International Journal of Differential Equations
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© 2015 , Hindawi Publishing Corporation . 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.

Existence of Solutions for TwoPoint Boundary Value Problem of Fractional Differential Equations at Resonance
Tue, 05 Aug 2014 12:30:02 +0000
http://www.hindawi.com/journals/ijde/2014/632434/
We establish the existence results for twopoint boundary value problem of fractional differential equations at resonance by means of the coincidence degree theory. Furthermore, a result on the uniqueness of solution is obtained. We give an example to demonstrate our results.
Lei Hu, Shuqin Zhang, and Ailing Shi
Copyright © 2014 Lei Hu et al. All rights reserved.

An Existence Theorem for a Nonlocal Global Pandemic Model for InsectBorne Diseases
Thu, 24 Jul 2014 07:14:43 +0000
http://www.hindawi.com/journals/ijde/2014/187685/
We construct and analyze a nonlocal global pandemic model that comprises a system of two nonlocal integrodifferential equations (functional differential equations) and
an ordinary differential equation. This model was constructed by considering a spherical coordinate transformation of a previously established epidemiology model that can be applied to insectborne
diseases, like yellow fever. This transformation amounts to a nonlocal boundary value problem on the unit sphere and therefore can be interpreted as a global pandemic model for insectborne diseases. We ultimately show that a weak solution to the weak formulation of this model exists using a fixed point argument, which calls upon the construction of a weak formulation and the existence
and uniqueness of an auxiliary problem.
John R. Cannon and Daniel J. Galiffa
Copyright © 2014 John R. Cannon and Daniel J. Galiffa. All rights reserved.

On Certain Class of NonBazilevič Functions of Order Defined by a Differential Subordination
Thu, 17 Jul 2014 11:27:14 +0000
http://www.hindawi.com/journals/ijde/2014/458090/
We introduce a new subclass of NonBazilevič functions of order . Some subordination relations and inequality properties are discussed. The results obtained generalize the related work of some authors. In addition, some other new results are also obtained.
A. G. Alamoush and M. Darus
Copyright © 2014 A. G. Alamoush and M. Darus. All rights reserved.

Solving Singular Boundary Value Problems by Optimal Homotopy Asymptotic Method
Sun, 29 Jun 2014 10:56:04 +0000
http://www.hindawi.com/journals/ijde/2014/287480/
In this paper, optimal homotopy asymptotic method (OHAM) for the semianalytic solutions of nonlinear singular twopoint boundary value problems has been applied to several problems. The solutions obtained by OHAM have been compared with the solutions of another method named as modified adomain decomposition (MADM). For testing the success of OHAM, both of the techniques have been analyzed against the exact solutions in all problems. It is proved by this paper that solutions of OHAM converge rapidly to the exact solution and show most effectiveness as compared to MADM.
S. Zuhra, S. Islam, M. Idrees, Rashid Nawaz, I. A. Shah, and H. Ullah
Copyright © 2014 S. Zuhra et al. All rights reserved.

Oscillation of SecondOrder Nonlinear Delay Dynamic Equations with Damping on Time Scales
Sun, 22 Jun 2014 13:08:21 +0000
http://www.hindawi.com/journals/ijde/2014/594376/
We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the secondorder nonlinear delay dynamic equation with damping on a time scale , , where , , and are positive right dense continuous (rdcontinuous) functions on . Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the secondorder nonlinear delay differential equation with damping and the secondorder nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results.
H. A. Agwa, Ahmed M. M. Khodier, and Heba A. Hassan
Copyright © 2014 H. A. Agwa et al. All rights reserved.

Existence of Mild and Classical Solutions for Nonlocal Impulsive Integrodifferential Equations in Banach Spaces with Measure of Noncompactness
Thu, 19 Jun 2014 07:51:38 +0000
http://www.hindawi.com/journals/ijde/2014/319250/
We study the existence of mild and classical solutions are proved for a class of impulsive integrodifferential equations with nonlocal conditions in Banach spaces. The main results are obtained by using measure of noncompactness and semigroup theory. An example is presented.
K. Karthikeyan, A. Anguraj, K. Malar, and Juan J. Trujillo
Copyright © 2014 K. Karthikeyan et al. All rights reserved.

On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems
Sun, 01 Jun 2014 07:34:24 +0000
http://www.hindawi.com/journals/ijde/2014/948597/
This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the SalamonWeiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type
on unconditional martingale differences UMD spaces, provided that the leading operator generates a semigroup. Next, a necessary and sufficient condition for admissibility of the system's control operator is given in terms of the UMDproperty of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).
Ahmed Fadili and Hamid Bounit
Copyright © 2014 Ahmed Fadili and Hamid Bounit. All rights reserved.

Stability of Solutions to a Free Boundary Problem for Tumor Growth
Wed, 21 May 2014 11:30:25 +0000
http://www.hindawi.com/journals/ijde/2014/427547/
We study the asymptotic behaviour of quasistationary solutions of a free boundary
problem which had been discussed by Bueno (2005). Using a simpler method we prove that the quasisteady solutions of the problem converge uniformly to the unique nontrivial steady solution.
Shihe Xu
Copyright © 2014 Shihe Xu. All rights reserved.

On the Oscillation of EvenOrder HalfLinear Functional Difference Equations with Damping Term
Mon, 19 May 2014 12:14:44 +0000
http://www.hindawi.com/journals/ijde/2014/791631/
We investigate the oscillatory behavior of solutions of the th order halflinear functional difference equations with damping term of the form , , where is even and , is a fixed real number. Our main results are obtained via employing the generalized Riccati transformation. We provide two examples to illustrate the effectiveness of the proposed results.
Yaşar Bolat and Jehad Alzabut
Copyright © 2014 Yaşar Bolat and Jehad Alzabut. All rights reserved.