Corrections on ‘Commutativity of second order time varying systems’

In a recent investigation of the conditions for the commutativity of time varying linear systems (Köksal 1982), the casθb ₁?αa ₁=0 was overlooked, and as a result of this some incomplete and wrong results were obtained. In this contribution, these results are corrected and summarized in the form of a main theorem, and an extension to higher order systems is indicated.     

On two linearized difference schemes for Burgers’ equation

Burgers’ equation can model several physical phenomena. In the first part of this work, we derive a three-level linearized difference scheme for Burgers’ equation, which is then proved to be energy conservative, unique solvable and unconditionally convergent in the maximum norm by the energy method combining with the inductive method. In the second part of the work, we prove the L_∞ unconditional convergence of a two-level linearized difference scheme for Burgers’ equation proposed by Sheng [A new difference scheme for Burgers equation, J. Jiangsu Normal Univ. 30 (2012), pp. 39–43], which was proved previously conditionally convergent.     

Bi-discontinuous time step integration algorithms––Part 1: first order equations

In this paper, time step integration algorithms for linear first order equations with both the initial and final conditions weakly enforced are investigated. Discontinuous jumps may appear at the beginning and at the end of a time interval under consideration. The initial conditions are usually given while the final conditions are artificial variables as in the hybrid finite element formulation. If the approximate solution within a time interval is assumed to be a polynomial of degree n, there are n+2 unknowns in the formulation. It is shown that the order of accuracy of the approximate solution would be at least n in general. If the weighting parameters (and hence the weighting functions) are chosen carefully, the order of accuracy of the approximate solution at the end of a time interval given by the final condition can be improved to 2n+2. Besides, unconditionally stable algorithms equivalent to the generalized Padé approximations can be constructed systematically. The time-discontinuous Galerkin and bi-discontinuous Galerkin methods are treated as special cases. The weighting parameters and the corresponding weighting functions are given explicitly. Furthermore, it is shown that the accuracy of the particular solutions is compatible with the homogenous solutions if the proposed weighting functions are employed.     

Volterra series for solving weakly non-linear partial differential equations: application to a dissipative Burgers’ equation

A method to solve weakly non-linear partial differential equations with Volterra series is presented in the context of single-input systems. The solution x(z,t) is represented as the output of a z-parameterized Volterra system, where z denotes the space variable, but z could also have a different meaning or be a vector. In place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems, the problem turns into solving linear differential equations. This paper introduces the method on an example: a dissipative Burgers'equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments. The kernels are computed analytically in the Laplace domain. As a new result, writing the Volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics. Furthermore, the ability of the Volterra system to treat other signals constitutes an improvement for the sound synthesis. It allows the simulation for any regime, including attacks and transients. Numerical simulations are presented and their validity are discussed.     

An eighth order exponentially-fitted method for the numerical integration of the Schrödinger equation

An eighth order exponentially-fitted method is developed for the numerical solution of the Schrödinger equation. The formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions. An error analysis is also given. Numerical and theoretical results indicate that the new method is much more accurate than other classical and exponentially fitted methods.     

High order Bessel fitting methods for the numerical integration of the Schrödinger equation

In this paper we show how to construct explicit multistep algorithms for an accurate and efficient numerical integration of the radial Schr?dinger equation. The proposed methods are Bessel fitting, that is to say, they integrate exactly any linear combination of Bessel and Newman functions and ordinary polynomials. They are the first of the like methods that can achieve any order.     

A novel numerical method for a class of problems with the transition layer and Burgers’ equation

A numerical method for solving a class of quasi-linear singular two-point boundary value problems with a transition layer is presented in this paper. For the problem ? u _xx +a(u+f(x))u _x +b(x, u)=0, we develop a multiple scales method. First, this method solves the location of the transition layer, then it approximates the singular problem with reduced problems in the non-layer domain and pluses a layer corrected problem which nearly has an effect in the layer domain. Both problems are transformed into first-order problems which can be solved easily. For the problem ? u _xx +b(x, u)=0, we establish a similar method which approximate the problem with reduced problems and a two-point boundary value problem. Unsteady problems are also considered in our paper. We extend our method to solve Burgers’ equation problems by catching the transition layer with the formula of shock wave velocity and approximating it by a similar process.     

A new high-accuracy method based on off-step cubic polynomial approximations for the solution of coupled Burgers’ equations and Burgers–Huxley equation

Engineering with Computers - Using two off-step points and a central point, we discuss a new two-time-level implicit method of order three based on polynomial cubic spline approximations for the...     

The adaptive fuzzy time series model with an application to Taiwan’s tourism demand

In this study, an adaptive fuzzy time series model for forecasting Taiwan’s tourism demand is proposed to further enhance the predicted accuracy. We first transfer fuzzy time series data to the fuzzy logic group, assign weights to each period, and then use the proposed adaptive fuzzy time series model for forecasting in which an enrollment forecasting values is applied to obtain the smallest forecasting error. Finally, an illustrated example for forecasting Taiwan’s tourism demand is used to verify the effectiveness of proposed model and confirmed the potential benefits of the proposed approach with a very small forecasting error MAPE and RMSE.     

A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation

In this paper, an efficient numerical scheme based on uniform Haar wavelets and the quasilinearization process is proposed for the numerical simulation of time dependent nonlinear Burgers’ equation. The equation has great importance in many physical problems such as fluid dynamics, turbulence, sound waves in a viscous medium etc. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. More accurate solutions are obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents a solution of boundary value problems. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are compared with existing numerical solutions found in the literature. The use of the uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and has small computation costs.     

Efficient and accurate finite difference schemes for solving one-dimensional Burgers’ equation

In this paper, two efficient fourth-order compact finite difference algorithms have been developed to solve the one-dimensional Burgers’ equation: u _t +u u _x =ε u _xx . The methods are based on the Hopf–Cole transformation, Richardson's extrapolation, and multilevel grids. In both methods, we first transform the original nonlinear Burgers’ equation into a linear heat equation: w _t =ε w _xx using the Hopf–Cole transformation, which is given as u=?2ε (w _x /w). In the first method, the resulted heat equation is solved by the second-order accurate Crank–Nicholson algorithm while w _x is approximated by central finite difference, which is also second-order accurate. Richardson's extrapolation technique is then applied in both time and space to obtain fourth-order accuracy. In the second method, to reduce the cancellation error in approximating w _x , we derive the heat equation satisfied by w _x , which is then solved by the Crank–Nicholson algorithm. The original Dirichlet boundary condition is transformed into the Robin boundary condition, which is also approximated using second-order central finite difference. Finally, Richardson's extrapolation and multilevel grid techniques are applied in both time and space to obtain fourth-order accuracy. To study the efficiency, accuracy and robustness, we solved two numerical examples and the results are compared with those of two other higher-order methods proposed in W. Liao [An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Appl. Math. Comput. 206(2) (2008), pp. 755–764] and I.A. Hassanien, A.A. Salama, and H.A. Hosham [Fourth-order finite difference method for solving Burgers’ equation, Appl. Math. Comput. 170 (2005), pp. 781–800].     

Sensitivity analysis of shock wave Burgers’ equation via a novel algorithm based on scale-3 Haar wavelets

In this paper, a novel technique is being formulated for the numerical solutions of Shock wave Burgers' equations for planar and non-planar geometry. It is well known that Burgers' equation is sensitive to the perturbations in the diffusion term. Thus we use robustness of wavelets generated by dilation and translation of Haar wavelets on third scale to capture the sensitivity information. The present approach is an improved form of the scale-2 Haar wavelet method. The scheme is based on the forward finite difference scheme for time integration, scale-3 Haar wavelets for space integration and the nonlinearity has been tackled via quasilinearzation technique. Through scale-3 Haar wavelet analysis once the wavelet coefficient is calculated then we can compute the solutions at near the perturbation point. The computation cost of the present scheme is negligible. The proposed method is tested on six test problems to check its computational efficiency where the convergence analysis of scale-3 Haar wavelet method is the proof of our computational arguments.     

Investigation progress on key photonic integration for application in optical communication network

Proceeding from the consideration of the demands from the functional architecture of high speed, high capacity optical communication network, this paper points out that photonic integrated devices, including high speed response laser source, narrow band response photodetector high speed wavelength converter, dense wavelength multi/demultiplexer, low loss high speed response photo-switch and multi-beam coupler are the key components in the system. The investigation progress in the laboratory will be introduced.     

A numerical study of stationary solution of viscous Burgers’ equation using wavelet

In this paper we have studied the numerical stationary solution of viscous Burgers’ equation with Neumann boundary conditions by applying wavelet Galerkin method. Burns et al. [J. Burns, A. Balogh, D.S. Gilliam, and V. I. Shubov, Numerical stationary solutions for a viscous Burgers’ equation, J. Maths. Sys. Est. Contl. 8 (1998), pp. 1–16] have reported that for moderately small viscosity and for certain initial conditions, numerical solution approaches non-constant shock-type stationary solution though only possible actual stationary solution is a constant. We found that the wavelet Galerkin method precisely captures the correct steady-state solution. The solutions obtained were impressive and verify theoretical results.     

New solitary wave solutions of (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation

In this article, via the improved fractional subequation method, the fully analytical solutions of the (2+1)-dimensional space–time fractional Burgers equation and Korteweg–de Vries equation involving Jumarie’s modified Riemann–Liouville derivative have been derived. As a result, with the aid of symbolic computation, many types of new analytical solutions are obtained, which include new solitary wave, periodic wave and rational wave solutions. The graphical representations show that these gained solutions have abundant structures.     

Synthesis of stabilizing spacecraft control based on generalized Ackermann’s formula

Problem of modal synthesis of controllers and observers using the generalized Ackermann’s formula is solved for a spacecraft as a complex dynamic system with high interconnections. All possible controller matrices (the whole set of controllers) are obtained for solution of the problem of stabilization of orbital orientation of the spacecraft in inseparable channels of bank and yaw angles.     

Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods—Part I: Formulation and application to the 2-D Burgers’ equation

The lattice Boltzmann method is being increasingly employed in the field of computational fluid dynamics due to its computational efficiency. Floating-point operations in the lattice Boltzmann method involve local data and therefore allow easy cache optimization and parallelization. Due to this, the cache-optimized lattice Boltzmann method has superior computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size and therefore the efficiency of the lattice Boltzmann method for steady flows. To quantify the computational performance of the lattice Boltzmann method for steady flows, a comparison study between the lattice Boltzmann method (LBM) and the alternating direction implicit (ADI) method was performed using the 2-D steady Burgers’ equation. The comparison study showed that the LBM performs comparatively poor on high-resolution meshes due to smaller time step sizes, while on coarser meshes where the time step size is similar for both methods, the cache-optimized LBM performance is superior. Because flow domains can be discretized with multiblock grids consisting of coarse and fine grid blocks, the cache-optimized LBM can be applied on the coarse grid block while the traditional implicit methods are applied on the fine grid blocks. This paper finds the coupled cache-optimized lattice Boltzmann-ADI method to be faster by a factor of 4.5 over the traditional methods while maintaining similar accuracy.     

Prediction of members’ repurchase rates with time weight function

Customer relationship management (CRM) leverages historical users’ behaviors to dawn effort of enhancing customer satisfaction and loyalty. Thus, constructing a successful customer profile plays a critical role in CRM. In this study, we are expected to predict the repurchase rates for the registered members at the specific category of e-shop. However, customers’ preferences change over time. To capture the preference drifts of the members, we propose a novel and simple time function to increase/decrease the weight of the old data in evaluating various members’ past behaviors. Then, we construct a repurchase index with time factor (RIT) model to effectively predict repurchase rates. The marketers of e-shop can thus target the members with high repurchase rates. Experimental results with a real dataset have demonstrated that this RIT model can be practically implemented and provide satisfactory results.