Abstract and Applied Analysis http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk Wed, 01 Apr 2015 14:10:23 +0000 http://www.hindawi.com/journals/aaa/2015/209307/ We consider the reducing subspaces of on , where , , and for . We prove that each reducing subspace of is a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases that and , respectively. Finally, we give a complete description of minimal reducing subspaces of on with . Yanyue Shi and Na Zhou Copyright © 2015 Yanyue Shi and Na Zhou. All rights reserved. Nonlinear Partial Differential Equations in Mathematics and Physics Wed, 01 Apr 2015 07:43:47 +0000 http://www.hindawi.com/journals/aaa/2015/593126/ Bo-Qing Dong, Caidi Zhao, Xiaohong Qin, Linghai Zhang, and Liangpan Li Copyright © 2015 Bo-Qing Dong et al. All rights reserved. On Unique Continuation for Navier-Stokes Equations Mon, 30 Mar 2015 11:51:50 +0000 http://www.hindawi.com/journals/aaa/2015/597946/ We study the unique continuation properties of solutions of the Navier-Stokes equations. We take advantage of rotation transformation of the Navier-Stokes equations to prove the “logarithmic convexity” of certain quantities, which measure the suitable Gaussian decay at infinity to obtain the Gaussian decay weighted estimates, as well as Carleman inequality. As a consequence we obtain sufficient conditions on the behavior of the solution at two different times and which guarantee the “global” unique continuation of solutions for the Navier-Stokes equations. Zhiwen Duan, Shuxia Han, and Peipei Sun Copyright © 2015 Zhiwen Duan et al. All rights reserved. Numerical Studies for Fractional Functional Differential Equations with Delay Based on BDF-Type Shifted Chebyshev Approximations Sun, 29 Mar 2015 13:12:26 +0000 http://www.hindawi.com/journals/aaa/2015/510875/ Fractional functional differential equations with delay (FDDEs) have recently played a significant role in modeling of many real areas of sciences such as physics, engineering, biology, medicine, and economics. FDDEs often cannot be solved analytically so the approximate and numerical methods should be adapted to solve these types of equations. In this paper we consider a new method of backward differentiation formula- (BDF-) type for solving FDDEs. This approach is based on the interval approximation of the true solution using the Clenshaw and Curtis formula that is based on the truncated shifted Chebyshev polynomials. It is shown that the new approach can be reformulated in an equivalent way as a Runge-Kutta method and the Butcher tableau of this method is given. Estimation of local and global truncating errors is deduced and this leads to the proof of the convergence for the proposed method. Illustrative examples of FDDEs are included to demonstrate the validity and applicability of the proposed approach. V. G. Pimenov and A. S. Hendy Copyright © 2015 V. G. Pimenov and A. S. Hendy. All rights reserved. Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014 Sun, 29 Mar 2015 13:03:29 +0000 http://www.hindawi.com/journals/aaa/2015/193030/ Santanu Saha Ray, Rasajit K. Bera, Adem Kılıçman, Om P. Agrawal, and Yasir Khan Copyright © 2015 Santanu Saha Ray et al. All rights reserved. Temperature Dependent Viscosity of a Third Order Thin Film Fluid Layer on a Lubricating Vertical Belt Thu, 26 Mar 2015 13:37:33 +0000 http://www.hindawi.com/journals/aaa/2015/386759/ This paper aims to study the influence of heat transfer on thin film flow of a reactive third order fluid with variable viscosity and slip boundary condition. The problem is formulated in the form of coupled nonlinear equations governing the flow together with appropriate boundary conditions. Approximate analytical solutions for velocity and temperature are obtained using Adomian Decomposition Method (ADM). Such solutions are also obtained by using Optimal Homotopy Asymptotic Method (OHAM) and are compared with ADM solutions. Both of these solutions are found identical as shown in graphs and tables. The graphical results for embedded flow parameters are also shown. T. Gul, S. Islam, R. A. Shah, I. Khan, and L. C. C. Dennis Copyright © 2015 T. Gul et al. All rights reserved. Classification of Multiply Travelling Wave Solutions for Coupled Burgers, Combined KdV-Modified KdV, and Schrödinger-KdV Equations Wed, 25 Mar 2015 12:34:49 +0000 http://www.hindawi.com/journals/aaa/2015/369294/ Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematical physics are presented. By applying a theory of Frobenius decompositions and, more precisely, by using a transformation method to the coupled Burgers, combined Korteweg-de Vries- (KdV-) modified KdV and Schrödinger-KdV equation is written as bilinear ordinary differential equations and two solutions to describing nonlinear interaction of travelling waves are generated. The properties of the multiple travelling wave solutions are shown by some figures. All solutions are stable and have applications in physics. A. R. Seadawy and K. El-Rashidy Copyright © 2015 A. R. Seadawy and K. El-Rashidy. All rights reserved. Recent Developments and Applications on Qualitative Theory of Fractional Equations and Related Topics Wed, 25 Mar 2015 12:27:59 +0000 http://www.hindawi.com/journals/aaa/2015/724396/ Shurong Sun, Luisa Morgado, Jehad Alzabut, and Ivanka Stamova Copyright © 2015 Shurong Sun et al. All rights reserved. Exact Solutions for Unsteady Free Convection Flow of Casson Fluid over an Oscillating Vertical Plate with Constant Wall Temperature Wed, 25 Mar 2015 11:54:36 +0000 http://www.hindawi.com/journals/aaa/2015/946350/ The unsteady free flow of a Casson fluid past an oscillating vertical plate with constant wall temperature has been studied. The Casson fluid model is used to distinguish the non-Newtonian fluid behaviour. The governing partial differential equations corresponding to the momentum and energy equations are transformed into linear ordinary differential equations by using nondimensional variables. Laplace transform method is used to find the exact solutions of these equations. Expressions for shear stress in terms of skin friction and the rate of heat transfer in terms of Nusselt number are also obtained. Numerical results of velocity and temperature profiles with various values of embedded flow parameters are shown graphically and their effects are discussed in detail. Asma Khalid, Ilyas Khan, and Sharidan Shafie Copyright © 2015 Asma Khalid et al. All rights reserved. A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation Wed, 25 Mar 2015 11:53:43 +0000 http://www.hindawi.com/journals/aaa/2015/539652/ The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method. Wenyuan Liao Copyright © 2015 Wenyuan Liao. All rights reserved. Corrigendum to “Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials” Wed, 25 Mar 2015 11:29:56 +0000 http://www.hindawi.com/journals/aaa/2015/108249/ Emran Tohidi, M. M. Ezadkhah, and S. Shateyi Copyright © 2015 Emran Tohidi et al. All rights reserved. Existence and Uniqueness of Fixed Point in Various Abstract Spaces and Related Applications Wed, 25 Mar 2015 08:40:16 +0000 http://www.hindawi.com/journals/aaa/2015/123984/ Erdal Karapınar, Wei-Shih Du, Poom Kumam, Adrian Petruşel, and Salvador Romaguera Copyright © 2015 Erdal Karapınar et al. All rights reserved. Stability and Bifurcation Analysis of Differential Equations and Its Applications Tue, 24 Mar 2015 11:36:48 +0000 http://www.hindawi.com/journals/aaa/2015/343528/ Yongli Song, Junling Ma, Yonghui Xia, Sanling Yuan, and Tonghua Zhang Copyright © 2015 Yongli Song et al. All rights reserved. Networked Systems with Incomplete Information Tue, 24 Mar 2015 09:15:23 +0000 http://www.hindawi.com/journals/aaa/2015/852103/ Zidong Wang, Bo Shen, Hongli Dong, Xiao He, and Jun Hu Copyright © 2015 Zidong Wang et al. All rights reserved. Recent Theory and Applications on Numerical Algorithms and Special Functions Tue, 24 Mar 2015 09:14:38 +0000 http://www.hindawi.com/journals/aaa/2015/101063/ Ali H. Bhrawy, Robert A. Van Gorder, Dumitru Baleanu, and Guo-Cheng Wu Copyright © 2015 Ali H. Bhrawy et al. All rights reserved. Bifurcation of Traveling Wave Solutions for (2+1)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy Mon, 23 Mar 2015 07:25:27 +0000 http://www.hindawi.com/journals/aaa/2015/820916/ Four (2+1)-dimensional nonlinear evolution equations, generated by the Jaulent-Miodek hierarchy, are investigated by the bifurcation method of planar dynamical systems. The bifurcation regions in different subsets of the parameters space are obtained. According to the different phase portraits in different regions, we obtain kink (antikink) wave solutions, solitary wave solutions, and periodic wave solutions for the third of these models by dynamical system method. Furthermore, the explicit exact expressions of these bounded traveling waves are obtained. All these wave solutions obtained are characterized by distinct physical structures. Yanping Ran, Jing Li, Xin Li, and Zheng Tian Copyright © 2015 Yanping Ran et al. All rights reserved. Monotone and Concave Positive Solutions to Three-Point Boundary Value Problems of Higher-Order Fractional Differential Equations Mon, 23 Mar 2015 07:02:43 +0000 http://www.hindawi.com/journals/aaa/2015/728491/ We study the three-point boundary value problem of higher-order fractional differential equations of the form , , , , , where is the Caputo fractional derivative of order , and the function is continuously differentiable. Here, , , . By virtue of some fixed point theorems, some sufficient criteria for the existence and multiplicity results of positive solutions are established and the obtained results also guarantee that the positive solutions discussed are monotone and concave. Wenyong Zhong and Lanfang Wang Copyright © 2015 Wenyong Zhong and Lanfang Wang. All rights reserved. Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model Sun, 22 Mar 2015 12:44:50 +0000 http://www.hindawi.com/journals/aaa/2015/354918/ We study a cancer model given by a three-dimensional system of ordinary differential equations, depending on eight parameters, which describe the interaction among healthy cells, tumour cells, and effector cells of immune system. The model was previously studied in the literature and was shown to have a chaotic attractor. In this paper we study how such a chaotic attractor is formed. More precisely, by varying one of the parameters, we prove that a supercritical Hopf bifurcation occurs, leading to the creation of a stable limit cycle. Then studying the continuation of this limit cycle we numerically found a cascade of period-doubling bifurcations which leads to the formation of the mentioned chaotic attractor. Moreover, analyzing the model dynamics from a biological point of view, we notice the possibility of both the tumour cells and the immune system cells to vanish and only the healthy cells survive, suggesting the possibility of cure, since the interactions with the immune system can eliminate tumour cells. Marluci Cristina Galindo, Cristiane Nespoli, and Marcelo Messias Copyright © 2015 Marluci Cristina Galindo et al. All rights reserved. Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials Sun, 22 Mar 2015 11:25:24 +0000 http://www.hindawi.com/journals/aaa/2015/672703/ The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms. W. M. Abd-Elhameed Copyright © 2015 W. M. Abd-Elhameed. All rights reserved. Analysis of the Structured Perturbation for the BCSCB Linear System Sun, 22 Mar 2015 10:55:58 +0000 http://www.hindawi.com/journals/aaa/2015/471362/ Circulant and block circulant type matrices are important tools in solving networked systems. In this paper, based on the style spectral decomposition of the basic circulant matrix and the basic skew circulant matrix, the block style spectral decomposition of the BCSCB matrix is obtained. And then, the structure perturbation is analysed, which includes the condition number and relative error of the BCSCB linear system. Then the optimal backward perturbation bound of the BCSCB linear system is discussed. Simultaneously, the algorithm for the optimal backward perturbation bound is given. Finally, a numerical example is provided to verify the effectiveness of the algorithm. Xia Tang and Zhaolin Jiang Copyright © 2015 Xia Tang and Zhaolin Jiang. All rights reserved. Fixed Point Theorems for Ćirić-Berinde Type Contractive Multivalued Mappings Sun, 22 Mar 2015 09:39:49 +0000 http://www.hindawi.com/journals/aaa/2015/768238/ We give a Ćirić-Berinde type contractive condition for multivalued mappings and analyze the existence of fixed point for these mappings. Seong-Hoon Cho Copyright © 2015 Seong-Hoon Cho. All rights reserved. Fixed Points for Multivalued Mappings in -Metric Spaces Sun, 22 Mar 2015 09:08:50 +0000 http://www.hindawi.com/journals/aaa/2015/718074/ In 2012, Samet et al. introduced the notion of α-ψ-contractive mapping and gave sufficient conditions for the existence of fixed points for this class of mappings. The purpose of our paper is to study the existence of fixed points for multivalued mappings, under an α-ψ-contractive condition of Ćirić type, in the setting of complete b-metric spaces. An application to integral equation is given. Mohamed Jleli, Bessem Samet, Calogero Vetro, and Francesca Vetro Copyright © 2015 Mohamed Jleli et al. All rights reserved. New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions Sun, 22 Mar 2015 09:06:13 +0000 http://www.hindawi.com/journals/aaa/2015/205452/ We consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form: , and . According to these conditions, the value of the unknown function at the left end point is proportional to its value at a nonlocal point while the value at an arbitrary (local) point is proportional to the contribution due to a substrip of arbitrary length . These conditions appear in the mathematical modelling of physical problems when different parts (nonlocal points and substrips of arbitrary length) of the domain are involved in the input data for the process under consideration. We discuss the existence of solutions for the given problem by means of the Sadovski fixed point theorem for condensing maps and a fixed point theorem due to O’Regan. Some illustrative examples are also presented. Ahmed Alsaedi, Sotiris K. Ntouyas, and Bashir Ahmad Copyright © 2015 Ahmed Alsaedi et al. All rights reserved. Best Proximity Points for Generalized Proximal Weak Contractions Satisfying Rational Expression on Ordered Metric Spaces Sun, 22 Mar 2015 09:00:59 +0000 http://www.hindawi.com/journals/aaa/2015/361657/ We introduce a generalized proximal weak contraction of rational type for the non-self-map and proved results to ensure the existence and uniqueness of best proximity point for such mappings in the setting of partially ordered metric spaces. Further, our results provides an extension of a result due to Luong and Thuan (2011) and also it provides an extension of Harjani (2010) to the case of self-mappings. V. Pragadeeswarar and M. Marudai Copyright © 2015 V. Pragadeeswarar and M. Marudai. All rights reserved. On Jacobsthal and Jacobsthal-Lucas Circulant Type Matrices Sun, 22 Mar 2015 08:56:32 +0000 http://www.hindawi.com/journals/aaa/2015/418293/ Circulant type matrices have become an important tool in solving fractional order differential equations. In this paper, we consider the circulant and left circulant and -circulant matrices with the Jacobsthal and Jacobsthal-Lucas numbers. First, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix. Furthermore, the invertibility of the left circulant and -circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and -circulant matrices by utilizing the relation between left circulant, -circulant matrices, and circulant matrix, respectively. Yanpeng Gong, Zhaolin Jiang, and Yun Gao Copyright © 2015 Yanpeng Gong et al. All rights reserved. Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time Scales Sun, 22 Mar 2015 08:56:00 +0000 http://www.hindawi.com/journals/aaa/2015/643706/ Tongxing Li, Josef Diblík, Alexander Domoshnitsky, Yuriy V. Rogovchenko, Felix Sadyrbaev, and Qi-Ru Wang Copyright © 2015 Tongxing Li et al. All rights reserved. Strong Convergence for the Split Common Fixed-Point Problem for Total Quasi-Asymptotically Nonexpansive Mappings in Hilbert Space Sun, 22 Mar 2015 08:50:06 +0000 http://www.hindawi.com/journals/aaa/2015/412318/ In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature. Lawan Bulama Mohammed and A. Kılıçman Copyright © 2015 Lawan Bulama Mohammed and A. Kılıçman. All rights reserved. Synchronization Transition and Traffic Congestion in One-Dimensional Traffic Model Sun, 22 Mar 2015 08:46:03 +0000 http://www.hindawi.com/journals/aaa/2015/167430/ A nonlinear car-following model with driver’s reaction time is studied from the synchronization transition viewpoint. We investigate the traffic congestion from the view of chaos system synchronization transition. Our result shows that the uniform flow corresponds to the complete synchronization and the stop-and-go congested state corresponds to the lag synchronization of the vehicles. An analytical criterion for synchronization manifolds stability is obtained; the analytical result and the numerical result are consistent. The synchronization transition is also trigged by the driver’s reaction time. We analyze the car-following model by the use of the nonlinear analysis method and derive the modified KdV equation describing the kink density wave. Zhi Xin and Jian Xu Copyright © 2015 Zhi Xin and Jian Xu. All rights reserved. Equalities and Inequalities for Norms of Block Imaginary Circulant Operator Matrices Thu, 19 Mar 2015 13:52:31 +0000 http://www.hindawi.com/journals/aaa/2015/521214/ Circulant, block circulant-type matrices and operator norms have become effective tools in solving networked systems. In this paper, the block imaginary circulant operator matrices are discussed. By utilizing the special structure of such matrices, several norm equalities and inequalities are presented. The norm in consideration is the weakly unitarily invariant norm, which satisfies . The usual operator norm and Schatten -norm are included. Furthermore, some special cases and examples are given. Xiaoyu Jiang and Kicheon Hong Copyright © 2015 Xiaoyu Jiang and Kicheon Hong. All rights reserved. Analysis, Filtering, and Control for Takagi-Sugeno Fuzzy Models in Networked Systems Thu, 19 Mar 2015 13:13:06 +0000 http://www.hindawi.com/journals/aaa/2015/856390/ The fuzzy logic theory has been proven to be effective in dealing with various nonlinear systems and has a great success in industry applications. Among different kinds of models for fuzzy systems, the so-called Takagi-Sugeno (T-S) fuzzy model has been quite popular due to its convenient and simple dynamic structure as well as its capability of approximating any smooth nonlinear function to any specified accuracy within any compact set. In terms of such a model, the performance analysis and the design of controllers and filters play important roles in the research of fuzzy systems. In this paper, we aim to survey some recent advances on the T-S fuzzy control and filtering problems with various network-induced phenomena. The network-induced phenomena under consideration mainly include communication delays, packet dropouts, signal quantization, and randomly occurring uncertainties (ROUs). With such network-induced phenomena, the developments on T-S fuzzy control and filtering issues are reviewed in detail. In addition, some latest results on this topic are highlighted. In the end, conclusions are drawn and some possible future research directions are pointed out. Sunjie Zhang, Zidong Wang, Jun Hu, Jinling Liang, and Fuad E. Alsaadi Copyright © 2015 Sunjie Zhang et al. All rights reserved.