First-order Space Research Articles

An accuracy-oriented self-splitting fuzzy classifier with support vector learning in high-order expanded consequent space

This paper proposes a self-splitting fuzzy classifier with support vector learning in expanded high-order consequent space (SFC-SVHC) for classification accuracy improvement. The SFC-SVHC expands the rule-mapped consequent space of a first-order Takagi-Sugeno (TS)-type fuzzy system by including high-order terms to enhance the rule discrimination capability. A novel structure and parameter learning approach is proposed to construct the SFC-SVHC. For structure learning, a variance-based self-splitting clustering (VSSC) algorithm is used to determine distributions of the fuzzy sets in the input space. There are no rules in the SFC-SVHC initially. The VSSC algorithm generates a new cluster by splitting an existing cluster into two according to a predefined cluster-variance criterion. The SFC-SVHC uses trigonometric functions to expand the rule-mapped first-order consequent space to a higher-dimensional space. For parameter optimization in the expanded rule-mapped consequent space, a support vector machine is employed to endow the SFC-SVHC with high generalization ability. Experimental results on several classification benchmark problems show that the SFC-SVHC achieves good classification results with a small number of rules. Comparisons with different classifiers demonstrate the superiority of the SFC-SVHC in classification accuracy.

Variational homotopy perturbation method for the approximate solution of the foam drainage equation with time and space fractional derivatives

In this paper, variational homotopy perturbation method (VHPM) is applied for solving the foam drainage equation with time and space-fractional derivatives. Numerical solutions are obtained for various values of the time and space-order derivative in (0,1]. For the first-order time and space derivative, compared with the exact solution, the result showed that the proposed method could be used as an alternative method for obtaining an analytic and approximate solution for different types of differential equations.

Note: A novel dual-channel time-of-flight mass spectrometer for photoelectron imaging spectroscopy

A novel dual-channel time-of-flight mass spectrometer (D-TOFMS) has been designed to select anions in the photoelectron imaging measurements. In this instrument, the radiation laser can be triggered precisely to overlap with the selected ion cloud at the first-order space focusing plane. Compared with that of the conventional single channel TOFMS, the in situ mass selection performance of D-TOFMS is significantly improved. Preliminary experiment results are presented for the mass-selected photodetachment spectrum of F(-) to demonstrate the capability of the instrument.

A fourth-order finite difference method based on spline in tension approximation for the solution of one-space dimensional second-order quasi-linear hyperbolic equations

In this paper, we propose a new three-level implicit nine-point compact finite difference formulation of order two in time and four in space directions, based on spline in tension approximation in x-direction and central finite difference approximation in t-direction for the numerical solution of one-space dimensional second-order quasi-linear hyperbolic equations with first-order space derivative term. We describe the mathematical formulation procedure in detail and also discuss how our formulation is able to handle a wave equation in polar coordinates. The proposed method, when applied to a general form of the telegrapher equation, is also shown to be unconditionally stable. Numerical examples are used to illustrate the usefulness of the proposed method. MSC: 65M06, 65M12.

Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation

Fractional advection-dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in porous medium. A space-time fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the first-order space derivative is replaced with Caputo or Riemann-Liouville derivative of order , and the second-order space derivative is replaced with the Caputo or the Riemann-Liouville fractional derivative of order . We derive the solution of the new equation in terms of Mittag-Leffler functions using Laplace transfrom. Some examples are given. The results from comparison let no doubt that the FADE is better in prediction than ADE.

Thermodynamical setting for gradient continuum theories with vectorial internal variables: Application to granular materials

The exploitation of the entropy principle in thermodynamics with Lagrange multipliers, the so–called Liu procedure, is based on the celebrated Liu theorem. In some recent papers, the Liu procedure has been generalized by considering the gradients of the governing equations as additional constraints. Here, we apply this generalized procedure to the equations of a continuum with vectorial internal variable and first-order non-local state space. Then, owing to the obtained results, we develop a thermodynamic model for continua with scalar microstructure.

Uniqueness of weighted Sobolev spaces with weakly differentiable weights

We prove that weakly differentiable weights w which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order p-Sobolev space, that isH1,p(Rd,wdx)=V1,p(Rd,wdx)=W1,p(Rd,wdx), where d∈N and p∈[1,∞). If w admits a (weak) logarithmic gradient ∇w/w which is in Llocq(wdx;Rd), q=p/(p−1), we propose an alternative definition of the weighted p-Sobolev space based on an integration by parts formula involving ∇w/w. We prove that weights of the form exp(−β|⋅|q−W−V) are p-admissible, in particular, satisfy a Poincaré inequality, where β∈(0,∞), W, V are convex and bounded below such that |∇W| satisfies a growth condition (depending on β and q) and V is bounded. We apply the uniqueness result to weights of this type. The associated nonlinear degenerate evolution equation is also discussed.

A POSTERIORI ERROR ESTIMATOR FOR LINEAR ELASTICITY BASED ON NONSYMMETRIC STRESS TENSOR APPROXIMATION

In this paper we present an a posteriori error estimator for the stabilized P1 nonconforming finite element method of the linear elasticity problem based on a nonsymmetric H(div)-conforming approximation of the stress tensor in the first-order Raviart-Thomas space. By combining the equilibrated residual method and the hypercircle method, it is shown that the error estimator gives a fully computable upper bound on the actual error. Numerical results are provided to confirm the theory and illustrate the effectiveness of our error estimator.

Minimal weak upper gradients in Newtonian spaces based on quasi-Banach function lattices

Properties of first-order Sobolev-type spaces on abstract metric measure spaces, so-called Newtonian spaces, based on quasi-Banach function lattices are investigated. The set of all weak upper gradients of a Newtonian function is of particular interest. Existence of minimal weak upper gradients in this general setting is proven and corresponding representation formulae are given. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.

Semi-analytical free-vibration analysis of piezoelectric adaptive beams using the distributed transfer function approach

SUMMARY A distributed transfer function (DTF) approach is proposed for the free-vibration analysis of moderately thick cantilever beams with symmetrically bonded piezoelectric patches. The piezoelectric adaptive structure is decomposed into three sections; the first and third sections contain only the bare parts of the beam before and after the patches, whereas the second section contains the beam part equipped with the piezoelectric patches bonded to its upper and lower surfaces. The analytical formulation assumes first-order shear deformation kinematics and linear electric potential through each patch thickness and uses the extended Hamilton's principle to derive the equations of motion and electromechanical boundary conditions. The latter are then transformed into a first-order state space equation using the DTF approach in order to get the short-circuit and open-circuit free-vibration problems that consider the equipotential physical conditions on the electroded upper and lower surfaces of the piezoelectric patches. The implemented DTF approach is validated by running three benchmarks from open literature. The comparison of the obtained results with those provided by three-dimensional finite elements using a commercial software shows reasonable correlation. Copyright © 2011 John Wiley & Sons, Ltd.

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

AbstractWe develop in this paper a discretization for the convection term in variable density unstationary Navier–Stokes equations, which applies to low‐order non‐conforming finite element approximations (the so‐called Crouzeix–Raviart or Rannacher–Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L2 stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L2 norm, second‐order space convergence for the velocity and first‐order space convergence for the pressure are observed. Copyright © 2010 John Wiley & Sons, Ltd.

Discovering latent topical structure by second-order similarity analysis

Computing document similarity directly from a “bag of words” vector space model can be problematic because term independence causes the relationships between synonymous terms and the contextual influences that determine the sense of polysemous terms to be ignored. This study compares two methods that potentially address these problems by deriving the higher order relationships that lie latent within the original first-order space. The first is latent semantic analysis (LSA), a dimension reduction method that is a well-known means of addressing the vocabulary mismatch problem in information retrieval systems. The second is the lesser known yet conceptually simple approach of second-order similarity (SOS) analysis, whereby latent similarity is measured in terms of mutual first-order similarity. Nearest neighbour tests show that SOS analysis derives similarity models that are superior to both first-order and LSA-derived models at both coarse and fine levels of semantic granularity. SOS analysis has been criticized for its computational complexity. A second contribution is the novel application of vector truncation to reduce run-time by a constant factor. Speed-ups of 4 to 10 times are achievable without compromising the structural gains achieved by full-vector SOS analysis.

Galerkin finite element approximation of symmetric space-fractional partial differential equations

In this paper, symmetric space-fractional partial differential equations (SSFPDE) with the Riesz fractional operator are considered. The SSFPDE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order 2 β ∈ (0, 1) and 2 α ∈ (1, 2], respectively. We prove that the variational solution of the SSFPDE exists and is unique. Using the Galerkin finite element method and a backward difference technique, a fully discrete approximating system is obtained, which has a unique solution according to the Lax–Milgram theorem. The stability and convergence of the fully discrete schemes are derived. Finally, some numerical experiments are given to confirm our theoretical analysis.

Diffusion with space memory modelled with distributed order space fractional differential equations

Distributed order fractional differential equations (Caputo, 1995, 2001; Bagley and Torvik, 2000a,b) were fi rst used in the time domain; they are here considered in the space domain and introduced in the constitutive equation of diffusion. The solution of the classic problems are obtained, with closed form formulae. In general, the Green functions act as low pass fi lters in the frequency domain. The major difference with the case when a single space fractional derivative is present in the constitutive equations of diffusion (Caputo and Plastino, 2002) is that the solutions found here are potentially more fl exible to represent more complex media (Caputo, 2001a). The difference between the space memory medium and that with the time memory is that the former is more fl exible to represent local phenomena while the latter is more fl exible to represent variations in space. Concerning the boundary value problem, the difference with the solution of the classic diffusion medium, in the case when a constant boundary pressure is assigned and in the medium the pressure is initially nil, is that one also needs to assign the fi rst order space derivative at the boundary.

Approximate solution of the fractional advection–dispersion equation

In this paper, we consider practical numerical method to solve a space–time fractional advection–dispersion equation with variable coefficients on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative, and the first-order and second-order space derivatives by the Riemann–Liouville fractional derivative, respectively. Here, a new method for solving this equation is proposed in the reproducing kernel space. The representation of solution is given by the form of series and the n-term approximation solution is obtained by truncating the series. The method is easy to implement and the numerical results show the accuracy of the method.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α ∈ ( 1 , 2 ] . The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order β ∈ ( 0 , 1 ) and of order α ∈ ( 1 , 2 ] , respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation

In this paper, we consider a Riesz fractional advection-dispersion equation (RFADE), which is derived from the kinetics of chaotic dynamics. The RFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives by the Riesz fractional derivatives of order a e (0, 1) and fi e (1, 2], respectively. We derive the fundamental solution for the Riesz fractional advection-dispersion equation with an initial condition (RFADE-IC). We investigate a discrete random walk model based on an explicit finite-difference approximation for the RFADE-IC and prove that the random walk model belongs to the domain of attraction of the corresponding stable distribution. We also present explicit and implicit difference approximations for the Riesz fractional advection-dispersion equation with initial and boundary conditions (RFADE-IBC) in a finite domain. Stability and convergence of these numerical methods for the RFADE-IBC are discussed. Some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Iterated elastic Brownian motions and fractional diffusion equations

Fractional diffusion equations of order ν ∈ ( 0 , 2 ) are examined and solved under different types of boundary conditions. In particular, for the fractional equation on the half-line [ 0 , + ∞ ) and with an elastic boundary condition at x = 0 , we are able to provide the general solution in terms of the density of the elastic Brownian motion. This permits us, for equations of order ν = 1 2 n , to write the solution as the density of the process obtained by composing the elastic Brownian motion with the ( n − 1 ) -times iterated Brownian motion. Also the limiting case for n → ∞ is investigated and the explicit form of the solution is expressed in terms of exponentials. Moreover, the fractional diffusion equations on the half-lines [ 0 , + ∞ ) and ( − ∞ , a ] with additional first-order space derivatives are analyzed also under reflecting or absorbing conditions. The solutions in this case lead to composed processes with general form X ( | I n − 1 ( t ) | ) , where only the driving process X is affected by drift, while the role of time is played by iterated Brownian motion I n − 1 .

Approximate analytical solution to fractional modified KdV equations

In this paper, the fractional modified Korteweg–de Vries equation (fmKdV) is introduced by replacing the first-order time and space derivatives by fractional derivatives of order α and β with 0 < α , β ≤ 1 . The homotopy-perturbation method (HPM) is applied to derive approximate analytical solutions to fmKdV.

Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation

In this paper, we consider a space–time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0, 1], and the first-order and second-order space derivatives by the Riemman–Liouville fractional derivatives of order β ∈ (0, 1] and of order γ ∈ (1, 2], respectively. For the space fractional derivatives D x β u ( x , t ) and D x γ u ( x , t ) , we adopted the Grünwald formula and the shift Grünwald formula, respectively. We propose an implicit difference method (IDM) and an explicit difference method (EDM) to solve this equation. Stability and convergence of these methods are discussed. Using mathematical induction, we prove that the IDM is unconditionally stable and convergent, but the EDM is conditionally stable and convergent. Numerical results are in good agreement with theoretical analysis.