A PIECEWISE–LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO–POINT BOUNDARY–VALUE PROBLEMS

Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left boundary of the interval and whose solution may be obtained by means of the Newton–Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearizat ion is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.     

Nitrous Oxide Adsorption and Decomposition on Zeolites and Zeolite-like Materials

Decomposition of N₂O on modified zeolites, crystalline titanosilicalites, and related amorphous systems is studied by the catalytic and spectroscopic methods. Zinc-containing HZSM-5 zeolites and titanosilicalites with moderate Ti/Si ratios are shown to exhibit a better catalytic performance in N₂O decomposition as compared with conventionally used Cu/HZSM-5 zeolites and amorphous Cu-containing catalysts. Dehydroxylation of the HZSM-5 zeolite by calcination at 1120 K results in an enhancement of the N₂O conversion. The mechanism of the reaction and the role of coordinatively unsaturated cations and Lewis acid sites in N₂O decomposition are discussed on the basis of the spectroscopic data.     

Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterers

We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet–Bloch waves through an infinite array of scatterers. The approach lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on topical examples from topological wave physics.     

Existence of Three Weak Solutions for a Class of Quasi-Linear Elliptic Operators with a Mixed Boundary Value Problem Containing $p(\cdot )$-Laplacian in a Variable Exponent Sobolev Space

In this paper, we consider a mixed boundary value problem to a class of nonlinear operators containing $p(\cdot )$-Laplacian. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of at least three weak solutions under some hypotheses on given functions and the values of parameters.     

Production of COx-Free Hydrogen and Few-Layer Graphene Nanoplatelets by Catalytic Decomposition of Methane over Ni-Lignin-Derived Nanoparticles

Nickel (Ni)-lignin nanocomposites were synthesized from nickel nitrate and kraft lignin then catalytically graphitized to few-layer graphene-encapsulated nickel nanoparticles (Ni@G). Ni@G nanoparticles were used for catalytic decomposition of methane (CDM) to produce COx-free hydrogen and graphene nanoplatelets. Ni@G showed high catalytic activity for methane decomposition at temperatures of 800 to 900 °C and exhibited long-term stability of 600 min time-on-stream (TOS) without apparent deactivation. The catalytic stability may be attributed to the nickel dispersion in the Ni@G sample. During the CDM reaction process, graphene shells over Ni@G nanoparticles were cracked and peeled off the nickel cores at high temperature. Both the exposed nickel nanoparticles and the cracked graphene shells may participate the CDM reaction, making Ni@G samples highly active for CDM reaction. The vacancy defects and edges in the cracked graphene shells serve as the active sites for methane decomposition. The edges are continuously regenerated by methane molecules through CDM reaction.     

Entropy Could Quantify Brain Activation Induced by Mechanical Impedance-Restrained Active Arm Motion: A Functional NIRS Study

Brain activation has been used to understand brain-level events associated with cognitive tasks or physical tasks. As a quantitative measure for brain activation, we propose entropy in place of signal amplitude and beta value, which are widely used, but sometimes criticized for their limitations and shortcomings as such measures. To investigate the relevance of our proposition, we provided 22 subjects with physical stimuli through elbow extension-flexion motions by using our exoskeleton robot, measured brain activation in terms of entropy, signal amplitude, and beta value; and compared entropy with the other two. The results show that entropy is superior, in that its change appeared in limited, well established, motor areas, while signal amplitude and beta value changes appeared in a widespread fashion, contradicting the modularity theory. Entropy can predict increase in brain activation with task duration, while the other two cannot. When stimuli shifted from the rest state to the task state, entropy exhibited a similar increase as the other two did. Although entropy showed only a part of the phenomenon induced by task strength, it showed superiority by showing a decrease in brain activation that the other two did not show. Moreover, entropy was capable of identifying the physiologically important location.     

水杨酸钴非等温热分解动力学研究

用热分析(TG/DTG)、X射线衍射(XRD)技术研究了固态物质水杨酸钴在空气中热分解的过程。热分析结果表明,水杨酸钴在空气中分两步分解,其失重率与理论计算失重率相吻合;XRD结果表明,水杨酸钴分解的终产物为Co3O4。用Friedman法和Flynn-Wall-Ozawa(FWO)法求取了分解过程的活化能E,并用多元线性回归和多元非线性回归法给出了可能的机理函数,由这些方法得到的动力学数据相互比较吻合。     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

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