International Journal of Differential Equations The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . 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. Existence of Solutions for Two-Point Boundary Value Problem of Fractional Differential Equations at Resonance Tue, 05 Aug 2014 12:30:02 +0000 We establish the existence results for two-point 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 Insect-Borne Diseases Thu, 24 Jul 2014 07:14:43 +0000 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 insect-borne 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 insect-borne 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 Non-Bazilevič Functions of Order Defined by a Differential Subordination Thu, 17 Jul 2014 11:27:14 +0000 We introduce a new subclass of Non-Bazilevič 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 In this paper, optimal homotopy asymptotic method (OHAM) for the semianalytic solutions of nonlinear singular two-point 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 Second-Order Nonlinear Delay Dynamic Equations with Damping on Time Scales Sun, 22 Jun 2014 13:08:21 +0000 We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale , , where , , and are positive right dense continuous (rd-continuous) functions on . Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order 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 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 This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss 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 UMD-property 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 We study the asymptotic behaviour of quasi-stationary solutions of a free boundary problem which had been discussed by Bueno (2005). Using a simpler method we prove that the quasi-steady 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 Even-Order Half-Linear Functional Difference Equations with Damping Term Mon, 19 May 2014 12:14:44 +0000 We investigate the oscillatory behavior of solutions of the th order half-linear 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.