Advances in Mathematical Physics The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Description of the Magnetic Field and Divergence of Multisolenoid Aharonov-Bohm Potential Tue, 09 Feb 2016 08:36:20 +0000 Explicit formulas for the magnetic field and divergence of multisolenoid Aharonov-Bohm potential are obtained; the mathematical essence of this potential is explained. It is shown that the magnetic field and divergence of this potential are very singular generalized functions concentrated at a finite number of thin solenoids. Deficiency index is found for the minimal operator generated by the Aharonov-Bohm differential expression. Araz R. Aliev, Elshad H. Eyvazov, Said F. M. Ibrahim, and Hassan A. Zedan Copyright © 2016 Araz R. Aliev et al. All rights reserved. Kubo Fluctuation Relations in the Generalized Elastic Model Sun, 31 Jan 2016 09:07:31 +0000 The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. In this paper we show that the Fractional Langevin Equation (FLE) is a suitable framework for the study of the tracer (probe) particle dynamics, when an external force acts only on a single point (tagged probe) belonging to the system. With the help of the Fox function formalism we study the scaling behaviour of the noise- and force-propagators for large and short times (distances). We show that the Kubo fluctuation relations are exactly fulfilled when a time periodic force is exerted on the tagged probe. Most importantly, by studying the large and low frequency behaviour of the complex mobility we illustrate surprising nontrivial physical scenarios. Our analysis shows that the system splits into two distinct regions whose size depends on the applied frequency, characterized by very different response to the periodic perturbation exerted, both in the phase shift and in the amplitude. Alessandro Taloni Copyright © 2016 Alessandro Taloni. All rights reserved. A Comparative Approach to the Solution of the Zabolotskaya-Khokhlov Equation by Iteration Methods Tue, 26 Jan 2016 13:58:04 +0000 We employed different iteration methods like Homotopy Analysis Method (HAM), Adomian Decomposition Method (ADM), and Variational Iteration Method (VIM) to find the approximate solution to the Zabolotskaya-Khokhlov (ZK) equation. Iteration methods are used to solve linear and nonlinear PDEs whose classical methods are either very complex or too limited to apply. A comparison study has been made to see which of these methods converges to the approximate solution rapidly. The result revealed that, amongst these methods, ADM is more effective and simpler tool in its nature which does not require any transformation or linearization. Saeed Ahmed and Muhammad Kalim Copyright © 2016 Saeed Ahmed and Muhammad Kalim. All rights reserved. Explicit Solution of Reinsurance-Investment Problem for an Insurer with Dynamic Income under Vasicek Model Tue, 26 Jan 2016 13:46:24 +0000 Unlike traditionally used reserves models, this paper focuses on a reserve process with dynamic income to study the reinsurance-investment problem for an insurer under Vasicek stochastic interest rate model. The insurer’s dynamic income is given by the remainder after a dynamic reward budget being subtracted from the insurer’s net premium which is calculated according to expected premium principle. Applying stochastic control technique, a Hamilton-Jacobi-Bellman equation is established and the explicit solution is obtained under the objective of maximizing the insurer’s power utility of terminal wealth. Some economic interpretations of the obtained results are explained in detail. In addition, numerical analysis and several graphics are given to illustrate our results more meticulous. De-Lei Sheng Copyright © 2016 De-Lei Sheng. All rights reserved. Tight --Frame and Its Novel Characterizations via Atomic Systems Tue, 26 Jan 2016 07:15:57 +0000 -frame is a generalization of -frame. We generalize the tight -frame to --frame via atomic systems. In this paper, the definition of tight --frame is put forward; equivalent characterizations and necessary conditions of tight --frame are given. In particular, the necessary and sufficient condition for tight --frame being tight -frame is obtained. Finally, by means of methods and techniques of frame theory, several properties of tight --frame are given. Yongdong Huang and Dingli Hua Copyright © 2016 Yongdong Huang and Dingli Hua. All rights reserved. Singularity Analysis for a Class of Porous Medium Equation with Time-Dependent Coefficients Tue, 19 Jan 2016 09:17:04 +0000 This paper concerns the singularity and global regularity for the porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary conditions. Firstly, some global regularity results are established. Furthermore, we investigate the blow-up solution to the boundary value problem. The upper and lower estimates to the lifespan of the singular solution are also obtained here. Anyin Xia, Xianxiang Pu, and Shan Li Copyright © 2016 Anyin Xia et al. All rights reserved. Finite Time Control for Fractional Order Nonlinear Hydroturbine Governing System via Frequency Distributed Model Sun, 17 Jan 2016 08:41:05 +0000 This paper studies the application of frequency distributed model for finite time control of a fractional order nonlinear hydroturbine governing system (HGS). Firstly, the mathematical model of HGS with external random disturbances is introduced. Secondly, a novel terminal sliding surface is proposed and its stability to origin is proved based on the frequency distributed model and Lyapunov stability theory. Furthermore, based on finite time stability and sliding mode control theory, a robust control law to ensure the occurrence of the sliding motion in a finite time is designed for stabilization of the fractional order HGS. Finally, simulation results show the effectiveness and robustness of the proposed scheme. Bin Wang, Lin Yin, Shaojie Wang, Shirui Miao, Tantan Du, and Chao Zuo Copyright © 2016 Bin Wang et al. All rights reserved. Inverse Uniqueness in Interior Transmission Problem and Its Eigenvalue Tunneling in Simple Domain Tue, 12 Jan 2016 09:25:16 +0000 We study inverse uniqueness with a knowledge of spectral data of an interior transmission problem in a penetrable simple domain. We expand the solution in a series of one-dimensional problems in the far-fields. We define an ODE by restricting the PDE along a fixed scattered direction. Accordingly, we obtain a Sturm-Liouville problem for each scattered direction. There exists the correspondence between the ODE spectrum and the PDE spectrum. We deduce the inverse uniqueness on the index of refraction from the discussion on the uniqueness anglewise of the Strum-Liouville problem. Lung-Hui Chen Copyright © 2016 Lung-Hui Chen. All rights reserved. Upper Bounds on the Degeneracy of the Ground State in Quantum Field Models Wed, 06 Jan 2016 14:05:08 +0000 Axiomatic abstract formulations are presented to derive upper bounds on the degeneracy of the ground state in quantum field models including massless ones. In particular, given is a sufficient condition under which the degeneracy of the ground state of the perturbed Hamiltonian is less than or equal to the degeneracy of the ground state of the unperturbed one. Applications of the abstract theory to models in quantum field theory are outlined. Asao Arai and Daiju Funakawa Copyright © 2016 Asao Arai and Daiju Funakawa. All rights reserved. Seiberg-Witten Like Equations on Pseudo-Riemannian Manifolds with Structure Wed, 06 Jan 2016 13:28:13 +0000 We consider 7-dimensional pseudo-Riemannian manifolds with structure group . On such manifolds, the space of 2-forms splits orthogonally into components . We define self-duality of a 2-form by considering the part as the bundle of self-dual 2-forms. We express the spinor bundle and the Dirac operator and write down Seiberg-Witten like equations on such manifolds. Finally we get explicit forms of these equations on and give some solutions. Nülifer Özdemir and Nedim Deǧirmenci Copyright © 2016 Nülifer Özdemir and Nedim Deǧirmenci. All rights reserved. MHD Flow due to the Nonlinear Stretching of a Porous Sheet Tue, 05 Jan 2016 14:13:45 +0000 The MHD flow due to the nonlinear stretching of a porous sheet is investigated. A closed form solution is obtained when the stretching rate is inversely proportional to the distance from the origin. Otherwise a uniformly valid asymptotic expansion, for large magnetic interaction number , is developed. It coincides with a homotopy perturbation expansion for the problem. The asymptotic/homotopy perturbation expansion gives results in excellent agreement with accurate numerical results, for large as well as small values of . For large , the expansion, being asymptotic, needs a small number of terms, regardless of the mass transfer rate or the degree of nonlinearity. For small , the expansion is a homotopy perturbation one. It needs considerably increasing number of terms with higher injection rates and/or with stretching rates approaching the inverse proportionality. It may even fail. Tarek M. A. El-Mistikawy Copyright © 2016 Tarek M. A. El-Mistikawy. All rights reserved. Bi-Integrable Couplings of a New Soliton Hierarchy Associated with Thu, 31 Dec 2015 08:25:20 +0000 Based on the six-dimensional real special orthogonal Lie algebra , a new Lax integrable hierarchy is obtained by constructing an isospectral problem. Furthermore, we construct bi-integrable couplings for this hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Hamiltonian structures of the obtained bi-integrable couplings are constructed by the variational identity. Yan Cao, Liangyun Chen, and Baiying He Copyright © 2015 Yan Cao et al. All rights reserved. Properties of Stark Resonant States in Exactly Solvable Systems Wed, 30 Dec 2015 12:47:51 +0000 Properties of Stark resonant states are studied in two exactly solvable systems. These resonances are shown to form a biorthogonal system with respect to a pairing defined by a contour integral that selects states with outgoing wave boundary conditions. Analytic expressions are derived for the pseudonorm, dipole moment, and coupling matrix elements which relate systems with different strengths of the external field. All results are based on explicit calculations made possible by a newly designed integration method for combinations of Airy functions representing resonant eigenstates. Generalizations for one-dimensional systems with short-range potentials are presented, and relations are identified which are likely to hold in systems with three spatial dimensions. Jeffrey M. Brown and Miroslav Kolesik Copyright © 2015 Jeffrey M. Brown and Miroslav Kolesik. All rights reserved. -Symmetric Laplace Operators on Star Graphs: Real Spectrum and Self-Adjointness Sun, 27 Dec 2015 13:00:55 +0000 How ideas of -symmetric quantum mechanics can be applied to quantum graphs is analyzed, in particular to the star graph. The class of rotationally symmetric vertex conditions is analyzed. It is shown that all such conditions can effectively be described by circulant matrices: real in the case of odd number of edges and complex having particular block structure in the even case. Spectral properties of the corresponding operators are discussed. Maria Astudillo, Pavel Kurasov, and Muhammad Usman Copyright © 2015 Maria Astudillo et al. All rights reserved. Stability of Negative Solitary Waves for a Generalized Camassa-Holm Equation with Quartic Nonlinearity Thu, 24 Dec 2015 15:18:23 +0000 We consider the stability of negative solitary waves to a generalized Camassa-Holm equation with quartic nonlinearity. We obtain the existence of negative solitary waves for any wave speed and some of their qualitative properties and then prove that they are orbitally stable by using a method proposed by Grillakis et al. Shan Zheng and Zhengyong Ouyang Copyright © 2015 Shan Zheng and Zhengyong Ouyang. All rights reserved. Random 2D Composites and the Generalized Method of Schwarz Mon, 21 Dec 2015 07:08:30 +0000 Two-phase composites with nonoverlapping inclusions randomly embedded in matrix are investigated. A straightforward approach is applied to estimate the effective properties of random 2D composites. First, deterministic boundary value problems are solved for all locations of inclusions, that is, for all events of the considered probabilistic space by the generalized method of Schwarz. Second, the effective properties are calculated in analytical form and averaged over . This method is related to the traditional method based on the average probabilistic values involving the -point correlation functions. However, we avoid computation of the correlation functions and compute their weighted moments of high orders by an indirect method which does not address the correlation functions. The effective properties are exactly expressed through these moments. It is proved that the generalized method of Schwarz converges for an arbitrary multiply connected doubly periodic domain and for an arbitrary contrast parameter. The proposed method yields an algorithm which can be applied with symbolic computations. The Torquato-Milton parameter is exactly written for circular inclusions. Vladimir Mityushev Copyright © 2015 Vladimir Mityushev. All rights reserved. Local Strong Solution for a Class of Shear Thickening Fluids with Non-Newtonian Potential and Heat-Conducting Wed, 16 Dec 2015 09:35:33 +0000 The aim of this paper is to discuss the model for a class of shear thickening fluids with non-Newtonian potential and heat-conducting. Existence and uniqueness of local strong solutions for the model are proved. In this paper, there exist two difficulties we have to overcome. One is the strong nonlinearity of the system. The other is that the state function is not fixed. Yunliang Zhang and Zhidong Guo Copyright © 2015 Yunliang Zhang and Zhidong Guo. All rights reserved. On Relations between One-Dimensional Quantum and Two-Dimensional Classical Spin Systems Tue, 15 Dec 2015 07:18:52 +0000 We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems characterised by long-range interactions and with critical properties equivalent to those of the class of one-dimensional quantum systems treated by the authors in a previous publication. In particular, we use three approaches: the Trotter-Suzuki mapping, the method of coherent states, and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in the companion paper for the classical systems identified. J. Hutchinson, J. P. Keating, and F. Mezzadri Copyright © 2015 J. Hutchinson et al. All rights reserved. Links between -KP Hierarchy, -mKP Hierarchy, and (2+1)--Harry Dym Hierarchy Mon, 14 Dec 2015 06:49:59 +0000 The new (2+1)--Harry Dym hierarchy and -mKP hierarchy with two new time series and , which consist of -flow, -flow, and mixed and evolution equations of eigenfunctions, are proposed. Gauge transformations and reciprocal transformations between -KP hierarchy, -mKP hierarchy, and (2+1)--Harry Dym hierarchy are studied. Their soliton solutions are presented. Yehui Huang, Yuqin Yao, and Yunbo Zeng Copyright © 2015 Yehui Huang et al. All rights reserved. On the Nonlinear Perturbation Rosenau-Hyman Equation: A Model of Nonlinear Scattering Wave Sun, 13 Dec 2015 11:39:14 +0000 We investigate a nonlinear wave phenomenon described by the perturbation Rosenau-Hyman equation within the concept of derivative with fractional order. We used the Caputo fractional derivative and we proposed an iteration method in order to find a particular solution of the extended perturbation equation. We proved the stability and the convergence of the suggested method for solving the extended equation without any restriction on and also on the perturbations terms. Using the inner product we proved the uniqueness of the special solution. By choosing randomly the fractional orders and , we presented the numerical solutions. Abdon Atangana, Dumitru Baleanu, Maysaa’ Mohamed Al Qurashi, and Xiao-Jun Yang Copyright © 2015 Abdon Atangana et al. All rights reserved. Exact Relativistic Magnetized Haloes around Rotating Disks Thu, 10 Dec 2015 14:27:02 +0000 The study of the dynamics of magnetic fields in galaxies is one of important problems in formation and evolution of galaxies. In this paper, we present the exact relativistic treatment of a rotating disk surrounded by a magnetized material halo. The features of the halo and disk are described by the distributional energy-momentum tensor of a general fluid in canonical form. All the relevant quantities and the metric and electromagnetic potentials are exactly determined by an arbitrary harmonic function only. For instance, the generalized Kuzmin-disk potential is used. The particular class of solutions obtained is asymptotically flat and satisfies all the energy conditions. Moreover, the motion of a charged particle on the halo is described. As far as we know, this is the first relativistic model describing analytically the magnetized halo of a rotating disk. Antonio C. Gutiérrez-Piñeres and Abraão J. S. Capistrano Copyright © 2015 Antonio C. Gutiérrez-Piñeres and Abraão J. S. Capistrano. All rights reserved. Kawahara-Burgers Equation on a Strip Thu, 10 Dec 2015 10:22:35 +0000 An initial-boundary value problem for the 2D Kawahara-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in the -norm. N. A. Larkin Copyright © 2015 N. A. Larkin. All rights reserved. System Reliability Evaluation Based on Convex Combination Considering Operation and Maintenance Strategy Tue, 08 Dec 2015 06:29:41 +0000 The approaches to the system reliability evaluation with respect to the cases, where the components are independent or the components have interactive relationships within the system, were proposed in this paper. Starting from the higher requirements on system operational safety and economy, the reliability focused optimal models of multiobjective maintenance strategies were built. For safety-critical systems, the pessimistic maintenance strategies are usually taken, and, in these cases, the system reliability evaluation has also to be tackled pessimistically. For safety-uncritical systems, the optimistic maintenance strategies were usually taken, and, in these circumstances, the system reliability evaluation had also to be tackled optimistically, respectively. Besides, the reasonable maintenance strategies and their corresponding reliability evaluation can be obtained through the convex combination of the above two cases. With a high-speed train system as the example background, the proposed method is verified by combining the actual failure data with the maintenance data. Results demonstrate that the proposed study can provide a new system reliability calculation method and solution to select and optimize the multiobjective operational strategies with the considerations of system safety and economical requirements. The theoretical basis is also provided for scientifically estimating the reliability of a high-speed train system and formulating reasonable maintenance strategies. Lijie Li, Limin Jia, and Yanhui Wang Copyright © 2015 Lijie Li et al. All rights reserved. The Interactions of -Soliton Solutions for the Generalized ()-Dimensional Variable-Coefficient Fifth-Order KdV Equation Thu, 26 Nov 2015 09:39:37 +0000 A generalized ()-dimensional variable-coefficient KdV equation is introduced, which can describe the interaction between a water wave and gravity-capillary waves better than the ()-dimensional KdV equation. The -soliton solutions of the ()-dimensional variable-coefficient fifth-order KdV equation are obtained via the Bell-polynomial method. Then the soliton fusion, fission, and the pursuing collision are analyzed depending on the influence of the coefficient ; when , the soliton fusion and fission will happen; when , the pursuing collision will occur. Moreover, the B├Ącklund transformation of the equation is gotten according to the binary Bell-polynomial and the period wave solutions are given by applying the Riemann theta function method. Xiangrong Wang, Xiaoen Zhang, Yong Zhang, and Huanhe Dong Copyright © 2015 Xiangrong Wang et al. All rights reserved. Role of Time Relaxation in a One-Dimensional Diffusion-Advection Model of Water and Salt Transport Wed, 25 Nov 2015 12:15:16 +0000 The transport of salt, necessarily coupled with the transport of water, through porous building materials may heavily limit their durability due to possible deterioration and structural damage. Usually, the binding of salt to the pore walls is assumed to occur instantly, as soon as the salt is transported by water to a given position. We consider the advection-diffusion model of the transport and generalize it to include possible delays in the binding. Applying the Boltzmann-Matano method, we calculate the diffusion coefficient of the salt in dependence on the salt concentration and show that it increases with the rate of binding. We apply our results to an example of the chloride transport in a lime plaster. Igor Medved’ and Robert Černý Copyright © 2015 Igor Medved’ and Robert Černý. All rights reserved. Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations Wed, 25 Nov 2015 08:26:13 +0000 We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis. X. Wang, F. Liu, and X. Chen Copyright © 2015 X. Wang et al. All rights reserved. Hypersurface Constrained Elasticae in Lorentzian Space Forms Tue, 24 Nov 2015 13:18:23 +0000 We study geodesics in hypersurfaces of a Lorentzian space form , which are critical curves of the -bending energy functional, for variations constrained to lie on the hypersurface. We characterize critical geodesics showing that they live fully immersed in a totally geodesic and that they must be of three different types. Finally, we consider the classification of surfaces in the Minkowski 3-space foliated by critical geodesics. Óscar J. Garay, Álvaro Pámpano, and Changhwa Woo Copyright © 2015 Óscar J. Garay et al. All rights reserved. A Regime Switching Model of Schooling Choice as a Job Search Process Tue, 24 Nov 2015 08:06:29 +0000 We propose a regime switching model of schooling choice as a job search process. We adopt a two-state Markov process and the derived coupled Bellman equations are solved by seeking the root of an auxiliary algebraic equation. Some numerical examples are also considered. Yong Hyun Shin and Ho-Seok Lee Copyright © 2015 Yong Hyun Shin and Ho-Seok Lee. All rights reserved. Dynamics of a General Stochastic Nonautonomous Lotka-Volterra Model with Delays and Impulsive Perturbations Sun, 22 Nov 2015 11:20:09 +0000 A stochastic nonautonomous N-species Lotka-Volterra model with delays and impulsive perturbations is investigated. For this model, sufficient conditions for extinction, nonpersistence in the mean, weak persistence and stochastic permanence are given, respectively. The influences of the stochastic noises, and the impulsive perturbations on the properties of the stochastic model are also discussed. The critical value between weak persistence and extinction is obtained. Finally, numerical simulations are given to support the theoretical analysis results. Qing Wang, Yongguang Yu, and Shuo Zhang Copyright © 2015 Qing Wang et al. All rights reserved. Construction of Analytic Solution for Time-Fractional Boussinesq Equation Using Iterative Method Sun, 22 Nov 2015 07:25:30 +0000 This paper is aimed at constructing analytical solution for both linear and nonlinear time-fractional Boussinesq equations by an iterative method. By the iterative process, we can obtain the analytic solution of the fourth-order time-fractional Boussinesq equation in , , and , the sixth-order time-fractional Boussinesq equation, and the th-order time-fractional Boussinesq equation in . Through these examples, it shows that the method is simple and effective. Fei Xu, Yixian Gao, and Weipeng Zhang Copyright © 2015 Fei Xu et al. All rights reserved.