Advances in Mathematical Physics The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . 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. A New Nonlinear Diffusion Equation Model for Noisy Image Segmentation Wed, 09 Mar 2016 13:44:46 +0000 Image segmentation and image denoising are two important and fundamental topics in the field of image processing. Geometric active contour model based on level set method can deal with the problem of image segmentation, but it does not consider the problem of image denoising. In this paper, a new diffusion equation model for noisy image segmentation is proposed by incorporating some classical diffusion equation denoising models into the segmental process. An assumption about the connection between the image intensity and level set function is given firstly. Some classical denoising models are employed to describe the evolution of level set function secondly. The final nonlinear diffusion equation model for noisy image segmentation is built thirdly. Then image segmentation and image denoising are combined in a united framework. The segmental results can be presented by level set function. Experimental results show that the new model has the advantage of noise resistance and is superior to traditional segmentation model. Bo Chen, Xiao-Hui Zhou, Li-Wei Zhang, Jie Wang, Wei-Qiang Zhang, and Chen Zhang Copyright © 2016 Bo Chen et al. All rights reserved. The Interval Slope Method for Long-Term Forecasting of Stock Price Trends Sun, 06 Mar 2016 14:19:07 +0000 A stock price is a typical but complex type of time series data. We used the effective prediction of long-term time series data to schedule an investment strategy and obtain higher profit. Due to economic, environmental, and other factors, it is very difficult to obtain a precise long-term stock price prediction. The exponentially segmented pattern (ESP) is introduced here and used to predict the fluctuation of different stock data over five future prediction intervals. The new feature of stock pricing during the subinterval, named the interval slope, can characterize fluctuations in stock price over specific periods. The cumulative distribution function (CDF) of MSE was compared to those of MMSE-BC and SVR. We concluded that the interval slope developed here can capture more complex dynamics of stock price trends. The mean stock price can then be predicted over specific time intervals relatively accurately, in which multiple mean values over time intervals are used to express the time series in the long term. In this way, the prediction of long-term stock price can be more precise and prevent the development of cumulative errors. Chun-xue Nie and Xue-bo Jin Copyright © 2016 Chun-xue Nie and Xue-bo Jin. All rights reserved. A Nonlinear Schrödinger Equation Resonating at an Essential Spectrum Wed, 02 Mar 2016 09:04:55 +0000 We consider the nonlinear Schrödinger equation . The potential function satisfies that the essential spectrum of the Schrödinger operator is and this Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. The nonlinearity satisfies the resonance type condition . Under some additional conditions on and , we prove that this equation has infinitely many solutions. Shaowei Chen and Haijun Zhou Copyright © 2016 Shaowei Chen and Haijun Zhou. All rights reserved. A Note on the Discrete Spectrum of Gaussian Wells (I): The Ground State Energy in One Dimension Mon, 29 Feb 2016 17:26:30 +0000 The ground state energy of is computed for small values of by means of an approximation of an integral operator in momentum space. Such an approximation leads to a transcendental equation for which is the root. G. Muchatibaya, S. Fassari, F. Rinaldi, and J. Mushanyu Copyright © 2016 G. Muchatibaya et al. All rights reserved. The Approximate Solution of Some Plane Boundary Value Problems of the Moment Theory of Elasticity Mon, 29 Feb 2016 17:23:44 +0000 We consider a two-dimensional system of differential equations of the moment theory of elasticity. The general solution of this system is represented by two arbitrary harmonic functions and solution of the Helmholtz equation. Based on the general solution, an algorithm of constructing approximate solutions of boundary value problems is developed. Using the proposed method, the approximate solutions of some problems on stress concentration on the contours of holes are constructed. The values of stress concentration coefficients obtained in the case of moment elasticity and for the classical elastic medium are compared. In the final part of the paper, we construct the approximate solution of a nonlocal problem whose exact solution is already known and compare our approximate solution with the exact one. Supposedly, the proposed method makes it possible to construct approximate solutions of quite a wide class of boundary value problems. Roman Janjgava Copyright © 2016 Roman Janjgava. All rights reserved. Variational Multiscale Element Free Galerkin Method Coupled with Low-Pass Filter for Burgers’ Equation with Small Diffusion Mon, 29 Feb 2016 12:58:29 +0000 Variational multiscale element free Galerkin (VMEFG) method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFG method still suffers from instability in the presence of boundary or interior layers. In order to overcome this problem, the high order low-pass filter is used to smooth the solution. Three test examples with very small diffusion are presented and the solutions obtained are compared with exact solutions and some other numerical methods. The numerical results are found in which the VMEFG coupled with low-pass filter works very well for Burgers’ equation with very small diffusivity coefficients. Ping Zhang, Xiaohua Zhang, and Laizhong Song Copyright © 2016 Ping Zhang et al. All rights reserved. Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation Mon, 22 Feb 2016 06:26:32 +0000 A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. The VE, which arises in modelling the propagation of high-frequency waves in a relaxing medium, has periodic and solitary traveling wave solutions some of which are loop-like in nature. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE), by a change of independent variables. The VPE has an -soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises -loop-like solitons. Aspects of the inverse scattering transform (IST) method, as applied originally to the KdV equation, are used to find one- and two-soliton solutions to the VPE even though the VPE’s spectral equation is third-order and not second-order. A Bäcklund transformation for the VPE is used to construct conservation laws. The standard IST method for third-order spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to -soliton solutions and -mode periodic solutions, respectively. Interactions between these types of solutions are investigated. V. O. Vakhnenko and E. J. Parkes Copyright © 2016 V. O. Vakhnenko and E. J. Parkes. All rights reserved. The Stability of Interbank Market Network: A Perspective on Contagion and Risk Sharing Thu, 18 Feb 2016 09:17:37 +0000 As an important part of the financial system, interbank market provides banks with liquidity and credit lending and also is the main channel for risk contagion. In this paper, we test the existence of systematic risk contagion within the Chinese interbank market. By building the networks of the Chinese interbank market for each year and using the measure of mutual information, we quantitatively detect the changes of interbank market networks and observe that the correlations between banks become increasingly tighter in recent years. With the bilateral risk exposure among Chinese listed commercial banks, we find that the possibility of systemic risk contagion in Chinese interbank market is fairly small. But of great concern on each individual bank, the matter is different. Our simulation shows that the failures of three special banks (i.e., Agricultural Bank of China and Bank of China and Industrial and Commercial Bank of China) most likely lead to systemic risk contagion. Furthermore, we test the antirisk ability of the Chinese interbank market from the perspective of risk sharing and discover that the interbank market is stable when the loss scale is lower than forty percent of banks’ total core capital. Chi Xie, Yang Liu, Gang-Jin Wang, and Yan Xu Copyright © 2016 Chi Xie et al. All rights reserved. Direct Scaling Analysis of Fermionic Multiparticle Correlated Anderson Models with Infinite-Range Interaction Thu, 18 Feb 2016 08:53:33 +0000 We adapt the method of direct scaling analysis developed earlier for single-particle Anderson models, to the fermionic multiparticle models with finite or infinite interaction on graphs. Combined with a recent eigenvalue concentration bound for multiparticle systems, the new method leads to a simpler proof of the multiparticle dynamical localization with optimal decay bounds in a natural distance in the multiparticle configuration space, for a large class of strongly mixing random external potentials. Earlier results required the random potential to be IID. Victor Chulaevsky Copyright © 2016 Victor Chulaevsky. All rights reserved. Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields Sun, 14 Feb 2016 11:24:25 +0000 We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearity of order for arbitrarily large initial data, where the lower bound is a positive root of for and for Our purpose is to extend the previous results for higher space dimensions concerning -time decay and to improve the lower bound of under the same dissipative condition on : and as in the previous works. Nakao Hayashi, Chunhua Li, and Pavel I. Naumkin Copyright © 2016 Nakao Hayashi et al. 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.