Abstract and Applied Analysis http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. Optimal Control of Renewable Resources Based on the Effective Utilization Rate Thu, 16 Apr 2015 14:15:30 +0000 http://www.hindawi.com/journals/aaa/2015/369493/ The effective utilization rate of exploited renewable resources affects the final total revenue and the further exploitation of renewable resources. Considering the effective utilization rate, we propose an optimal control model for the exploitation of the renewable resources in this study. Firstly, we can prove that the novel model is nonsingular compared with the singular basic model. Secondly, we solve the novel model and obtain the optimal solution by Bang-Bang theory. Furthermore, we can determine the optimal total resources and the maximal total revenue. Finally, a numerical example is provided to verify the obtained theoretical results. Rui Wu, Zhengwei Shen, and Fucheng Liao Copyright © 2015 Rui Wu et al. All rights reserved. Corrigendum to “Krasnosel’skii Type Hybrid Fixed Point Theorems and Their Applications to Fractional Integral Equations” Thu, 16 Apr 2015 13:03:00 +0000 http://www.hindawi.com/journals/aaa/2015/467569/ H. M. Srivastava, Sachin V. Bedre, S. M. Khairnar, and B. S. Desale Copyright © 2015 H. M. Srivastava et al. All rights reserved. Reliable Control for Uncertain Singular Systems with Randomly Occurring Time-Varying Delay and Actuator Faults Thu, 16 Apr 2015 11:49:59 +0000 http://www.hindawi.com/journals/aaa/2015/238692/ The problem of reliable control is investigated for uncertain continuous singular systems with randomly occurring time-varying delay and actuator faults in this work. The delay occurs in a random way, and such randomly occurring delay obeys certain mutually uncorrelated Bernoulli distributed white noise sequences. The uncertainties under consideration are norm-bounded, and may vary with time. Then, with the constructed Lyapunov function, a sufficient condition is given to ensure the unforced system is mean-square exponentially stable and the corresponding controller can be derived from such condition, and the actuator faults problem is guaranteed. A numerical example is provided to show the effectiveness of the methods. Yu-Lin Li, Lin-Sheng Li, Zhi-Cheng Zhao, and Jing-Gang Zhang Copyright © 2015 Yu-Lin Li et al. All rights reserved. Approximate Controllability of Semilinear Impulsive Evolution Equations Thu, 16 Apr 2015 11:24:47 +0000 http://www.hindawi.com/journals/aaa/2015/797439/ We prove the approximate controllability of the following semilinear impulsive evolution equation: where , and are Hilbert spaces, , is a bounded linear operator, are smooth functions, and is an unbounded linear operator in which generates a strongly continuous semigroup . We suppose that is bounded and the linear system is approximately controllable on for all . Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on . Hugo Leiva Copyright © 2015 Hugo Leiva. All rights reserved. Fractional Cauchy Problem with Caputo Nabla Derivative on Time Scales Thu, 16 Apr 2015 08:22:52 +0000 http://www.hindawi.com/journals/aaa/2015/486054/ The definition of Caputo fractional derivative is given and some of its properties are discussed in detail. After then, the existence of the solution and the dependency of the solution upon the initial value for Cauchy type problem with fractional Caputo nabla derivative are studied. Also the explicit solutions to homogeneous equations and nonhomogeneous equations are derived by using Laplace transform method. Jiang Zhu and Ling Wu Copyright © 2015 Jiang Zhu and Ling Wu. All rights reserved. On the Fourier-Transformed Boltzmann Equation with Brownian Motion Thu, 16 Apr 2015 07:10:53 +0000 http://www.hindawi.com/journals/aaa/2015/318618/ We establish a global existence theorem, and uniqueness and stability of solutions of the Cauchy problem for the Fourier-transformed Fokker-Planck-Boltzmann equation with singular Maxwellian kernel, which may be viewed as a kinetic model for the stochastic time-evolution of characteristic functions governed by Brownian motion and collision dynamics. Yong-Kum Cho and Eunsil Kim Copyright © 2015 Yong-Kum Cho and Eunsil Kim. All rights reserved. Nonlinear Analysis: Optimization Methods, Convergence Theory, and Applications Thu, 16 Apr 2015 06:28:03 +0000 http://www.hindawi.com/journals/aaa/2015/429595/ Gonglin Yuan, Gaohang Yu, Neculai Andrei, Yunhai Xiao, and Li Zhang Copyright © 2015 Gonglin Yuan et al. All rights reserved. The Existence and Uniqueness of Global Solutions to the Initial Value Problem for the System of Nonlinear Integropartial Differential Equations in Spatial Economics: The Dynamic Continuous Dixit-Stiglitz-Krugman Model in an Urban-Rural Setting Wed, 15 Apr 2015 14:10:19 +0000 http://www.hindawi.com/journals/aaa/2015/760136/ Assume that economic activities are conducted in a bounded continuous domain where workers move toward regions that offer higher real wages and away from regions that offer below-average real wages. The density of real wages is calculated by solving the nominal wage equation of the continuous Dixit-Stiglitz-Krugman model in an urban-rural setting. The evolution of the density of workers is described by an unknown function of the replicator equation whose growth rate is equal to the difference between the density of real wages and the average real wage. Hence, the evolution of the densities of workers and real wages is described by the system of the nominal wage equation and the replicator equation. This system of equations is an essentially new kind of system of nonlinear integropartial differential equations in the theory of functional equations. The purpose of this paper is to obtain a sufficient condition for the initial value problem for this system to have a unique global solution. Minoru Tabata and Nobuoki Eshima Copyright © 2015 Minoru Tabata and Nobuoki Eshima. All rights reserved. A Smoothing Inexact Newton Method for Nonlinear Complementarity Problems Wed, 15 Apr 2015 11:15:02 +0000 http://www.hindawi.com/journals/aaa/2015/731026/ A smoothing inexact Newton method is presented for solving nonlinear complementarity problems. Different from the existing exact methods, the associated subproblems are not necessary to be exactly solved to obtain the search directions. Under suitable assumptions, global convergence and superlinear convergence are established for the developed inexact algorithm, which are extensions of the exact case. On the one hand, results of numerical experiments indicate that our algorithm is effective for the benchmark test problems available in the literature. On the other hand, suitable choice of inexact parameters can improve the numerical performance of the developed algorithm. Zhong Wan, HuanHuan Li, and Shuai Huang Copyright © 2015 Zhong Wan et al. All rights reserved. A Note on Continuity of Solution Set for Parametric Weak Vector Equilibrium Problems Wed, 15 Apr 2015 07:06:05 +0000 http://www.hindawi.com/journals/aaa/2015/503091/ We consider the parametric weak vector equilibrium problem. By using a weaker assumption of Peng and Chang (2014), the sufficient conditions for continuity of the solution mappings to a parametric weak vector equilibrium problem are established. Examples are provided to illustrate the essentialness of imposed assumptions. As advantages of the results, we derive the continuity of solution mappings for vector optimization problems. Pakkapon Preechasilp and Rabian Wangkeeree Copyright © Pakkapon Preechasilp and Rabian Wangkeeree. All rights reserved. On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation Tue, 14 Apr 2015 17:02:54 +0000 http://www.hindawi.com/journals/aaa/2015/219616/ We develop the Newton-Kantorovich method to solve the system of nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method. Hameed Husam Hameed, Z. K. Eshkuvatov, Anvarjon Ahmedov, and N. M. A. Nik Long Copyright © 2015 Hameed Husam Hameed et al. All rights reserved. On the Relation between Phase-Type Distributions and Positive Systems Tue, 14 Apr 2015 11:51:02 +0000 http://www.hindawi.com/journals/aaa/2015/731261/ The relation between phase-type representation and positive system realization in both the discrete and continuous time is discussed. Using the Perron-Frobenius theorem of nonnegative matrix theory, a transformation from positive realization to phase-type realization is derived under the excitability condition. In order to explain the connection, some useful properties and characteristics such as irreducibility, excitability, transparency, and order reduction for positive realization and phase-type representation are discussed. In addition, the connection between the phase-type renewal process and the feedback positive system is discussed in the stabilization concept. Kyungsup Kim Copyright © 2015 Kyungsup Kim. 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.