International Journal of Differential Equations
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© 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
http://www.hindawi.com/journals/ijde/2015/347864/
We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the stepsize for directly solving general secondorder 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 twostep hybrid trigonometrically fitted method with two offgrid points. The BHT is implemented in a blockbyblock fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictorcorrector 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
http://www.hindawi.com/journals/ijde/2015/919608/
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
http://www.hindawi.com/journals/ijde/2015/407930/
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 IllPosed Cauchy Elliptic Problem
Mon, 23 Nov 2015 06:44:14 +0000
http://www.hindawi.com/journals/ijde/2015/468918/
We give a characterization of the control for illposed 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 noregret 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
http://www.hindawi.com/journals/ijde/2015/690519/
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 TimeDependent Partial Differential Equations
Thu, 05 Nov 2015 14:20:32 +0000
http://www.hindawi.com/journals/ijde/2015/762034/
Meshless method of line is a powerful device to solve timedependent 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 illconditioning. 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 quasiuniform and locally bounded meshes. These avoid the illconditioning in applying radial basis functions. Moreover, to generate more smooth adaptive meshes another modification is investigated, such as using modified arclength 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
http://www.hindawi.com/journals/ijde/2015/230405/
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
http://www.hindawi.com/journals/ijde/2015/241402/
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 squaresummable sequences.
MihaiGabriel Babuţia and Nicolae Marian Seimeanu
Copyright © 2015 MihaiGabriel 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
http://www.hindawi.com/journals/ijde/2015/494907/
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 InitialBoundaryValue Problem for the TimeFractional Diffusion Equation on the Real Positive Semiaxis
Wed, 07 Oct 2015 06:04:19 +0000
http://www.hindawi.com/journals/ijde/2015/439419/
We consider the timefractional 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 initialboundaryvalue problems for the timefractional 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 boundaryvalue 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
http://www.hindawi.com/journals/ijde/2015/805625/
We establish existence and uniqueness of solutions to the Cauchy problem associated with a new onedimensional weaklynonlinear, weaklydispersive 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 BoundaryValue Problem in a Perforated Domain
Wed, 30 Sep 2015 16:26:08 +0000
http://www.hindawi.com/journals/ijde/2015/392479/
We consider a family with respect to a small parameter of nonlinear boundaryvalue 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
http://www.hindawi.com/journals/ijde/2015/748607/
We consider nonlinear impulsive differential equations with ψexponential and ψordinary dichotomous linear part in a Banach space.
By the help of Banach’s fixedpoint principle sufficient conditions are found for the existence
of ψbounded solutions of these equations on and .
Hristo Kiskinov and Andrey Zahariev
Copyright © 2015 Hristo Kiskinov and Andrey Zahariev. All rights reserved.

MeanSquare Asymptotically Almost Automorphic Solutions to Fractional Stochastic Relaxation Equations
Mon, 28 Sep 2015 13:58:13 +0000
http://www.hindawi.com/journals/ijde/2015/143591/
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 squaremean 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
http://www.hindawi.com/journals/ijde/2015/580741/
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 nearfield approximation derived from the Adomian decomposition method with the farfield 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 nearfield 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 ThirdOrder Delay Differential Equations with Multiple Deviating Arguments
Wed, 16 Sep 2015 13:12:42 +0000
http://www.hindawi.com/journals/ijde/2015/213935/
The behaviour of solutions for certain thirdorder 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
http://www.hindawi.com/journals/ijde/2015/263837/
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 SecondOrder Nonlinear Difference Equations
Thu, 10 Sep 2015 06:32:44 +0000
http://www.hindawi.com/journals/ijde/2015/679017/
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.

SelfSimilar BlowUp Solutions of the KPZ Equation
Wed, 26 Aug 2015 14:05:22 +0000
http://www.hindawi.com/journals/ijde/2015/572841/
Selfsimilar blowup solutions for the generalized deterministic KPZ equation with are considered. The asymptotic behavior of selfsimilar 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
http://www.hindawi.com/journals/ijde/2015/721503/
We will investigate
the decay estimate of the energy of the global
solutions to the pLaplacian 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
http://www.hindawi.com/journals/ijde/2015/910124/
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 TwoPoint Fuzzy Boundary Value Problems
Sun, 14 Jun 2015 13:00:13 +0000
http://www.hindawi.com/journals/ijde/2015/346036/
Iterative methods particularly the TwoParameter Alternating Group Explicit
(TAGE) methods are used to solve system of linear equations generated from the
discretization of twopoint 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 SturmLiouville Operator with a BesselType Singularity
Tue, 19 May 2015 13:30:31 +0000
http://www.hindawi.com/journals/ijde/2015/647396/
The inverse problem by the Weyl matrix is studied for the matrix SturmLiouville equation on a finite interval with a Besseltype 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 FourthOrder Delay Differential Equation
Thu, 02 Apr 2015 14:10:01 +0000
http://www.hindawi.com/journals/ijde/2015/618359/
The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourthorder vector delay differential equation of the following form: By constructing a Lyapunov functional, we obtained the result of stability.
A. M. A. AbouElEla, A. I. Sadek, A. M. Mahmoud, and R. O. A. Taie
Copyright © 2015 A. M. A. AbouElEla 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
http://www.hindawi.com/journals/ijde/2015/325102/
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
http://www.hindawi.com/journals/ijde/2015/340715/
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,
powerlogarithmic, exotic, and complicated expansions. Here we develop 3D Power
Geometry and apply it for calculation powerelliptic expansions of solutions to an
ODE. Among them we select regular powerelliptic 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
http://www.hindawi.com/journals/ijde/2015/203715/
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
http://www.hindawi.com/journals/ijde/2015/138629/
We consider the nonlinear eigenvalue problem , , , , where is a cubiclike 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
http://www.hindawi.com/journals/ijde/2014/383254/
We investigate the large time behavior of the global weak entropy solutions to the symmetric KeyfitzKranzer 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
http://www.hindawi.com/journals/ijde/2014/245350/
The purpose of the present paper is to investigate the mixed DirichletNeumann boundary value problems for the anisotropic LaplaceBeltrami 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 LaxMilgram 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.