International Journal of Differential Equations The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm Tue, 24 Nov 2015 09:28:08 +0000 We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented 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 BHT 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 © 2015 F. F. Ngwane and S. N. Jator. All rights reserved. Entropy Solution for Doubly Nonlinear Elliptic Anisotropic Problems with Robin Boundary Conditions Tue, 24 Nov 2015 07:27:45 +0000 We study in this paper nonlinear anisotropic problems with Robin boundary conditions. We prove, by using the technic of monotone operators in Banach spaces, the existence of a sequence of weak solutions of approximation problems associated with the anisotropic Robin boundary value problem. For the existence and uniqueness of entropy solutions, we prove that the sequence of weak solutions converges to a measurable function which is the entropy solution of the anisotropic Robin boundary value problem. I. Ibrango and S. Ouaro Copyright © 2015 I. Ibrango and S. Ouaro. All rights reserved. Numerical Analysis of a Distributed Optimal Control Problem Governed by an Elliptic Variational Inequality Mon, 23 Nov 2015 13:38:06 +0000 The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energy . The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positive (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter goes to zero. Mariela Olguín and Domingo A. Tarzia Copyright © 2015 Mariela Olguín and Domingo A. Tarzia. All rights reserved. Optimal Control of the Ill-Posed Cauchy Elliptic Problem Mon, 23 Nov 2015 06:44:14 +0000 We give a characterization of the control for ill-posed problems of oscillating solutions. More precisely, we study the control of Cauchy elliptic problems via a regularization approach which generates incomplete information. We obtain a singular optimality system characterizing the no-regret control for the Cauchy problem. A. Berhail and A. Omrane Copyright © 2015 A. Berhail and A. Omrane. All rights reserved. Bounds for Products of Zeros of Solutions to Nonhomogeneous ODE with Polynomial Coefficients Wed, 18 Nov 2015 06:52:19 +0000 We consider the equation , where is a polynomial and is an entire function. Let be the zeros of a solution to that equation. Lower estimates for the products are derived. In particular, they give us a bound for the zero free domain. Applications of the obtained estimates to the counting function of the zeros of solutions are also discussed. Michael Gil’ Copyright © 2015 Michael Gil’. All rights reserved. Redistribution of Nodes with Two Constraints in Meshless Method of Line to Time-Dependent Partial Differential Equations Thu, 05 Nov 2015 14:20:32 +0000 Meshless method of line is a powerful device to solve time-dependent partial differential equations. In integrating step, choosing a suitable set of points, such as adaptive nodes in spatial domain, can be useful, although in some cases this can cause ill-conditioning. In this paper, to produce smooth adaptive points in each step of the method, two constraints are enforced in Equidistribution algorithm. These constraints lead to two different meshes known as quasi-uniform and locally bounded meshes. These avoid the ill-conditioning in applying radial basis functions. Moreover, to generate more smooth adaptive meshes another modification is investigated, such as using modified arc-length monitor function in Equidistribution algorithm. Influence of them in growing the accuracy is investigated by some numerical examples. The results of consideration of two constraints are compared with each other and also with uniform meshes. Jafar Biazar and Mohammad Hosami Copyright © 2015 Jafar Biazar and Mohammad Hosami. All rights reserved. Solvability of Nth Order Linear Boundary Value Problems Thu, 29 Oct 2015 11:23:03 +0000 This paper presents a method that provides necessary and sufficient conditions for the existence of solutions of nth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists. P. Almenar and L. Jódar Copyright © 2015 P. Almenar and L. Jódar. All rights reserved. Connections between Some Concepts of Polynomial Trichotomy for Noninvertible Evolution Operators in Banach Spaces Thu, 29 Oct 2015 06:50:29 +0000 The present paper treats three concepts of nonuniform polynomial trichotomies for noninvertible evolution operators acting on Banach spaces. The connections between these concepts are established through numerous examples and counterexamples for systems defined on the Banach space of square-summable sequences. Mihai-Gabriel Babuţia and Nicolae Marian Seimeanu Copyright © 2015 Mihai-Gabriel Babuţia and Nicolae Marian Seimeanu. All rights reserved. Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent Wed, 28 Oct 2015 08:08:49 +0000 We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland’s variational principle. Mohammed El Mokhtar Ould El Mokhtar Copyright © 2015 Mohammed El Mokhtar Ould El Mokhtar. All rights reserved. On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis Wed, 07 Oct 2015 06:04:19 +0000 We consider the time-fractional derivative in the Caputo sense of order . Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in , two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation. D. Goos, G. Reyero, S. Roscani, and E. Santillan Marcus Copyright © 2015 D. Goos et al. All rights reserved. Propagation of Water Waves over Uneven Bottom under the Effect of Surface Tension Wed, 30 Sep 2015 16:36:25 +0000 We establish existence and uniqueness of solutions to the Cauchy problem associated with a new one-dimensional weakly-nonlinear, weakly-dispersive system which arises as an asymptotical approximation of the full potential theory equations for modelling propagation of small amplitude water waves on the surface of a shallow channel with variable depth, taking into account the effect of surface tension. Furthermore, numerical schemes of spectral type are introduced for approximating the evolution in time of solutions of this system and its travelling wave solutions, in both the periodic and nonperiodic case. Juan Carlos Muñoz Grajales Copyright © 2015 Juan Carlos Muñoz Grajales. All rights reserved. On the Convergence of a Nonlinear Boundary-Value Problem in a Perforated Domain Wed, 30 Sep 2015 16:26:08 +0000 We consider a family with respect to a small parameter of nonlinear boundary-value problems as well as the corresponding spectral problems in a domain perforated periodically along a part of the boundary. We prove the convergence of solution of the original problems to the solution of the respective homogenized problem in this domain. Yulia Koroleva Copyright © 2015 Yulia Koroleva. All rights reserved. Nonlinear Impulsive Differential Equations with Weighted Exponential or Ordinary Dichotomous Linear Part in a Banach Space Tue, 29 Sep 2015 11:10:10 +0000 We consider nonlinear impulsive differential equations with ψ-exponential and ψ-ordinary dichotomous linear part in a Banach space. By the help of Banach’s fixed-point principle sufficient conditions are found for the existence of ψ-bounded solutions of these equations on and . Hristo Kiskinov and Andrey Zahariev Copyright © 2015 Hristo Kiskinov and Andrey Zahariev. All rights reserved. Mean-Square Asymptotically Almost Automorphic Solutions to Fractional Stochastic Relaxation Equations Mon, 28 Sep 2015 13:58:13 +0000 Mild solutions generated by a -regularized family to fractional stochastic relaxation equations are studied. The main objective is to establish the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to linear and semilinear case of these equations. Under different hypotheses, some new theorems concerning the main objective are derived. Qiong Wu Copyright © 2015 Qiong Wu. All rights reserved. Numerical Solution of Riccati Equations by the Adomian and Asymptotic Decomposition Methods over Extended Domains Sun, 20 Sep 2015 10:37:37 +0000 We combine the Adomian decomposition method (ADM) and Adomian’s asymptotic decomposition method (AADM) for solving Riccati equations. We investigate the approximate global solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from Adomian’s asymptotic decomposition method for Riccati equations and in such cases when we do not find any region of overlap between the obtained approximate solutions by the two proposed methods, we connect the two approximations by the Padé approximant of the near-field approximation. We illustrate the efficiency of the technique for several specific examples of the Riccati equation for which the exact solution is known in advance. Jafar Biazar and Mohsen Didgar Copyright © 2015 Jafar Biazar and Mohsen Didgar. All rights reserved. Stability, Boundedness, and Existence of Periodic Solutions to Certain Third-Order Delay Differential Equations with Multiple Deviating Arguments Wed, 16 Sep 2015 13:12:42 +0000 The behaviour of solutions for certain third-order nonlinear differential equations with multiple deviating arguments is considered. By employing Lyapunov’s second method, a complete Lyapunov functional is constructed and used to establish sufficient conditions that guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results not only are new but also include many outstanding results in the literature. Finally, the correctness and effectiveness of the results are justified with examples. A. T. Ademola, B. S. Ogundare, M. O. Ogundiran, and O. A. Adesina Copyright © 2015 A. T. Ademola et al. All rights reserved. Periodic Solutions of Some Polynomial Differential Systems in Dimension 3 via Averaging Theory Wed, 16 Sep 2015 12:47:49 +0000 We provide sufficient conditions for the existence of periodic solutions of the polynomial third order differential system ,  ,  and  , where , , and are polynomials in the variables , , and of degree with being periodic functions, is a real number, and is a small parameter. Amar Makhlouf and Lilia Bousbiat Copyright © 2015 Amar Makhlouf and Lilia Bousbiat. All rights reserved. Dynamical Behavior of a System of Second-Order Nonlinear Difference Equations Thu, 10 Sep 2015 06:32:44 +0000 This paper is concerned with local stability, oscillatory character of positive solutions to the system of the two nonlinear difference equations , , where , , , and , . Hongmei Bao Copyright © 2015 Hongmei Bao. All rights reserved. Self-Similar Blow-Up Solutions of the KPZ Equation Wed, 26 Aug 2015 14:05:22 +0000 Self-similar blow-up solutions for the generalized deterministic KPZ equation with are considered. The asymptotic behavior of self-similar solutions is studied. Alexander Gladkov Copyright © 2015 Alexander Gladkov. All rights reserved. The Rate at Which the Energy of Solutions for a Class of -Laplacian Wave Equation Decays Wed, 12 Aug 2015 14:13:50 +0000 We will investigate the decay estimate of the energy of the global solutions to the p-Laplacian wave equation with dissipation of the form under suitable assumptions on the positive function . For this end we use the multiplier method combined with nonlinear integral inequalities given by Martinez; the proof is based on the construction of a special weight function that depends on the behavior of . Soufiane Mokeddem and Khaled Ben Walid Mansour Copyright © 2015 Soufiane Mokeddem and Khaled Ben Walid Mansour. All rights reserved. On Certain Subclasses of Analytic Multivalent Functions Using Generalized Salagean Operator Tue, 07 Jul 2015 08:02:12 +0000 We introduce and study two subclasses of multivalent functions denoted by and . Further, by using the method of differential subordination, certain inclusion relations between the two subclasses aforementioned are given. Moreover, several consequences of the main results are also discussed. Adnan Ghazy Alamoush and Maslina Darus Copyright © 2015 Adnan Ghazy Alamoush and Maslina Darus. All rights reserved. Implementation of TAGE Method Using Seikkala Derivatives Applied to Two-Point Fuzzy Boundary Value Problems Sun, 14 Jun 2015 13:00:13 +0000 Iterative methods particularly the Two-Parameter Alternating Group Explicit (TAGE) methods are used to solve system of linear equations generated from the discretization of two-point fuzzy boundary value problems (FBVPs). The formulation and implementation of the TAGE method are also presented. Then numerical experiments are carried out onto two example problems to verify the effectiveness of the method. The results show that TAGE method is superior compared to GS method in the aspect of number of iterations, execution time, and Hausdorff distance. A. A. Dahalan and J. Sulaiman Copyright © 2015 A. A. Dahalan and J. Sulaiman. All rights reserved. An Inverse Spectral Problem for the Matrix Sturm-Liouville Operator with a Bessel-Type Singularity Tue, 19 May 2015 13:30:31 +0000 The inverse problem by the Weyl matrix is studied for the matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval. We construct special fundamental systems of solutions for this equation and prove the uniqueness theorem of the inverse problem. Natalia Bondarenko Copyright © 2015 Natalia Bondarenko. All rights reserved. A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation Thu, 02 Apr 2015 14:10:01 +0000 The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form: By constructing a Lyapunov functional, we obtained the result of stability. A. M. A. Abou-El-Ela, A. I. Sadek, A. M. Mahmoud, and R. O. A. Taie Copyright © 2015 A. M. A. Abou-El-Ela et al. All rights reserved. On the Limit Cycles of a Class of Generalized Kukles Polynomial Differential Systems via Averaging Theory Sun, 15 Mar 2015 09:14:19 +0000 We apply the averaging theory of first and second order to a class of generalized Kukles polynomial differential systems to study the maximum number of limit cycles of these systems. Amar Makhlouf and Amor Menaceur Copyright © 2015 Amar Makhlouf and Amor Menaceur. All rights reserved. Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations Thu, 05 Feb 2015 10:07:07 +0000 We consider an ordinary differential equation (ODE) which can be written as a polynomial in variables and derivatives. Several types of asymptotic expansions of its solutions can be found by algorithms of 2D Power Geometry. They are power, power-logarithmic, exotic, and complicated expansions. Here we develop 3D Power Geometry and apply it for calculation power-elliptic expansions of solutions to an ODE. Among them we select regular power-elliptic expansions and give a survey of all such expansions in solutions of the Painlevé equations . Alexander D. Bruno Copyright © 2015 Alexander D. Bruno. All rights reserved. Nonlocal Boundary Value Problems for -Difference Equations and Inclusions Tue, 27 Jan 2015 11:53:23 +0000 We study boundary value problems for -difference equations and inclusions with nonlocal and integral boundary conditions which have different quantum numbers. Some new existence and uniqueness results are obtained by using fixed point theorems. Examples are given to illustrate the results. Sotiris K. Ntouyas and Jessada Tariboon Copyright © 2015 Sotiris K. Ntouyas and Jessada Tariboon. All rights reserved. Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems Tue, 06 Jan 2015 07:13:07 +0000 We consider the nonlinear eigenvalue problem ,  ,  ,  , where is a cubic-like nonlinear term and is a parameter. It is known by Korman et al. (2005) that, under the suitable conditions on , there exist exactly three bifurcation branches (), and these curves are parameterized by the maximum norm of the solution corresponding to . In this paper, we establish the precise global structures for (), which can be applied to the inverse bifurcation problems. The precise local structures for () are also discussed. Furthermore, we establish the asymptotic shape of the spike layer solution , which corresponds to , as . Tetsutaro Shibata Copyright © 2015 Tetsutaro Shibata. All rights reserved. Asymptotic Behavior of Global Entropy Solutions for Nonstrictly Hyperbolic Systems with Linear Damping Tue, 18 Nov 2014 06:30:56 +0000 We investigate the large time behavior of the global weak entropy solutions to the symmetric Keyfitz-Kranzer system with linear damping. It is proved that as the entropy solutions tend to zero in the norm. Richard Alexander De la Cruz Guerrero, Juan Carlos Juajibioy Otero, and Leonardo Rendon Copyright © 2014 Richard Alexander De la Cruz Guerrero et al. All rights reserved. Mixed Boundary Value Problem on Hypersurfaces Sun, 17 Aug 2014 12:45:26 +0000 The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation on a smooth hypersurface with the boundary in . is an bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts and on the Dirichlet boundary conditions are prescribed, while on the Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution to is proved, which is interpreted as the invertibility of this operator in the setting , where is a subspace of the Bessel potential space and consists of functions with mean value zero. R. DuDuchava, M. Tsaava, and T. Tsutsunava Copyright © 2014 R. DuDuchava et al. All rights reserved.