Unconditionally stable high accuracy compact difference schemes for multi-space dimensional vibration problems with simply supported boundary conditions

In this paper, the fourth order parabolic partial differential equation, that governs the behavior of a vibrating beam, is solved by using the Adomian Decomposition Method. The solution is derived in the form of a power series with easily computable components. The nonhomogeneous problem is quickly solved by observing the self-canceling noise terms whose sum vanishes in the limit. Comparing this methodology with some known techniques shows that the present approach is highly accurate.

We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic partial differential equations (PDE’s). We then study their exit problem as they exit nice domains in $\mathbb{R}^d$ , and connect it to elliptic PDE’s. We show that these processes have the peculiar property that they solve fourth order parabolic PDE’s, but their exit distribution—at least in the standard Brownian time process case—solves the usual second order Dirichlet problem. We recover fourth order PDE’s in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense.

Some discrete Galerkin methods are formulated for approximately solving a linear fourth order parabolic partial differential equation in $\Omega \times (0,T]$, where $\Omega $ is an arbitrary bounded domain in $\mathbb{R}^2 $. Error estimates are obtained which show that these methods are second order correct in time. When $\Omega $ is a rectangle, alternating-direction Galerkin methods are discussed.

Asymptotic behavior of solutions of second order parabolic partial differential equations with unbounded coefficients

A class of two- and three-level implicit methods of order two in time and four in space based on half-step discretization for two-dimensional fourth order quasi-linear parabolic equations

Domain decomposition and expanded mixed method for parabolic partial differential equations

In this research a comprehensive modeling framework for a piezoelectrically-actuated cantilever beam is developed and a detailed model and vibration analyses is performed. To achieve this goal, the governing dynamics for the system as well as boundary conditions are derived using the extended Hamilton's principle. The equations of motion of cantilever beam are derived according to the Euler-Bernoulli, Rayleigh and Timoshenko theories separately. The Euler-Bernoulli theory neglects the effects of rotary inertia and shear deformation and is only applicable to analysis of thin beams. The Rayleigh theory considers the effect of rotary inertia, while the Timoshenko theory considers the effects of both rotary inertia and shear deformation for thick beams. It is evident from the nature of discontinuous geometry of system, equation of stress-strain relationship are modified as shown in theory subsection meanwhile the natural surface in the composite (beam-piezoelectric layer) portion of the cantilever beam must be considered in this stage of calculations. Then the first five natural frequencies of this composite system are obtained by those three different theories and the results are compared. Relevant mode shapes are also drawn and effects of including rotary inertia and shear deformation are discussed for slender and stocky beams. Then, the forced vibration problem is solved and the cantilever tip deflection is obtained in which applied voltage to the piezoelectric layer is considered to be a unit-step input. The results are compared again for slender and stocky beams.

An arms race with deteriorating armaments

Chebyshev series solution of the two dimensional heat equation

MATLAB Distributed Computing Server (MDCS) and Parallel Virtual Machine (PVM) software are two types of distributed computing environments. MDCS is recently used in selecting the best network training algorithm and assessing the effect of parallelization. Since it is practical and user friendly, it gives PVM the opportunity to become the communication paradigm of choice. The PVM provides a powerful set of process control and dynamic resource management features. In the distributed parallel computing (DPC), however, both solutions have different strengths and limitations. Based on these concerns, this paper compares the numerical analysis for mathematical modeling of large sparse 2D and 3D of second order parabolic partial differential equations (PDE) on MDCS and PVM based on parallel performance indicators (PPI). The PDE geometry is discretized into a sparse grid structure using the FDM method. In using a method with the highest accuracy, Parallel Alternating Group Explicit (PAGE) scheme was chosen. The parallel strategies focus on the PAGE's convergence speed and various domain de-composition techniques, as well as a block iterative scheme and load balance using fine granular techniques. Furthermore, comparison of distributed computing environments also relies on multiple processors running on Unix-like operating systems with Fedora installed to support large-scale simulations. The analysis and validation of PPI for both communication software is also investigated in this paper. For a sequential algorithm, accuracy, estimation of error, and stability are employed as indicators. As a conclusion, based on the PPI and numerical analysis, the tables and graphs show that compared to MDCS, PVM is a better environment for solving multidimensional parabolic PDE modeling.KeywordsDistributed computingParabolic PDEsMDCSPVM

Chaos synchronization is a topic of great interest, due to its observation in a huge variety of phenomena of different nature. We study synchronization of two chaotic oscillators in a Master- Slave configuration. The two dynamic systems are coupled via a directed feedback that randomly switches among a finite set of given constant function at a prescribed time rate. And we use Lyapunov stability theory. This paper discussed the using of lag synchronization approach, and provided the equilibrium solutions of a new class of higher order parabolic partial differential equations to be applicable for Lorenz chaotic system in order to minimize the dynamical error of large Lorenz chaotic system

In this paper, dynamic response of the horizontal concrete beam subjected to seismic ground excitation is investigated. The structure is reinforced by Fe2O3 nanoparticles which have the magnetic properties. The hyperbolic shear deformation beam theory (HSDBT) is used for mathematical modeling of the structure. Based on the Mori-Tanaka model, the effective material properties of concrete beam is calculated considering the agglomeration of Fe2O3 nanoparticles. Applying energy method and Hamilton\'s principle, the motion equations are derived. Harmonic differential quadrature method (HDQM) along with Newmark method is utilized for numerical solution of the motion equations. The effects of different parameters such as volume fraction and agglomeration of Fe2O3 nanoparticles, magnetic field, boundary conditions and geometrical parameters of concrete beam are studied on the dynamic response of the structure. In order to validation of this work, an exact solution is used for comparing the numerical and analytical results. The results indicated that applying magnetic field decreases the of the structure up to 54 percent. In addition, increase too much the magnetic field (Hx>5e8 A/m) does not considerable effect on the reduction of the maximum dynamic displacement.

A unified modeling approach for rotating flexible shaft-disk systems with general boundary and coupling conditions

In this study, free vibration of square cross-sectioned aluminum beams is investigated analytically and numerically under four different boundary conditions: Clamped-Clamped (C-C), Clamped-Free (C-F), Clamped-Simply Supported (C-SS) and Simply Supported-Simply Supported (SS-SS). Analytical solution is carried out using Euler-Bernoulli beam theory and Newton Raphson Method. First, the equations of motion are provided. Then, solutions including the effects of the geometric characteristics, and boundary conditions are obtained and discussed for the natural frequencies of the first three modes. To confirm the reliability of the vibration analysis carried out in the present paper as well, all the analytical results are checked with the corresponding numerical results obtained from the finite-element-method (FEM) based software called ANSYS. Numerical and analytical results are found to be good agreement.

In practical engineering applications, convection diffusion equations are generally used to describe the transport processes involving fluid motion, heat transfer, astrophysics, oceanography, meteorology, semiconductors, hydraulics, pollutant & sediment transport and chemical engineering. In this paper, a high order compact difference scheme based on the fourth order compact difference scheme in spatial discretization and the fourth order Runge-Kutta method in time integration is proposed for the numerical simulation of the unsteady convection-diffusion equation. The validity and effectiveness of the proposed method is firstly tested by a two-dimensional convection-diffusion equation with a Gaussian pulse type concentration. The L 2 error norms are used to measure differences between the exact and numerical solutions and compared to those obtained by other methods. It is shown that the results obtained by proposed method agree very well with the analytical solutions and is more accurate than other methods. Then, a two-dimensional non-linear Burgers equation is used to validate the effectiveness of the proposed method used to solve the non-linear convection-diffusion equation, which also models well. Finally, the Taylor’s vortex problem is investigated by the proposed method and good agreement is obtained with the exact solutions. From the three test problems, it is shown that the proposed high order compact difference scheme is an efficient and accurate method to simulate the transport problems and also can be applied to many engineering problems.