Advances in Mathematical Physics https://www.hindawi.com The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. Guaranteed Cost Finite-Time Control of Fractional-Order Positive Switched Systems Wed, 24 May 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/2695894/ The problem of guaranteed cost finite-time control of fractional-order positive switched systems (FOPSS) is considered in this paper. Firstly, a new cost function is defined. Then, by constructing linear copositive Lyapunov functions and using the average dwell time (ADT) approach, a state feedback controller and a static output feedback controller are constructed, respectively, and sufficient conditions are derived to guarantee that the corresponding closed-loop systems are guaranteed cost finite-time stable (GCFTS). Such conditions can be easily solved by linear programming. Finally, two examples are given to illustrate the effectiveness of the proposed method. Leipo Liu, Xiangyang Cao, Zhumu Fu, and Shuzhong Song Copyright © 2017 Leipo Liu et al. All rights reserved. Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping Tue, 23 May 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/2708483/ We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of -norm. Furthermore, if, additionally, -norm () of the initial perturbation is finite, we also prove the optimal decay rates for such a solution without the additional technical assumptions for the nonlinear damping given by Li and Saxton. Hongjun Qiu and Yinghui Zhang Copyright © 2017 Hongjun Qiu and Yinghui Zhang. All rights reserved. On Certain Translation Invariant Properties of Interior Transmission Spectra and Their Doppler’s Effect Sun, 21 May 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/3838507/ We study the translation invariant properties of the eigenvalues of scattering transmission problem. We examine the functional derivative of the eigenvalue density function to the defining index of refraction . By the limit behaviors in frequency sphere, we prove some results on the inverse uniqueness of index of refraction. In physics, Doppler’s effect connects the variation of the frequency/eigenvalue and the motion velocity/variation of position variable. In this paper, we proved the functional derivative Lung-Hui Chen Copyright © 2017 Lung-Hui Chen. All rights reserved. The Adaptive-Clustering and Error-Correction Method for Forecasting Cyanobacteria Blooms in Lakes and Reservoirs Thu, 04 May 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/9037358/ Globally, cyanobacteria blooms frequently occur, and effective prediction of cyanobacteria blooms in lakes and reservoirs could constitute an essential proactive strategy for water-resource protection. However, cyanobacteria blooms are very complicated because of the internal stochastic nature of the system evolution and the external uncertainty of the observation data. In this study, an adaptive-clustering algorithm is introduced to obtain some typical operating intervals. In addition, the number of nearest neighbors used for modeling was optimized by particle swarm optimization. Finally, a fuzzy linear regression method based on error-correction was used to revise the model dynamically near the operating point. We found that the combined method can characterize the evolutionary track of cyanobacteria blooms in lakes and reservoirs. The model constructed in this paper is compared to other cyanobacteria-bloom forecasting methods (e.g., phase space reconstruction and traditional-clustering linear regression), and, then, the average relative error and average absolute error are used to compare the accuracies of these models. The results suggest that the proposed model is superior. As such, the newly developed approach achieves more precise predictions, which can be used to prevent the further deterioration of the water environment. Xiao-zhe Bai, Hui-yan Zhang, Xiao-yi Wang, Li Wang, Ji-ping Xu, and Jia-bin Yu Copyright © 2017 Xiao-zhe Bai et al. All rights reserved. Error Estimates on Hybridizable Discontinuous Galerkin Methods for Parabolic Equations with Nonlinear Coefficients Wed, 03 May 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/9736818/ HDG method has been widely used as an effective numerical technique to obtain physically relevant solutions for PDE. In a practical setting, PDE comes with nonlinear coefficients. Hence, it is inevitable to consider how to obtain an approximate solution for PDE with nonlinear coefficients. Research on using HDG method for PDE with nonlinear coefficients has been conducted along with results obtained from computer simulations. However, error analysis on HDG method for such settings has been limited. In this research, we give error estimations of the hybridizable discontinuous Galerkin (HDG) method for parabolic equations with nonlinear coefficients. We first review the classical HDG method and define notions that will be used throughout the paper. Then, we will give bounds for our estimates when nonlinear coefficients obey “Lipschitz” condition. We will then prove our main result that the errors for our estimations are bounded. Minam Moon, Hyung Kyu Jun, and Tay Suh Copyright © 2017 Minam Moon et al. All rights reserved. Remarks on Pressure Blow-Up Criterion of the 3D Zero-Diffusion Boussinesq Equations in Margin Besov Spaces Mon, 24 Apr 2017 06:02:54 +0000 http://www.hindawi.com/journals/amp/2017/6754780/ This study is focused on the pressure blow-up criterion for a smooth solution of three-dimensional zero-diffusion Boussinesq equations. With the aid of Littlewood-Paley decomposition together with the energy methods, it is proved that if the pressure satisfies the following condition on margin Besov spaces, for then the smooth solution can be continually extended to the interval for some . The findings extend largely the previous results. Min Fu and Chao Cai Copyright © 2017 Min Fu and Chao Cai. All rights reserved. Recent Advances in Symmetry Analysis and Exact Solutions in Nonlinear Mathematical Physics Sun, 16 Apr 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/8325854/ Maria Bruzón, Chaudry M. Khalique, Maria L. Gandarias, Rita Traciná, and Mariano Torrisi Copyright © 2017 Maria Bruzón et al. All rights reserved. Color Confinement and Spatial Dimensions in the Complex-Sedenion Space Wed, 12 Apr 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/9876464/ The paper aims to apply the complex-sedenions to explore the wave functions and field equations of non-Abelian gauge fields, considering the spatial dimensions of a unit vector as the color degrees of freedom in the complex-quaternion wave functions, exploring the physical properties of the color confinement essentially. J. C. Maxwell was the first to employ the quaternions to study the electromagnetic fields. His method inspires subsequent scholars to introduce the quaternions, octonions, and sedenions to research the electromagnetic field, gravitational field, and nuclear field. The application of complex-sedenions is capable of depicting not only the field equations of classical mechanics, but also the field equations of quantum mechanics. The latter can be degenerated into the Dirac equation and Yang-Mills equation. In contrast to the complex-number wave function, the complex-quaternion wave function possesses three new degrees of freedom, that is, three color degrees of freedom. One complex-quaternion wave function is equivalent to three complex-number wave functions. It means that the three spatial dimensions of unit vector in the complex-quaternion wave function can be considered as the “three colors”; naturally the color confinement will be effective. In other words, in the complex-quaternion space, the “three colors” are only the spatial dimensions, rather than any property of physical substance. Zi-Hua Weng Copyright © 2017 Zi-Hua Weng. All rights reserved. Computation of the Stability and Complexity about Triopoly Price Game Model with Delay Decision Mon, 10 Apr 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/2717352/ We develop the price game model based on the entropy theory and chaos theory, considering the three enterprises are bounded rationality and using the cost function under the resource constraints; that is, the yield increase will bring increased costs. The enterprises of new model adopt the delay decision with the delay parameters and , respectively. According to the change of delay parameters and , the bifurcation, stability, and chaos of the system are discussed, and the change of entropy when the system is far away from equilibrium is considered. Prices and profits are found to lose stability and the evolution of the system tends to the equilibrium state of maximum entropy. And it has a big fluctuation with the increase of and . In the end, the chaos is controlled effectively. The entropy of the system decreases, and the interior reverts to order. The results of this study are of great significance for avoiding the chaos when the enterprises make price decisions. Yuling Wang, Shubing Guo, and Shide Duan Copyright © 2017 Yuling Wang et al. All rights reserved. The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam Mon, 10 Apr 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/5214616/ The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems. Y. H. Qian, J. L. Pan, S. P. Chen, and M. H. Yao Copyright © 2017 Y. H. Qian et al. All rights reserved. Robust Genetic Circuit Design: A Mixed and IQC Analysis Thu, 06 Apr 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/5757943/ This paper considers the problem of designing a genetic circuit which is robust to noise effect. To achieve this goal, a mixed and Integral Quadratic Constraints (IQC) approach is proposed. In order to minimize the effects of external noise on the genetic regulatory network in terms of norm, a design procedure of Hill coefficients in the promoters is presented. The IQC approach is introduced to analyze and guarantee the stability of the designed circuit. Ruijuan Chen, Wei Pan, Dongfei Fu, and Xin He Copyright © 2017 Ruijuan Chen et al. All rights reserved. Robustness of Supercavitating Vehicles Based on Multistability Analysis Wed, 05 Apr 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/6894041/ Supercavity can increase speed of underwater vehicles greatly. However, external interferences always lead to instability of vehicles. This paper focuses on robustness of supercavitating vehicles. Based on a 4-dimensional dynamic model, the existence of multistability is verified in supercavitating system through simulation, and the robustness of vehicles varying with parameters is analyzed by basins of attraction. Results of the research disclose that the supercavitating system has three stable states in some regions of parameters space, namely, stable, periodic, and chaotic states, while in other regions it has various multistability, such as coexistence of two types of stable equilibrium points, coexistence of a limit cycle with a chaotic attractor, and coexistence of 1-periodic cycle with 2-periodic cycle. Provided that cavitation number varies within a small range, with increase of the feedback control gain of fin deflection angle, size of basin of attraction becomes smaller and robustness of the system becomes weaker. In practical application, robustness of supercavitating vehicles can be improved by setting parameters of system or adjusting initial launching conditions. Yipin Lv, Tianhong Xiong, Wenjun Yi, and Jun Guan Copyright © 2017 Yipin Lv et al. All rights reserved. Modeling Anomalous Diffusion by a Subordinated Integrated Brownian Motion Tue, 04 Apr 2017 06:45:08 +0000 http://www.hindawi.com/journals/amp/2017/7246865/ We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times are correlated. In a continuum limit, the process can be defined by an integrated Brownian motion subordinated by an inverse -stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when , normal diffusion when , and superdiffusion when . The time-averaged mean square displacement is also employed to show weak ergodicity breaking occurring in the proposed process. An extension to the fractional case is also considered. Long Shi and Aiguo Xiao Copyright © 2017 Long Shi and Aiguo Xiao. All rights reserved. Application of Adjoint Data Assimilation Method to Atmospheric Aerosol Transport Problems Thu, 30 Mar 2017 06:17:05 +0000 http://www.hindawi.com/journals/amp/2017/5865403/ We propose combining the adjoint assimilation method with characteristic finite difference scheme (CFD) to solve the aerosol transport problems, which can predict the distribution of atmospheric aerosols efficiently by using large time steps. Firstly, the characteristic finite difference scheme (CFD) is tested to compute the Gaussian hump using large time step sizes and is compared with the first-order upwind scheme (US1) using small time steps; the US1 method gets error of 0.2887 using , while CFD method gets a much smaller of 0.2280 using a much larger time step . Then, the initial distribution of concentration is inverted by the adjoint assimilation method with CFD and US1. The adjoint assimilation method with CFD gets better accuracy than adjoint assimilation method with US1 while adjoint assimilation method with CFD costs much less computational time. Further, a real case of concentration distribution in China during the APEC 2014 is simulated by using adjoint assimilation method with CFD. The simulation results are in good agreement with the observed values. The adjoint assimilation method with CFD can solve large scale aerosol transport problem efficiently. Minjie Xu, Kai Fu, and Xianqing Lv Copyright © 2017 Minjie Xu et al. All rights reserved. Adaptive Modified Function Projective Lag Synchronization of Memristor-Based Five-Order Chaotic Circuit Systems Thu, 23 Mar 2017 09:17:17 +0000 http://www.hindawi.com/journals/amp/2017/1843179/ The modified function projective lag synchronization of the memristor-based five-order chaotic circuit system with unknown bounded disturbances is investigated. Based on the LMI approach and Lyapunov stability theorem, an adaptive control law is established to make the states of two different memristor-based five-order chaotic circuit systems asymptotically synchronized up to a desired scaling function matrix, while the parameter controlling strength update law is designed to estimate the parameters well. Finally, the simulation is put forward to demonstrate the correctness and effectiveness of the proposed methods. The control method involved is simple and practical. Qiaoping Li and Sanyang Liu Copyright © 2017 Qiaoping Li and Sanyang Liu. All rights reserved. Numerical Simulations of the Square Lid Driven Cavity Flow of Bingham Fluids Using Nonconforming Finite Elements Coupled with a Direct Solver Mon, 20 Mar 2017 08:56:53 +0000 http://www.hindawi.com/journals/amp/2017/5210708/ In this paper, numerical simulations are performed in a single and double lid driven square cavity to study the flow of a Bingham viscoplastic fluid. The governing equations are discretized with the help of finite element method in space and the nonconforming Stokes element is utilized which gives 2nd-order accuracy for velocity and 1st-order accuracy for pressure. The discretized systems of nonlinear equations are treated by using the Newton method and the inner linear subproablems are solved by the direct solver UMFPACK. A qualitative comparison is done with the results reported in the literature. In addition to these comparisons, some new reference data for the kinetic energy is generated. All these implementations are done in the open source software package FEATFLOW which is a general purpose finite element based solver package for solving partial differential equations. R. Mahmood, N. Kousar, M. Yaqub, and K. Jabeen Copyright © 2017 R. Mahmood et al. All rights reserved. Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation Mon, 20 Mar 2017 07:45:46 +0000 http://www.hindawi.com/journals/amp/2017/6743276/ The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalized -expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations. Dianchen Lu, Chen Yue, and Muhammad Arshad Copyright © 2017 Dianchen Lu et al. All rights reserved. Existence and Uniqueness of Periodic Solutions for a Kind of Second-Order Neutral Functional Differential Equation with Delays Sun, 19 Mar 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/9591087/ We consider a kind of second-order neutral functional differential equation. On the basis of Mawhin’s coincidence degree, the existence and uniqueness of periodic solutions are proved. It is indicated that the result is related to the deviating arguments. Moreover, we present two simulations to demonstrate the validity of analytical conclusion. Na Wang Copyright © 2017 Na Wang. All rights reserved. The Study of Seismic Response on Accelerated Contained Fluid Sun, 19 Mar 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/1492035/ Ground-supported cylindrical tanks are strategically very important structures used to store a variety of liquids. This paper presents the theoretical background of fluid effect on tank when a fluid container is subjected to horizontal acceleration. Fluid excites the hydrodynamic (impulsive and convective) pressures, impulsive and convective (sloshing) actions. Seismic response of cylindrical fluid filling tanks fixed to rigid foundations was calculated for variation of the tank slenderness parameter. The calculating procedure has been adopted in Eurocode 8. Kamila Kotrasova and Eva Kormanikova Copyright © 2017 Kamila Kotrasova and Eva Kormanikova. All rights reserved. Fully Discrete Local Discontinuous Galerkin Approximation for Time-Space Fractional Subdiffusion/Superdiffusion Equations Thu, 16 Mar 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/4961797/ We focus on developing the finite difference (i.e., backward Euler difference or second-order central difference)/local discontinuous Galerkin finite element mixed method to construct and analyze a kind of efficient, accurate, flexible, numerical schemes for approximately solving time-space fractional subdiffusion/superdiffusion equations. Discretizing the time Caputo fractional derivative by using the backward Euler difference for the derivative parameter () or second-order central difference method for (), combined with local discontinuous Galerkin method to approximate the spatial derivative which is defined by a fractional Laplacian operator, two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusion/superdiffusion equations are proposed, respectively. Through the mathematical induction method, we show the concrete analysis for the stability and the convergence under the norm of the LDG schemes. Several numerical experiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes. The numerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerful for solving fractional partial differential equations. Meilan Qiu, Liquan Mei, and Dewang Li Copyright © 2017 Meilan Qiu et al. All rights reserved. The Dynamical Behavior of a Rigid Body Relative Equilibrium Position Wed, 08 Mar 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/8070525/ In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures. T. S. Amer Copyright © 2017 T. S. Amer. All rights reserved. The Convergence Ball and Error Analysis of the Relaxed Secant Method Wed, 08 Mar 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/6976205/ A relaxed secant method is proposed. Radius estimate of the convergence ball of the relaxed secant method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given. Rongfei Lin, Qingbiao Wu, Minhong Chen, and Lu Liu Copyright © 2017 Rongfei Lin et al. All rights reserved. A New Analysis Method for Chemotaxis-Induced Instability in Multispecies Host-Parasitoid Systems Tue, 28 Feb 2017 14:20:05 +0000 http://www.hindawi.com/journals/amp/2017/9685013/ We propose a new method applying matrix theory to analyse the instability conditions of unique homogeneous coexistent state of multispecies host-parasitoid systems. We consider the eigenvalues of linearized operator of systems, and by dimensionality reduction, this infinite dimensional eigenproblem will be reduced to a parametrized finite dimensional eigenproblem, thereby applying combinatorial matrix theory to analyse the linear instability of such constant steady-state. Huaihuo Cao Copyright © 2017 Huaihuo Cao. All rights reserved. Quasi-Exact Coulomb Dynamics of Charges of Which Are Equal Tue, 28 Feb 2017 10:12:07 +0000 http://www.hindawi.com/journals/amp/2017/6751025/ For point charges of which are negative and equal quasi-exact periodic solutions of their Coulomb equation of motion are found. These solutions describe a motion of the negative charges around a coordinate axis in such a way that their coordinates coincide with vertices of a regular polygon in planes perpendicular to the axis along which the positive charge moves. The Weinstein and center Lyapunov theorems are utilized. Wolodymyr Skrypnik Copyright © 2017 Wolodymyr Skrypnik. All rights reserved. Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions Wed, 22 Feb 2017 13:37:29 +0000 http://www.hindawi.com/journals/amp/2017/5691452/ A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method. Bashar Zogheib, Emran Tohidi, and Stanford Shateyi Copyright © 2017 Bashar Zogheib et al. All rights reserved. Third-Order Approximate Solution of Chemical Reaction-Diffusion Brusselator System Using Optimal Homotopy Asymptotic Method Wed, 22 Feb 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/3895701/ The objective of this paper is to investigate the effectiveness and performance of optimal homotopy asymptotic method in solving a system of nonlinear partial differential equations. Since mathematical modeling of certain chemical reaction-diffusion experiments leads to Brusselator equations, it is worth demanding a new technique to solve such a system. We construct a new efficient recurrent relation to solve nonlinear Brusselator system of equations. It is observed that the method is easy to implement and quite valuable for handling nonlinear system of partial differential equations and yielding excellent results at minimum computational cost. Analytical solutions of Brusselator system are presented to demonstrate the viability and practical usefulness of the method. The results reveal that the method is explicit, effective, and easy to use. Salem Alkhalaf Copyright © 2017 Salem Alkhalaf. All rights reserved. Hyers-Ulam Stability and Existence of Solutions for Nigmatullin’s Fractional Diffusion Equation Tue, 21 Feb 2017 00:00:00 +0000 http://www.hindawi.com/journals/amp/2017/9692685/ We discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability of time-fractional order heat conduction equations via fractional Green function involving Wright function. In addition, an interesting existence result for solution is given. Zhuoyan Gao and JinRong Wang Copyright © 2017 Zhuoyan Gao and JinRong Wang. All rights reserved. The Cauchy Problem for Space-Time Monopole Equations in Temporal and Spatial Gauge Mon, 20 Feb 2017 06:39:57 +0000 http://www.hindawi.com/journals/amp/2017/4109645/ We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition and the temporal gauge condition . Hyungjin Huh and Jihyun Yim Copyright © 2017 Hyungjin Huh and Jihyun Yim. All rights reserved. Turing Bifurcation and Pattern Formation of Stochastic Reaction-Diffusion System Sun, 12 Feb 2017 09:22:50 +0000 http://www.hindawi.com/journals/amp/2017/9648538/ Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction-Diffusion System (RDS), Stochastic Reaction-Diffusion System (SRDS) is more complex and it is very difficult to deal with the noise function. In this paper, we have presented a method to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation. Qianiqian Zheng, Zhijie Wang, Jianwei Shen, and Hussain Muhammad Ather Iqbal Copyright © 2017 Qianiqian Zheng et al. All rights reserved. Third-Order Conditional Lie–Bäcklund Symmetries of Nonlinear Reaction-Diffusion Equations Thu, 09 Feb 2017 13:19:02 +0000 http://www.hindawi.com/journals/amp/2017/2825416/ The third-order conditional Lie–Bäcklund symmetries of nonlinear reaction-diffusion equations are constructed due to the method of linear determining equations. As a consequence, the exact solutions of the resulting equations are derived due to the compatibility of the governing equations and the admitted differential constraints, which are resting on the characteristic of the admitted conditional Lie–Bäcklund symmetries to be zero. Keqin Su and Jie Cao Copyright © 2017 Keqin Su and Jie Cao. All rights reserved.