Advances in Mathematical Physics The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. The Periodic Boundary Value Problem for the Weakly Dissipative -Hunter-Saxton Equation Mon, 05 Oct 2015 08:44:47 +0000 We study the periodic boundary value problem for the weakly dissipative -Hunter-Saxton equation. We establish the local well-posedness in Besov space , which extends the previous regularity range to the critical case. Zhengyong Ouyang, Xiangdong Wang, and Haiwu Rong Copyright © 2015 Zhengyong Ouyang et al. All rights reserved. A Numerical Method for Solving Fractional Differential Equations by Using Neural Network Sun, 04 Oct 2015 07:06:37 +0000 We present a new method for solving the fractional differential equations of initial value problems by using neural networks which are constructed from cosine basis functions with adjustable parameters. By training the neural networks repeatedly the numerical solutions for the fractional differential equations were obtained. Moreover, the technique is still applicable for the coupled differential equations of fractional order. The computer graphics and numerical solutions show that the proposed method is very effective. Haidong Qu and Xuan Liu Copyright © 2015 Haidong Qu and Xuan Liu. All rights reserved. Existence of Multiple Positive Solutions for Choquard Equation with Perturbation Wed, 30 Sep 2015 13:58:25 +0000 This paper is concerned with the following Choquard equation with perturbation: , , where , , and . This kind of equations is well known as the Choquard or nonlinear Schrödinger-Newton equation. Under some assumptions for the functions , we prove the existence of multiple positive solutions of the equation. Moreover, we also show that these results still hold for more generalized Choquard equation with perturbation. Tao Xie, Lu Xiao, and Jun Wang Copyright © 2015 Tao Xie et al. All rights reserved. Generalized Bilinear Differential Operators Application in a (3+1)-Dimensional Generalized Shallow Water Equation Thu, 10 Sep 2015 11:16:13 +0000 The relations between -operators and multidimensional binary Bell polynomials are explored and applied to construct the bilinear forms with -operators of nonlinear equations directly and quickly. Exact periodic wave solution of a (3+1)-dimensional generalized shallow water equation is obtained with the help of the -operators and a general Riemann theta function in terms of the Hirota method, which illustrate that bilinear -operators can provide a method for seeking exact periodic solutions of nonlinear integrable equations. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions. Jingzhu Wu, Xiuzhi Xing, and Xianguo Geng Copyright © 2015 Jingzhu Wu et al. All rights reserved. The Thermal Statistics of Quasi-Probabilities’ Analogs in Phase Space Tue, 08 Sep 2015 16:38:55 +0000 We focus attention upon the thermal statistics of the classical analogs of quasi-probabilities (QP) in phase space for the important case of quadratic Hamiltonians. We consider the three more important OPs: Wigner’s, -, and Husimi’s. We show that, for all of them, the ensuing semiclassical entropy is a function only of the fluctuation product . We ascertain that the semiclassical analog of -distribution seems to become unphysical at very low temperatures. The behavior of several other information quantifiers reconfirms such an assertion in manifold ways. We also examine the behavior of the statistical complexity and of thermal quantities like the specific heat. F. Pennini, A. Plastino, and M. C. Rocca Copyright © 2015 F. Pennini et al. All rights reserved. Mechanics and Geometry of Solids and Surfaces Thu, 03 Sep 2015 08:43:27 +0000 J. D. Clayton, M. A. Grinfeld, T. Hasebe, and J. R. Mayeur Copyright © 2015 J. D. Clayton et al. All rights reserved. Structures and Low Dimensional Classifications of Hom-Poisson Superalgebras Wed, 02 Sep 2015 06:16:27 +0000 Hom-Poisson superalgebras can be considered as a deformation of Poisson superalgebras. We prove that Hom-Poisson superalgebras are closed under tensor products. Moreover, we show that Hom-Poisson superalgebras can be described using only the twisting map and one binary operation. Finally, all algebra endomorphisms on 2-dimensional complex Poisson superalgebras are computed, and their associated Hom-Poisson superalgebras are described explicitly. Qingcheng Zhang, Chunyue Wang, and Zhu Wei Copyright © 2015 Qingcheng Zhang et al. All rights reserved. The Relationship between Focal Surfaces and Surfaces at a Constant Distance from the Edge of Regression on a Surface Tue, 01 Sep 2015 13:17:34 +0000 We investigate the relationship between focal surfaces and surfaces at a constant distance from the edge of regression on a surface. We show that focal surfaces F1 and F2 of the surface M can be obtained by means of some special surfaces at a constant distance from the edge of regression on the surface M. Semra Yurttancikmaz and Omer Tarakci Copyright © 2015 Semra Yurttancikmaz and Omer Tarakci. All rights reserved. The Steiner Formula and the Polar Moment of Inertia for the Closed Planar Homothetic Motions in Complex Plane Tue, 01 Sep 2015 07:33:45 +0000 The Steiner area formula and the polar moment of inertia were expressed during one-parameter closed planar homothetic motions in complex plane. The Steiner point or Steiner normal concepts were described according to whether rotation number was different from zero or equal to zero, respectively. The moving pole point was given with its components and its relation between Steiner point or Steiner normal was specified. The sagittal motion of a winch was considered as an example. This motion was described by a double hinge consisting of the fixed control panel of winch and the moving arm of winch. The results obtained in the second section of this study were applied for this motion. Ayhan Tutar and Onder Sener Copyright © 2015 Ayhan Tutar and Onder Sener. All rights reserved. Optimal Homotopy Asymptotic Solution for Exothermic Reactions Model with Constant Heat Source in a Porous Medium Tue, 01 Sep 2015 06:19:46 +0000 The heat flow patterns profiles are required for heat transfer simulation in each type of the thermal insulation. The exothermic reaction models in porous medium can prescribe the problems in the form of nonlinear ordinary differential equations. In this research, the driving force model due to the temperature gradients is considered. A governing equation of the model is restricted into an energy balance equation that provides the temperature profile in conduction state with constant heat source on the steady state. The proposed optimal homotopy asymptotic method (OHAM) is used to compute the solutions of the exothermic reactions equation. Fazle Mabood and Nopparat Pochai Copyright © 2015 Fazle Mabood and Nopparat Pochai. All rights reserved. Weyl-Euler-Lagrange Equations of Motion on Flat Manifold Tue, 01 Sep 2015 06:12:03 +0000 This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra software. Additionally, the improvements, obtained in this study, will be presented. Zeki Kasap Copyright © 2015 Zeki Kasap. All rights reserved. On Finsler Geometry and Applications in Mechanics: Review and New Perspectives Mon, 31 Aug 2015 14:15:02 +0000 In Finsler geometry, each point of a base manifold can be endowed with coordinates describing its position as well as a set of one or more vectors describing directions, for example. The associated metric tensor may generally depend on direction as well as position, and a number of connections emerge associated with various covariant derivatives involving affine and nonlinear coefficients. Finsler geometry encompasses Riemannian, Euclidean, and Minkowskian geometries as special cases, and thus it affords great generality for describing a number of phenomena in physics. Here, descriptions of finite deformation of continuous media are of primary focus. After a review of necessary mathematical definitions and derivations, prior work involving application of Finsler geometry in continuum mechanics of solids is reviewed. A new theoretical description of continua with microstructure is then outlined, merging concepts from Finsler geometry and phase field theories of materials science. J. D. Clayton Copyright © 2015 J. D. Clayton. All rights reserved. A Variational Approach to Electrostatics of Polarizable Heterogeneous Substances Mon, 31 Aug 2015 14:06:18 +0000 We discuss equilibrium conditions for heterogeneous substances subject to electrostatic or magnetostatic effects. We demonstrate that the force-like aleph tensor and the energy-like beth tensor for polarizable deformable substances are divergence-free: and . We introduce two additional tensors: the divergence-free energy-like gimel tensor for rigid dielectrics and the general electrostatic gamma tensor which is not divergence-free. Our approach is based on a logically consistent extension of the Gibbs energy principle that takes into account polarization effects. While the model is mathematically rigorous, we caution against the assumption that it can reliably predict physical phenomena. On the contrary, clear models often lead to conclusions that are at odds with experiment and therefore should be treated as physical paradoxes that deserve the attention of the scientific community. Michael Grinfeld and Pavel Grinfeld Copyright © 2015 Michael Grinfeld and Pavel Grinfeld. All rights reserved. Comparison of Optimal Homotopy Asymptotic and Adomian Decomposition Methods for a Thin Film Flow of a Third Grade Fluid on a Moving Belt Mon, 31 Aug 2015 14:02:05 +0000 We have investigated a thin film flow of a third grade fluid on a moving belt using a powerful and relatively new approximate analytical technique known as optimal homotopy asymptotic method (OHAM). The variation of velocity profile for different parameters is compared with the numerical values obtained by Runge-Kutta Fehlberg fourth-fifth order method and with Adomian Decomposition Method (ADM). An interesting result of the analysis is that the three terms OHAM solution is more accurate than five terms of the ADM solution and this thus confirms the feasibility of the proposed method. Fazle Mabood and Nopparat Pochai Copyright © 2015 Fazle Mabood and Nopparat Pochai. All rights reserved. Leaky Modes of Waveguides as a Classical Optics Analogy of Quantum Resonances Sun, 30 Aug 2015 12:18:03 +0000 A classical optics waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics. The analogy is based on the well-known resemblance between the guided and radiation modes of a waveguide with the bound and scattering states of a quantum well. As resonances are scattering states that spend some time in the zone of influence of the scatterer, we associate them with the leaky modes of a waveguide, the latter characterized by suffering attenuation in the direction of propagation but increasing exponentially in the transverse directions. The resemblance is complete because resonances (leaky modes) can be interpreted as bound states (guided modes) with definite lifetime (longitudinal shift). As an immediate application we calculate the leaky modes (resonances) associated with a dielectric homogeneous slab (square well potential) and show that these modes are attenuated as they propagate. Sara Cruz y Cruz and Oscar Rosas-Ortiz Copyright © 2015 Sara Cruz y Cruz and Oscar Rosas-Ortiz. All rights reserved. The Intersection Probability of Brownian Motion and SLEκ Wed, 26 Aug 2015 08:53:34 +0000 By using excursion measure Poisson kernel method, we obtain a second-order differential equation of the intersection probability of Brownian motion and . Moreover, we find a transformation such that the second-order differential equation transforms into a hypergeometric differential equation. Then, by solving the hypergeometric differential equation, we obtain the explicit formula of the intersection probability for the trace of the chordal and planar Brownian motion started from distinct points in an upper half-plane . Shizhong Zhou and Shiyi Lan Copyright © 2015 Shizhong Zhou and Shiyi Lan. All rights reserved. Similarity Measures of Sequence of Fuzzy Numbers and Fuzzy Risk Analysis Tue, 25 Aug 2015 08:39:21 +0000 We present the methods to evaluate the similarity measures between sequence of triangular fuzzy numbers for making contributions to fuzzy risk analysis. Firstly, we calculate the COG (center of gravity) points of sequence of triangular fuzzy numbers. After, we present the methods to measure the degree of similarity between sequence of triangular fuzzy numbers. In addition, we give an example to compare the methods mentioned in the text. Furthermore, in this paper, we deal with the type fuzzy number. By defining the algebraic operations on the type fuzzy numbers we can solve the equations in the form , where and are fuzzy number. By this way, we can build an algebraic structure on fuzzy numbers. Additionally, the generalized difference sequence spaces of triangular fuzzy numbers , , and , consisting of all sequences such that is in the spaces , , and , have been constructed, respectively. Furthermore, some classes of matrix transformations from the space and to and are characterized, respectively, where is any sequence space. Zarife Zararsız Copyright © 2015 Zarife Zararsız. All rights reserved. Automorphism Properties and Classification of Adinkras Mon, 24 Aug 2015 07:54:08 +0000 Adinkras are graphical tools for studying off-shell representations of supersymmetry. In this paper we efficiently classify the automorphism groups of Adinkras relative to a set of local parameters. Using this, we classify Adinkras according to their equivalence and isomorphism classes. We extend previous results dealing with characterization of Adinkra degeneracy via matrix products and present algorithms for calculating the automorphism groups of Adinkras and partitioning Adinkras into their isomorphism classes. B. L. Douglas, S. James Gates Jr., B. L. Segler, and J. B. Wang Copyright © 2015 B. L. Douglas et al. All rights reserved. Lie Symmetry Analysis of a First-Order Feedback Model of Option Pricing Sun, 23 Aug 2015 11:39:15 +0000 A first-order feedback model of option pricing consisting of a coupled system of two PDEs, a nonliner generalised Black-Scholes equation and the classical Black-Scholes equation, is studied using Lie symmetry analysis. This model arises as an extension of the classical Black-Scholes model when liquidity is incorporated into the market. We compute the admitted Lie point symmetries of the system and construct an optimal system of the associated one-dimensional subalgebras. We also construct some invariant solutions of the model. Winter Sinkala and Tembinkosi F. Nkalashe Copyright © 2015 Winter Sinkala and Tembinkosi F. Nkalashe. All rights reserved. Infinitely Many Standing Waves for the Nonlinear Chern-Simons-Schrödinger Equations Wed, 19 Aug 2015 12:13:14 +0000 We prove the existence of infinitely many solutions of the nonlinear Chern-Simons-Schrödinger equations under a wide class of nonlinearities. This class includes the standard power-type nonlinearity with exponent . This extends the previous result which covers the exponent . Jinmyoung Seok Copyright © 2015 Jinmyoung Seok. All rights reserved. Hom--Operators and Hom-Yang-Baxter Equations Wed, 12 Aug 2015 08:07:00 +0000 In Hom-Lie set, we introduce the concept of Hom--operators and study its relation with classical Hom-Yang-Baxter equation, as well as left-symmetric Hom-algebras. We construct the corresponding relation between left-symmetric Hom-algebras and Hom-1-cocycles, which are both related to classical Hom-Yang-Baxter equation. Moreover, in Hom-algebra setting, we establish the equivalent relationship between AHYBE (associative Hom-Yang-Baxter equations) and -operators on Frobenius monoidal Hom-algebras. Yuanyuan Chen and Liangyun Zhang Copyright © 2015 Yuanyuan Chen and Liangyun Zhang. All rights reserved. Second-Order Integrals for Systems in Involving Spin Thu, 30 Jul 2015 06:09:59 +0000 In two-dimensional Euclidean plane, existence of second-order integrals of motion is investigated for integrable Hamiltonian systems involving spin (e.g., those systems describing interaction between two particles with spin 0 and spin 1/2) and it has been shown that no nontrivial second-order integrals of motion exist for such systems. İsmet Yurduşen Copyright © 2015 İsmet Yurduşen. All rights reserved. Spatial Rotation of the Fractional Derivative in Two-Dimensional Space Mon, 27 Jul 2015 08:32:47 +0000 The transformations of the partial fractional derivatives under spatial rotation in are derived for the Riemann-Liouville and Caputo definitions. These transformation properties link the observation of physical quantities, expressed through fractional derivatives, with respect to different coordinate systems (observers). It is the hope that such understanding could shed light on the physical interpretation of fractional derivatives. Also it is necessary to be able to construct interaction terms that are invariant with respect to equivalent observers. Ehab Malkawi Copyright © 2015 Ehab Malkawi. All rights reserved. Numerical Solutions for the Eighth-Order Initial and Boundary Value Problems Using the Second Kind Chebyshev Wavelets Tue, 14 Jul 2015 06:55:49 +0000 A collocation method based on the second kind Chebyshev wavelets is proposed for the numerical solution of eighth-order two-point boundary value problems (BVPs) and initial value problems (IVPs) in ordinary differential equations. The second kind Chebyshev wavelets operational matrix of integration is derived and used to transform the problem to a system of algebraic equations. The uniform convergence analysis and error estimation for the proposed method are given. Accuracy and efficiency of the suggested method are established through comparing with the existing quintic B-spline collocation method, homotopy asymptotic method, and modified decomposition method. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literatures. Xiaoyong Xu and Fengying Zhou Copyright © 2015 Xiaoyong Xu and Fengying Zhou. All rights reserved. Higher-Stage Noether Identities and Second Noether Theorems Mon, 13 Jul 2015 06:20:03 +0000 The direct and inverse second Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Such Lagrangian theory is characterized by a hierarchy of nontrivial higher-stage Noether identities which is described in the homology terms. If a certain homology regularity condition holds, one can associate with a reducible degenerate Lagrangian the exact Koszul–Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete nontrivial Noether and higher-stage Noether identities. The second Noether theorems associate with the above-mentioned Koszul–Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this operator is extended to the nilpotent BRST operator which brings the above-mentioned cochain sequence into the BRST complex and provides a BRST extension of an original Lagrangian. G. Sardanashvily Copyright © 2015 G. Sardanashvily. All rights reserved. Bound-State Solution of s-Wave Klein-Gordon Equation for Woods-Saxon Potential Sun, 12 Jul 2015 06:59:29 +0000 The bound-state solution of s-wave Klein-Gordon equation is calculated for Woods-Saxon potential by using the asymptotic iteration method (AIM). The energy eigenvalues and eigenfunctions are obtained for the required condition of bound-state solutions. Eser Olğar and Haydar Mutaf Copyright © 2015 Eser Olğar and Haydar Mutaf. All rights reserved. Simple Modules for Modular Lie Superalgebras , , and Thu, 09 Jul 2015 14:24:26 +0000 This paper constructs a series of modules from modular Lie superalgebras , , and over a field of prime characteristic . Cartan subalgebras, maximal vectors of these modular Lie superalgebras, can be solved. With certain properties of the positive root vectors, we obtain that the sufficient conditions of these modules are irreducible -modules, where , , and . Zhu Wei, Qingcheng Zhang, Yongzheng Zhang, and Chunyue Wang Copyright © 2015 Zhu Wei et al. All rights reserved. Formal Pseudodifferential Operators in One and Several Variables, Central Extensions, and Integrable Systems Sun, 05 Jul 2015 11:51:02 +0000 We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in one and several variables in a general algebraic context. We focus mainly on the construction and classification of nontrivial central extensions. As applications, we construct hierarchies of centrally extended Lie algebras of formal differential operators in one and several variables, Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form. Jarnishs Beltran and Enrique G. Reyes Copyright © 2015 Jarnishs Beltran and Enrique G. Reyes. All rights reserved. A Simpler GMRES Method for Oscillatory Integrals with Irregular Oscillations Thu, 02 Jul 2015 11:17:37 +0000 A simpler GMRES method for computing oscillatory integral is presented. Theoretical analysis shows that this method is mathematically equivalent to the GMRES method proposed by Olver (2009). Moreover, the simpler GMRES does not require upper Hessenberg matrix factorization, which leads to much simpler program and requires less work. Numerical experiments are conducted to illustrate the performance of the new method and show that in some cases the simpler GMRES method could achieve higher accuracy than GMRES. Qinghua Wu and Meiying Xiang Copyright © 2015 Qinghua Wu and Meiying Xiang. All rights reserved. Existence of Exponential -Stability Nonconstant Equilibrium of Markovian Jumping Nonlinear Diffusion Equations via Ekeland Variational Principle Thu, 25 Jun 2015 07:09:37 +0000 The authors obtained a delay-dependent exponential -stability criterion for a class of Markovian jumping nonlinear diffusion equations by employing the Lyapunov stability theory and some variational methods. As far as we know, it is the first time to apply Ekeland variational principle to obtain the existence of exponential stability equilibrium of -Laplacian dynamic system so that some methods used in this paper are different from those methods of many previous related literatures. In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay. Ruofeng Rao and Shouming Zhong Copyright © 2015 Ruofeng Rao and Shouming Zhong. All rights reserved.