic variants, characterised by time delay. Israel Ncube Copyright © 2016 Israel Ncube. All rights reserved. A New Result Concerning the Solvability of a Class of General Systems of Variational Equations with Nonmonotone Operators: Applications to Dirichlet and Neumann Nonlinear Problems Tue, 19 Jan 2016 12:25:46 +0000 A new result of solvability for a wide class of systems of variational equations depending on parameters and governed by nonmonotone operators is found in a Banach real and reflexive space with applications to Dirichlet and Neumann problems related to nonlinear elliptic systems. Luisa Toscano and Speranza Toscano Copyright © 2016 Luisa Toscano and Speranza Toscano. All rights reserved. Existence and Permanence in a Diffusive KiSS Model with Robust Numerical Simulations Thu, 24 Dec 2015 09:24:59 +0000 We have given an extension to the study of Kierstead, Slobodkin, and Skellam (KiSS) model. We present the theoretical results based on the survival and permanence of the species. To guarantee the long-term existence and permanence, the patch size denoted as must be greater than the critical patch size . It was also observed that the reaction-diffusion problem can be split into two parts: the linear and nonlinear terms. Hence, the use of two classical methods in space and time is permitted. We use spectral method in the area of mathematical community to remove the stiffness associated with the linear or diffusive terms. The resulting system is advanced with a modified exponential time-differencing method whose formulation was based on the fourth-order Runge-Kutta scheme. With high-order method, this extends the one-dimensional work and presents experiments for two-dimensional problem. The complexity of the dynamical model is discussed theoretically and graphically simulated to demonstrate and compare the behavior of the time-dependent density function. Kolade M. Owolabi and Kailash C. Patidar Copyright © 2015 Kolade M. Owolabi and Kailash C. Patidar. All rights reserved. On Stability of Basis Property of Root Vectors System of the Sturm-Liouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular Samarskii-Ionkin Type Problems Mon, 14 Dec 2015 12:57:19 +0000 We study a question on stability and instability of basis property of system of eigenfunctions and associated functions of the double differentiation operator with an integral perturbation of Samarskii-Ionkin type boundary conditions. N. S. Imanbaev Copyright © 2015 N. S. Imanbaev. All rights reserved. On -Anisotropic Problems with Neumann Boundary Conditions Sun, 13 Dec 2015 13:52:32 +0000 This work is devoted to the study of a general class of anisotropic problems involving -Laplace operator. Based on the variational method, we establish the existence of a nontrivial solution without Ambrosetti-Rabinowitz type conditions. Anass Ourraoui Copyright © 2015 Anass Ourraoui. All rights reserved. On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method Mon, 07 Dec 2015 14:23:48 +0000 The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique. Jerico B. Bacani and Julius Fergy T. Rabago Copyright © 2015 Jerico B. Bacani and Julius Fergy T. Rabago. All rights reserved.