Advances in Mathematical Physics
[SJR: 0.332] [H-I: 10] [6 followers] Follow
Open Access journal
ISSN (Print) 1687-9120 - ISSN (Online) 1687-9139
Published by Hindawi [333 journals]
- Numerical Simulations of the Square Lid Driven Cavity Flow of Bingham
Fluids Using Nonconforming Finite Elements Coupled with a Direct Solver
Abstract: In this paper, numerical simulations are performed in a single and double lid driven square cavity to study the flow of a Bingham viscoplastic fluid. The governing equations are discretized with the help of finite element method in space and the nonconforming Stokes element is utilized which gives 2nd-order accuracy for velocity and 1st-order accuracy for pressure. The discretized systems of nonlinear equations are treated by using the Newton method and the inner linear subproablems are solved by the direct solver UMFPACK. A qualitative comparison is done with the results reported in the literature. In addition to these comparisons, some new reference data for the kinetic energy is generated. All these implementations are done in the open source software package FEATFLOW which is a general purpose finite element based solver package for solving partial differential equations.
PubDate: Mon, 20 Mar 2017 08:56:53 +000
- Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order
Abstract: The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalized -expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations.
PubDate: Mon, 20 Mar 2017 07:45:46 +000
- Existence and Uniqueness of Periodic Solutions for a Kind of Second-Order
Neutral Functional Differential Equation with Delays
Abstract: We consider a kind of second-order neutral functional differential equation. On the basis of Mawhin’s coincidence degree, the existence and uniqueness of periodic solutions are proved. It is indicated that the result is related to the deviating arguments. Moreover, we present two simulations to demonstrate the validity of analytical conclusion.
PubDate: Sun, 19 Mar 2017 00:00:00 +000
- The Study of Seismic Response on Accelerated Contained Fluid
Abstract: Ground-supported cylindrical tanks are strategically very important structures used to store a variety of liquids. This paper presents the theoretical background of fluid effect on tank when a fluid container is subjected to horizontal acceleration. Fluid excites the hydrodynamic (impulsive and convective) pressures, impulsive and convective (sloshing) actions. Seismic response of cylindrical fluid filling tanks fixed to rigid foundations was calculated for variation of the tank slenderness parameter. The calculating procedure has been adopted in Eurocode 8.
PubDate: Sun, 19 Mar 2017 00:00:00 +000
- Fully Discrete Local Discontinuous Galerkin Approximation for Time-Space
Fractional Subdiffusion/Superdiffusion Equations
Abstract: We focus on developing the finite difference (i.e., backward Euler difference or second-order central difference)/local discontinuous Galerkin finite element mixed method to construct and analyze a kind of efficient, accurate, flexible, numerical schemes for approximately solving time-space fractional subdiffusion/superdiffusion equations. Discretizing the time Caputo fractional derivative by using the backward Euler difference for the derivative parameter () or second-order central difference method for (), combined with local discontinuous Galerkin method to approximate the spatial derivative which is defined by a fractional Laplacian operator, two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusion/superdiffusion equations are proposed, respectively. Through the mathematical induction method, we show the concrete analysis for the stability and the convergence under the norm of the LDG schemes. Several numerical experiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes. The numerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerful for solving fractional partial differential equations.
PubDate: Thu, 16 Mar 2017 00:00:00 +000
- The Convergence Ball and Error Analysis of the Relaxed Secant Method
Abstract: A relaxed secant method is proposed. Radius estimate of the convergence ball of the relaxed secant method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given.
PubDate: Wed, 08 Mar 2017 00:00:00 +000
- The Dynamical Behavior of a Rigid Body Relative Equilibrium Position
Abstract: In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.
PubDate: Wed, 08 Mar 2017 00:00:00 +000
- A New Analysis Method for Chemotaxis-Induced Instability in Multispecies
Abstract: We propose a new method applying matrix theory to analyse the instability conditions of unique homogeneous coexistent state of multispecies host-parasitoid systems. We consider the eigenvalues of linearized operator of systems, and by dimensionality reduction, this infinite dimensional eigenproblem will be reduced to a parametrized finite dimensional eigenproblem, thereby applying combinatorial matrix theory to analyse the linear instability of such constant steady-state.
PubDate: Tue, 28 Feb 2017 14:20:05 +000
- Quasi-Exact Coulomb Dynamics of Charges of Which Are Equal
Abstract: For point charges of which are negative and equal quasi-exact periodic solutions of their Coulomb equation of motion are found. These solutions describe a motion of the negative charges around a coordinate axis in such a way that their coordinates coincide with vertices of a regular polygon in planes perpendicular to the axis along which the positive charge moves. The Weinstein and center Lyapunov theorems are utilized.
PubDate: Tue, 28 Feb 2017 10:12:07 +000
- Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion
and Wave Equations with Dirichlet Boundary Conditions
Abstract: A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method.
PubDate: Wed, 22 Feb 2017 13:37:29 +000
- Third-Order Approximate Solution of Chemical Reaction-Diffusion
Brusselator System Using Optimal Homotopy Asymptotic Method
Abstract: The objective of this paper is to investigate the effectiveness and performance of optimal homotopy asymptotic method in solving a system of nonlinear partial differential equations. Since mathematical modeling of certain chemical reaction-diffusion experiments leads to Brusselator equations, it is worth demanding a new technique to solve such a system. We construct a new efficient recurrent relation to solve nonlinear Brusselator system of equations. It is observed that the method is easy to implement and quite valuable for handling nonlinear system of partial differential equations and yielding excellent results at minimum computational cost. Analytical solutions of Brusselator system are presented to demonstrate the viability and practical usefulness of the method. The results reveal that the method is explicit, effective, and easy to use.
PubDate: Wed, 22 Feb 2017 00:00:00 +000
- Hyers-Ulam Stability and Existence of Solutions for Nigmatullin’s
Fractional Diffusion Equation
Abstract: We discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability of time-fractional order heat conduction equations via fractional Green function involving Wright function. In addition, an interesting existence result for solution is given.
PubDate: Tue, 21 Feb 2017 00:00:00 +000
- The Cauchy Problem for Space-Time Monopole Equations in Temporal and
Abstract: We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition and the temporal gauge condition .
PubDate: Mon, 20 Feb 2017 06:39:57 +000
- Turing Bifurcation and Pattern Formation of Stochastic Reaction-Diffusion
Abstract: Noise is ubiquitous in a system and can induce some spontaneous pattern formations on a spatially homogeneous domain. In comparison to the Reaction-Diffusion System (RDS), Stochastic Reaction-Diffusion System (SRDS) is more complex and it is very difficult to deal with the noise function. In this paper, we have presented a method to solve it and obtained the conditions of how the Turing bifurcation and Hopf bifurcation arise through linear stability analysis of local equilibrium. In addition, we have developed the amplitude equation with a pair of wave vector by using Taylor series expansion, multiscaling, and further expansion in powers of small parameter. Our analysis facilitates finding regions of bifurcations and understanding the pattern formation mechanism of SRDS. Finally, the simulation shows that the analytical results agree with numerical simulation.
PubDate: Sun, 12 Feb 2017 09:22:50 +000
- Third-Order Conditional Lie–Bäcklund Symmetries of Nonlinear
Abstract: The third-order conditional Lie–Bäcklund symmetries of nonlinear reaction-diffusion equations are constructed due to the method of linear determining equations. As a consequence, the exact solutions of the resulting equations are derived due to the compatibility of the governing equations and the admitted differential constraints, which are resting on the characteristic of the admitted conditional Lie–Bäcklund symmetries to be zero.
PubDate: Thu, 09 Feb 2017 13:19:02 +000
- The General Solution of Differential Equations with Caputo-Hadamard
Fractional Derivatives and Noninstantaneous Impulses
Abstract: Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.
PubDate: Thu, 09 Feb 2017 06:12:58 +000
- Unsteady Helical Flows of a Size-Dependent Couple-Stress Fluid
Abstract: The helical flows of couple-stress fluids in a straight circular cylinder are studied in the framework of the newly developed, fully determinate linear couple-stress theory. The fluid flow is generated by the helical motion of the cylinder with time-dependent velocity. Also, the couple-stress vector is given on the cylindrical surface and the nonslip condition is considered. Using the integral transform method, analytical solutions to the axial velocity, azimuthal velocity, nonsymmetric force-stress tensor, and couple-stress vector are obtained. The obtained solutions incorporate the characteristic material length scale, which is essential to understand the fluid behavior at microscales. If characteristic length of the couple-stress fluid is zero, the results to the classical fluid are recovered. The influence of the scale parameter on the fluid velocity, axial flow rate, force-stress tensor, and couple-stress vector is analyzed by numerical calculus and graphical illustrations. It is found that the small values of the scale parameter have a significant influence on the flow parameters.
PubDate: Wed, 08 Feb 2017 12:46:06 +000
- Dirichlet Forms Constructed from Annihilation Operators on Bernoulli
Abstract: The annihilation operators on Bernoulli functionals (Bernoulli annihilators, for short) and their adjoint operators satisfy a canonical anticommutation relation (CAR) in equal-time. As a mathematical structure, Dirichlet forms play an important role in many fields in mathematical physics. In this paper, we apply the Bernoulli annihilators to constructing Dirichlet forms on Bernoulli functionals. Let be a nonnegative function on . By using the Bernoulli annihilators, we first define in a dense subspace of -space of Bernoulli functionals a positive, symmetric, bilinear form associated with . And then we prove that is closed and has the contraction property; hence, it is a Dirichlet form. Finally, we consider an interesting semigroup of operators associated with on -space of Bernoulli functionals, which we call the -Ornstein-Uhlenbeck semigroup, and, by using the Dirichlet form, we show that the -Ornstein-Uhlenbeck semigroup is a Markov semigroup.
PubDate: Wed, 08 Feb 2017 00:00:00 +000
- A Non-Convex Partition of Unity and Stress Analysis of a Cracked Elastic
Abstract: A stress analysis using a mesh-free method on a cracked elastic medium needs a partition of unity for a non-convex domain whether it is defined explicitly or implicitly. Constructing such partition of unity is a nontrivial task when we choose to create a partition of unity explicitly. We further extend the idea of the almost everywhere partition of unity and apply it to linear elasticity problem. We use a special mapping to build a partition of unity on a non-convex domain. The partition of unity that we use has a unique feature: the mapped partition of unity has a curved shape in the physical coordinate system. This novel feature is especially useful when the enrichment function has polar form, , because we can partition the physical domain in radial and angular directions to perform a highly accurate numerical integration to deal with edge-cracked singularity. The numerical test shows that we obtain a highly accurate result without refining the background mesh.
PubDate: Tue, 07 Feb 2017 10:00:31 +000
- Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator
Abstract: Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered. The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution. The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at are also considered.
PubDate: Thu, 02 Feb 2017 06:23:35 +000
- Stability Monitoring of Batch Processes with Iterative Learning Control
Abstract: In recent years, the iterative learning control (ILC) is widely used in batch processes to improve the quality of the products. Stability is a preoccupation of batch processes when the ILC is applied. Focusing on the stability monitoring of batch processes with ILC, a method based on innerwise matrix with considering the uncertainty of the model and disturbance was proposed. First, the batch process with ILC was derived as a two-dimensional autoregressive and moving average (2D-ARMA) model. Then two kinds of stability indices are constructed based on the innerwise matrix through the identification of the 2D-ARMA. Finally, the statistical process control (SPC) chart was adopted to monitor those stability indices. Numerical results are presented to demonstrate the effectiveness of the proposed method.
PubDate: Tue, 31 Jan 2017 10:04:13 +000
- The Analytical Form of the Dispersion Equation of Elastic Waves in
Periodically Inhomogeneous Medium of Different Classes of Crystals
Abstract: The investigation of thermoelastic wave propagation in elastic media is bound to have much influence in the fields of material science, geophysics, seismology, and so on. The heat conduction equations and bound equations of motions differ by the difficulty level and presence of many physical and mechanical parameters in them. Therefore thermoelasticity is being extensively studied and developed. In this paper by using analytical matrizant method set of equation of motions in elastic media are reduced to equivalent set of first-order differential equations. Moreover, for given set of equations, the structure of fundamental solutions for the general case has been derived and also dispersion relations are obtained.
PubDate: Sun, 29 Jan 2017 13:57:18 +000
- Asymptotic Stability and Asymptotic Synchronization of Memristive
Abstract: Memristive regulatory-type networks are recently emerging as a potential successor to traditional complementary resistive switch models. Qualitative analysis is useful in designing and synthesizing memristive regulatory-type networks. In this paper, we propose several succinct criteria to ensure global asymptotic stability and global asymptotic synchronization for a general class of memristive regulatory-type networks. The experimental simulations also show the performance of theoretical results.
PubDate: Thu, 26 Jan 2017 14:36:30 +000
- Calculations on Lie Algebra of the Group of Affine Symplectomorphisms
Abstract: We find the image of the affine symplectic Lie algebra from the Leibniz homology to the Lie algebra homology . The result shows that the image is the exterior algebra generated by the forms . Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such a map is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology is isomorphic to . Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.
PubDate: Mon, 23 Jan 2017 00:00:00 +000
- Symmetries and Properties of the Energy-Casimir Mapping in the Ball-Plate
Abstract: In this paper a system derived by an optimal control problem for the ball-plate dynamics is considered. Symplectic and Lagrangian realizations are given and some symmetries are studied. The image of the energy-Casimir mapping is described and some connections with the dynamics of the considered system are presented.
PubDate: Wed, 18 Jan 2017 10:11:55 +000
- Principal Component Analysis in the Nonlinear Dynamics of Beams:
Purification of the Signal from Noise Induced by the Nonlinearity of Beam
Abstract: The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA).
PubDate: Mon, 16 Jan 2017 08:55:58 +000
- New Operational Matrix via Genocchi Polynomials for Solving
Fredholm-Volterra Fractional Integro-Differential Equations
Abstract: It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function . This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval . This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution . A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods.
PubDate: Mon, 16 Jan 2017 05:59:15 +000
- Analysis of Waterman’s Method in the Case of Layered Scatterers
Abstract: The method suggested by Waterman has been widely used in the last years to solve various light scattering problems. We analyze the mathematical foundations of this method when it is applied to layered nonspherical (axisymmetric) particles in the electrostatic case. We formulate the conditions under which Waterman’s method is applicable, that is, when it gives an infinite system of linear algebraic equations relative to the unknown coefficients of the field expansions which is solvable (i.e., the inverse matrix exists) and solutions of the truncated systems used in calculations converge to the solution of the infinite system. The conditions obtained are shown to agree with results of numerical computations. Keeping in mind the strong similarity of the electrostatic and light scattering cases and the agreement of our conclusions with the numerical calculations available for homogeneous and layered scatterers, we suggest that our results are valid for light scattering as well.
PubDate: Mon, 16 Jan 2017 00:00:00 +000
- On the Symmetries and Conservation Laws of the Multidimensional Nonlinear
Damped Wave Equations
Abstract: We carry out a classification of Lie symmetries for the ()-dimensional nonlinear damped wave equation with variable damping. Similarity reductions of the equation are performed using the admitted Lie symmetries of the equation and some interesting solutions are presented. Employing the multiplier approach, admitted conservation laws of the equation are constructed for some new, interesting cases.
PubDate: Thu, 12 Jan 2017 00:00:00 +000
- The Neumann Problem for a Degenerate Elliptic System Near Resonance
Abstract: This paper studies the following system of degenerate equations , , , , , Here is a bounded domain, and is the exterior normal vector on . The coefficient function may vanish in , with . We show that the eigenvalues of the operator are discrete. Secondly, when the linear part is near resonance, we prove the existence of at least two different solutions for the above degenerate system, under suitable conditions on , and .
PubDate: Wed, 11 Jan 2017 12:44:35 +000