Advances in Mathematical Physics The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Application of Perturbation Theory to a Master Equation Thu, 30 Jun 2016 14:54:48 +0000 We develop a matrix perturbation method for the Lindblad master equation. The first- and second-order corrections are obtained and the method is generalized for higher orders. The perturbation method developed is applied to the problem of a lossy cavity filled with a Kerr medium; the second-order corrections are estimated and compared with the known exact analytic solution. The comparison is done by calculating the -function, the average number of photons, and the distance between density matrices. B. M. Villegas-Martínez, F. Soto-Eguibar, and H. M. Moya-Cessa Copyright © 2016 B. M. Villegas-Martínez et al. All rights reserved. Computing Coherence Vectors and Correlation Matrices with Application to Quantum Discord Quantification Wed, 29 Jun 2016 12:10:23 +0000 Coherence vectors and correlation matrices are important functions frequently used in physics. The numerical calculation of these functions directly from their definitions, which involves Kronecker products and matrix multiplications, may seem to be a reasonable option. Notwithstanding, as we demonstrate in this paper, some algebraic manipulations before programming can reduce considerably their computational complexity. Besides, we provide Fortran code to generate generalized Gell-Mann matrices and to compute the optimized and unoptimized versions of associated Bloch’s vectors and correlation matrix in the case of bipartite quantum systems. As a code test and application example, we consider the calculation of Hilbert-Schmidt quantum discords. Jonas Maziero Copyright © 2016 Jonas Maziero. All rights reserved. Homotopy Analysis Solution for Magnetohydrodynamic Squeezing Flow in Porous Medium Wed, 22 Jun 2016 12:24:47 +0000 The aim of the present work is to analyze the magnetohydrodynamic (MHD) squeezing flow through porous medium using homotopy analysis method (HAM). Fourth-order boundary value problem is modeled through stream function and transformation . Absolute residuals are used to check the efficiency and consistency of HAM. Other analytical techniques are compared with the present work. It is shown that results of good agreement can be obtained by choosing a suitable value of convergence control parameter in the valid region . The influence of different parameters on the flow is argued theoretically as well as graphically. Inayat Ullah, M. T. Rahim, Hamid Khan, and Mubashir Qayyum Copyright © 2016 Inayat Ullah et al. All rights reserved. Analytical Investigation of Magnetohydrodynamic Flow over a Nonlinear Porous Stretching Sheet Thu, 16 Jun 2016 06:35:17 +0000 We investigated the magnetohydrodynamic (MHD) boundary layer flow over a nonlinear porous stretching sheet with the help of semianalytical method known as optimal homotopy asymptotic method (OHAM). The effects of different parameters on fluid flow are investigated and discussed. The obtained results are compared with numerical Runge-Kutta-Fehlberg fourth-fifth-order method. It is found that the OHAM solution agrees well with numerical as well as published data for different assigned values of parameters; this thus indicates the feasibility of the proposed method (OHAM). Fazle Mabood and Nopparat Pochai Copyright © 2016 Fazle Mabood and Nopparat Pochai. All rights reserved. Interactions of Delta Shock Waves for Zero-Pressure Gas Dynamics with Energy Conservation Law Wed, 15 Jun 2016 15:54:04 +0000 We study the interactions of delta shock waves and vacuum states for the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. The Riemann problems with initial data of three piecewise constant states are solved case by case, and four different configurations of Riemann solutions are constructed. Furthermore, the numerical simulations completely coinciding with theoretical analysis are shown. Wei Cai and Yanyan Zhang Copyright © 2016 Wei Cai and Yanyan Zhang. All rights reserved. An Operational Matrix Technique for Solving Variable Order Fractional Differential-Integral Equation Based on the Second Kind of Chebyshev Polynomials Wed, 15 Jun 2016 12:17:56 +0000 An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient. Jianping Liu, Xia Li, and Limeng Wu Copyright © 2016 Jianping Liu et al. All rights reserved. CRE Solvability, Exact Soliton-Cnoidal Wave Interaction Solutions, and Nonlocal Symmetry for the Modified Boussinesq Equation Wed, 15 Jun 2016 06:33:43 +0000 It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special soliton-cnoidal wave interaction solution of the equation are explicitly given, which is difficult to be found by other traditional methods. Moreover, the nonlocal symmetry related to the consistent tanh expansion (CTE) and the residual symmetry from the truncated Painlevé expansion, as well as the relationship between them, are obtained. The residual symmetry is localized after embedding the original system in an enlarged one. The symmetry group transformation of the enlarged system is derived by applying the Lie point symmetry approach. Wenguang Cheng and Biao Li Copyright © 2016 Wenguang Cheng and Biao Li. All rights reserved. A Subdivision Based Iterative Collocation Algorithm for Nonlinear Third-Order Boundary Value Problems Thu, 09 Jun 2016 09:29:21 +0000 We construct an algorithm for the numerical solution of nonlinear third-order boundary value problems. This algorithm is based on eight-point binary subdivision scheme. Proposed algorithm is stable and convergent and gives more accurate results than fourth-degree B-spline algorithm. Syeda Tehmina Ejaz and Ghulam Mustafa Copyright © 2016 Syeda Tehmina Ejaz and Ghulam Mustafa. All rights reserved. -Complexity and Tilting Modules Tue, 07 Jun 2016 09:52:29 +0000 Let be a finite dimensional algebra over an algebraic closed field . In this note, we will show that if is a separating and splitting tilting -module, then -complexities of and are equal, where . Lijing Zheng, Chonghui Huang, and Qianhong Wan Copyright © 2016 Lijing Zheng et al. All rights reserved. Optimal Stable Approximation for the Cauchy Problem for Laplace Equation Mon, 06 Jun 2016 10:27:04 +0000 Cauchy problem for Laplace equation in a strip is considered. The optimal error bounds between the exact solution and its regularized approximation are given, which depend on the noise level either in a Hölder continuous way or in a logarithmic continuous way. We also provide two special regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, which realize the optimal error bounds. Hongfang Li and Feng Zhou Copyright © 2016 Hongfang Li and Feng Zhou. All rights reserved. Null Curve Evolution in Four-Dimensional Pseudo-Euclidean Spaces Thu, 02 Jun 2016 10:08:51 +0000 We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinitely many local symmetries for the null localized induction equation is provided. José del Amor, Ángel Giménez, and Pascual Lucas Copyright © 2016 José del Amor et al. All rights reserved. Mixed Initial-Boundary Value Problem for the Capillary Wave Equation Wed, 01 Jun 2016 08:09:33 +0000 We study the mixed initial-boundary value problem for the capillary wave equation: , where . We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions. B. Juarez Campos, Elena Kaikina, and Hector F. Ruiz Paredes Copyright © 2016 B. Juarez Campos et al. All rights reserved. Analysis of the Stability of the Riemann Problem for a Simplified Model in Magnetogasdynamics Wed, 25 May 2016 13:24:01 +0000 The generalized Riemann problem for a simplified model of one-dimensional ideal gas in magnetogasdynamics in a neighborhood of the origin in the plane is considered. According to the different cases of the corresponding Riemann solutions, we construct the perturbed solutions uniquely with the characteristic method. We find that, for some case, the contact discontinuity appears after perturbation while there is no contact discontinuity of the corresponding Riemann solution. For most cases, the Riemann solutions are stable and the perturbation can not affect the corresponding Riemann solutions. While, for some few cases, the forward (backward) rarefaction wave can be transformed into the forward (backward) shock wave which shows that the Riemann solutions are unstable under such local small perturbations of the Riemann initial data. Yujin Liu and Wenhua Sun Copyright © 2016 Yujin Liu and Wenhua Sun. All rights reserved. A Method of Finding Source Function for Inverse Diffusion Problem with Time-Fractional Derivative Wed, 25 May 2016 08:51:59 +0000 The Homotopy Perturbation Method is developed to find a source function for inverse diffusion problem with time-fractional derivative. The inverse problem is with variable coefficients and initial and boundary conditions. The analytical solutions to the inverse problems are obtained in the form of a finite convergent power series with easily obtainable components. Vildan Gülkaç Copyright © 2016 Vildan Gülkaç. All rights reserved. Iterative Methods for Solving the Fractional Form of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries Wed, 25 May 2016 06:25:34 +0000 An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is proposed with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations of motion is reduced to a single fourth-order nonlinear ordinary differential equation. By using the basic definitions of fractional calculus, we introduced the fractional order form of the fourth-order nonlinear ordinary differential equation. The resulting boundary value fractional problems are solved by the new iterative and Picard methods. Convergence of the considered methods is confirmed by obtaining absolute residual errors for approximate solutions for various Reynolds number. The comparisons of the solutions for various Reynolds number and various values of the fractional order confirm that the two methods are identical and therefore are suitable for solving this kind of problems. Finally, the effects of various Reynolds number on the solution are also studied graphically. A. A. Hemeda and E. E. Eladdad Copyright © 2016 A. A. Hemeda and E. E. Eladdad. All rights reserved. Existence of Positive Solutions for Two-Point Boundary Value Problems of Nonlinear Finite Discrete Fractional Differential Equations and Its Application Tue, 17 May 2016 13:31:01 +0000 This paper is concerned with the two-point boundary value problems of nonlinear finite discrete fractional differential equations. On one hand, we discuss some new properties of the Green function. On the other hand, by using the main properties of Green function and the Krasnoselskii fixed point theorem on cones, some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established. Caixia Guo, Jianmin Guo, Ying Gao, and Shugui Kang Copyright © 2016 Caixia Guo et al. All rights reserved. Equivalent Circuits Applied in Electrochemical Impedance Spectroscopy and Fractional Derivatives with and without Singular Kernel Mon, 16 May 2016 14:23:08 +0000 We present an alternative representation of integer and fractional electrical elements in the Laplace domain for modeling electrochemical systems represented by equivalent electrical circuits. The fractional derivatives considered are of Caputo and Caputo-Fabrizio type. This representation includes distributed elements of the Cole model type. In addition to maintaining consistency in adjusted electrical parameters, a detailed methodology is proposed to build the equivalent circuits. Illustrative examples are given and the Nyquist and Bode graphs are obtained from the numerical simulation of the corresponding transfer functions using arbitrary electrical parameters in order to illustrate the methodology. The advantage of our representation appears according to the comparison between our model and models presented in the paper, which are not physically acceptable due to the dimensional incompatibility. The Markovian nature of the models is recovered when the order of the fractional derivatives is equal to 1. J. F. Gómez-Aguilar, J. E. Escalante-Martínez, C. Calderón-Ramón, L. J. Morales-Mendoza, M. Benavidez-Cruz, and M. Gonzalez-Lee Copyright © 2016 J. F. Gómez-Aguilar et al. All rights reserved. On the Accuracy and Efficiency of Transient Spectral Element Models for Seismic Wave Problems Thu, 12 May 2016 15:28:41 +0000 This study concentrates on transient multiphysical wave problems for simulating seismic waves. The presented models cover the coupling between elastic wave equations in solid structures and acoustic wave equations in fluids. We focus especially on the accuracy and efficiency of the numerical solution based on higher-order discretizations. The spatial discretization is performed by the spectral element method. For time discretization we compare three different schemes. The efficiency of the higher-order time discretization schemes depends on several factors which we discuss by presenting numerical experiments with the fourth-order Runge-Kutta and the fourth-order Adams-Bashforth time-stepping. We generate a synthetic seismogram and demonstrate its function by a numerical simulation. Sanna Mönkölä Copyright © 2016 Sanna Mönkölä. All rights reserved. Similarity Solutions for Multiterm Time-Fractional Diffusion Equation Wed, 11 May 2016 11:12:35 +0000 Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on the obtained results, we propose a definition for a multiterm error function with generalized coefficients. A. Elsaid, M. S. Abdel Latif, and M. Maneea Copyright © 2016 A. Elsaid et al. All rights reserved. Flows with Slip of Oldroyd-B Fluids over a Moving Plate Mon, 09 May 2016 16:42:05 +0000 A general investigation has been made and analytic solutions are provided corresponding to the flows of an Oldroyd-B fluid, under the consideration of slip condition at the boundary. The fluid motion is generated by the flat plate which has a translational motion in its plane with a time-dependent velocity. The adequate integral transform approach is employed to find analytic solutions for the velocity field. Solutions for the flows corresponding to Maxwell fluid, second-grade fluid, and Newtonian fluid are also determined in both cases, namely, flows with slip on the boundary and flows with no slip on the boundary, respectively. Some of our results were compared with other results from the literature. The effects of several emerging dimensionless and pertinent parameters on the fluid velocity have been studied theoretically as well as graphically in the paper. Abdul Shakeel, Sohail Ahmad, Hamid Khan, Nehad Ali Shah, and Sami Ul Haq Copyright © 2016 Abdul Shakeel et al. All rights reserved. A Consistent Immersed Finite Element Method for the Interface Elasticity Problems Mon, 09 May 2016 12:34:11 +0000 We propose a new scheme for elasticity problems having discontinuity in the coefficients. In the previous work (Kwak et al., 2014), the authors suggested a method for solving such problems by finite element method using nonfitted grids. The proposed method is based on the -nonconforming finite element methods with stabilizing terms. In this work, we modify the method by adding the consistency terms, so that the estimates of consistency terms are not necessary. We show optimal error estimates in and divergence norms under minimal assumptions. Various numerical experiments also show optimal rates of convergence. Sangwon Jin, Do Y. Kwak, and Daehyeon Kyeong Copyright © 2016 Sangwon Jin et al. All rights reserved. Integrable 2D Time-Irreversible Systems with a Cubic Second Integral Wed, 04 May 2016 09:05:41 +0000 We construct a very rare integrable 2D mechanical system which admits a complementary integral of motion cubic in the velocities in the presence of conservative potential and velocity-dependent (gyroscopic) forces. Special cases are given interpretation as a motion of a particle on a sphere endowed with a Riemannian metric, a particle in the Euclidean plane, and new generalizations of two cases of motion of a rigid body with a cubic integral, known by names of Goriachev-Chaplygin and Goriachev. H. M. Yehia and A. A. Elmandouh Copyright © 2016 H. M. Yehia and A. A. Elmandouh. All rights reserved. Quantization of a 3D Nonstationary Harmonic plus an Inverse Harmonic Potential System Tue, 19 Apr 2016 14:36:11 +0000 The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions. Salim Medjber, Hacene Bekkar, Salah Menouar, and Jeong Ryeol Choi Copyright © 2016 Salim Medjber et al. All rights reserved. The Effect of Initial State Error for Nonlinear Systems with Delay via Iterative Learning Control Thu, 07 Apr 2016 14:08:47 +0000 An iterative learning control problem for nonlinear systems with delays is studied in detail in this paper. By introducing the λ-norm and being inspired by retarded Gronwall-like inequality, the novel sufficient conditions for robust convergence of the tracking error, whose initial states are not zero, with time delays are obtained. Finally, simulation example is given to illustrate the effectiveness of the proposed method. Zhang Qunli Copyright © 2016 Zhang Qunli. All rights reserved. A Soliton Hierarchy Associated with a Spectral Problem of 2nd Degree in a Spectral Parameter and Its Bi-Hamiltonian Structure Wed, 30 Mar 2016 13:18:31 +0000 Associated with , a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability. Yuqin Yao, Shoufeng Shen, and Wen-Xiu Ma Copyright © 2016 Yuqin Yao et al. All rights reserved. Firm Growth Function and Extended-Gibrat’s Property Tue, 22 Mar 2016 08:49:25 +0000 We analytically show that the logarithmic average sales of firms first follow power-law growth and subsequently follow exponential growth, if the growth-rate distributions of the sales obey the extended-Gibrat’s property and Gibrat’s law. Here, the extended-Gibrat’s property and Gibrat’s law are statistically observed in short-term data, which denote the dependence of the growth-rate distributions on the initial values. In the derivation, we analytically show that the parameter of the extended-Gibrat’s property is identical to the power-law growth exponent and that it also decides the parameter of the exponential growth. By employing around one million bits of exhaustive sales data of Japanese firms in the ORBIS database, we confirmed our analytic results. Atushi Ishikawa, Shouji Fujimoto, Takayuki Mizuno, and Tsutomu Watanabe Copyright © 2016 Atushi Ishikawa et al. All rights reserved. Comparing First-Order Microscopic and Macroscopic Crowd Models for an Increasing Number of Massive Agents Sun, 20 Mar 2016 10:14:42 +0000 A comparison between first-order microscopic and macroscopic differential models of crowd dynamics is established for an increasing number of pedestrians. The novelty is the fact of considering massive agents, namely, particles whose individual mass does not become infinitesimal when grows. This implies that the total mass of the system is not constant but grows with . The main result is that the two types of models approach one another in the limit , provided the strength and/or the domain of pedestrian interactions are properly modulated by at either scale. This is consistent with the idea that pedestrians may adapt their interpersonal attitudes according to the overall level of congestion. Alessandro Corbetta and Andrea Tosin Copyright © 2016 Alessandro Corbetta and Andrea Tosin. All rights reserved. On Pseudo-Petrov Symmetric Riemannian Manifolds Tue, 15 Mar 2016 12:22:51 +0000 The present paper deals with pseudo-Petrov symmetric Riemannian manifolds whose space-matter tensor satisfies a special condition. Firstly, basic results of pseudo-Petrov symmetric Riemannian manifolds are obtained. Then, pseudo-Petrov symmetric manifolds which are Einstein, quasi-Einstein, and locally decomposable are examined and some theorems involving these manifolds are proved. Finally, two examples proving the existence of pseudo-Petrov symmetric Riemannian manifolds are given. Sanjib Kumar Jana, Fusun Nurcan, Amit Kumar Debnath, and Joydeep Sengupta Copyright © 2016 Sanjib Kumar Jana et al. All rights reserved. The Ritz Method for Boundary Problems with Essential Conditions as Constraints Sun, 13 Mar 2016 12:22:04 +0000 We give an elementary derivation of an extension of the Ritz method to trial functions that do not satisfy essential boundary conditions. As in the Babuška-Brezzi approach boundary conditions are treated as variational constraints and Lagrange multipliers are used to remove them. However, we avoid the saddle point reformulation of the problem and therefore do not have to deal with the Babuška-Brezzi inf-sup condition. In higher dimensions boundary weights are used to approximate the boundary conditions, and the assumptions in our convergence proof are stated in terms of completeness of the trial functions and of the boundary weights. These assumptions are much more straightforward to verify than the Babuška-Brezzi condition. We also discuss limitations of the method and implementation issues that follow from our analysis and examine a number of examples, both analytic and numerical. Vojin Jovanovic and Sergiy Koshkin Copyright © 2016 Vojin Jovanovic and Sergiy Koshkin. All rights reserved. On Functions of Several Split-Quaternionic Variables Sun, 13 Mar 2016 07:43:23 +0000 Alesker studied a relation between the determinant of a quaternionic Hessian of a function and a specific complex volume form. In this note we show that similar relation holds for functions of several split-quaternionic variables and point to some relations with geometry. Gueo Grantcharov and Camilo Montoya Copyright © 2016 Gueo Grantcharov and Camilo Montoya. All rights reserved.