Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive wavelet space‐time SUPG method for hyperbolic conservation laws

This article concerns with incorporating wavelet bases into existing streamline upwind Petrov‐Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space‐time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusion. Furthermore, we add the hierarchical diffusion only in the vicinity of discontinuities using the feature of wavelet bases in detection of location of discontinuities. Also, we again use the last feature of the wavelet bases to perform a postprocessing using a denosing technique based on a minimization formulation to reduce Gibbs oscillations near discontinuities while keeping other regions intact. Finally, we show the performance of the proposed combination through some numerical examples including Burgers’, transport, and wave equations as well as systems of shallow water equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2062–2089, 2017     

A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in two‐dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi‐implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element‐wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third‐order in time. Copyright © 2014 John Wiley & Sons, Ltd.     

Formation of Phosphorylated 3H‐Pyrroles from NefIsocyanidePerkow Adducts and Tosylmethyl Isocyanide

Imidoyl chlorides, generated from isocyanides and acyl chlorides, react with trialkyl phosphites, in a Perkow‐type reaction, to afford 3‐(alkylimino)‐2‐[(dialkyloxyphosphoryl)oxy]acrylates, which undergo a smooth reaction with tosylmethyl isocyanide (TsMIC) to furnish 4‐(alkylamino)‐3‐[(dialkyloxyphosphoryl)oxy]‐5‐[(4‐methylphenyl)sulfonyl]‐3H‐pyrrole‐3‐carboxylates in moderate‐to‐good yields.     

Thermal Decomposition Kinetics of Ethane Halides and Derivatives（C_2H_（6-n）X_n（n = 1～3）;X = F,Cl,Br）：DFT Study and NBO Analysis

A theoretical study of the thermal decomposition kinetics of ethane halides(C2H6-nXn(n = 1～3);X = F,Cl,Br) has been carried out at the B3LYP/6-31++G** and B3PW91/631++G** levels of theory.Among these methods and comparison of activation parameters with available experimental values,the B3PW91/6-31++G** method is in good agreement with the experimental data.The analysis of bond order and natural bond orbital(NBO) charges,bond indexes,and synchronicity parameters suggest the elimination of HX in reactions 1～9(HF:compounds 1～3,HCl:compounds 4～6,and HBr:compounds 7～9) occur through a concerted and slightly asynchronous four-membered cyclic transition state type of mechanism.     

Fabrication of carbon nanofiber by pyrolysis of freeze-dried cellulose nanofiber

The possibility of fabricating carbon nanofibers from cellulose nanofibers was investigated. Cellulose nanofiber of ~50 nm in diameter was produced using ball milling in an eco-friendly manner. The effect of the drying techniques of cellulose nanofibers on the morphology of carbon residue was studied. After pyrolysis of freeze-dried cellulose nanofibers below 600 °C, amorphous carbon fibers of ~20 nm in diameter were obtained. The pyrolysis of oven-dried precursors resulted in the loss of original fibrous structures. The different results arising from the two drying techniques are attributed to the difference in the spatial distance between cellulose nanofiber precursors.     

Phytochemical analysis and antimicrobial activities of methanolic extracts of leaf, stem and root from different varieties of Labisa pumila Benth

A local herb, Kacip Fatimah, is famous amongst Malay women for its uses in parturition; however, its phytochemical contents have not been fully documented. Therefore, a study was performed to evaluate the phenolics, flavonoids, and total saponin contents, and antibacterial and antifungal properties of the leaf, stem and root of three varieties of Labisia pumila Benth. Total saponins were found to be higher in the leaves of all three varieties, compared to the roots and stems. Leaves of var. pumila exhibited significantly higher total saponin content than var. alata and lanceolata, with values of 56.4, 43.6 and 42.3 mg diosgenin equivalent/g dry weight, respectively. HPLC analyses of phenolics and flavonoids in all three varieties revealed the presence of gallic acid, caffeic acid, rutin, and myricetin in all plant parts. Higher levels of flavonoids (rutin, quercitin, kaempferol) were observed in var. pumila compared with alata and lanceolata, whereas higher accumulation of phenolics (gallic acid, pyrogallol) was recorded in var. alata, followed by pumila and lanceolata. Antibacterial activities of leaf, stem and root extracts of all varieties determined against both Gram positive (Micrococcus luteus, Bacillus subtilis B145, Bacillus cereus B43, Staphylococcus aureus S1431) and Gram negative (Enterobacter aerogenes, Klebsiella pneumonia K36, Escherichia coli E256, Pseudomonas aeruginosa PI96) pathogens showed that crude methanolic extracts are active against these bacteria at low concentrations, albeit with lower antibacterial activity compared to kanamycin used as the control. Antifungal activity of methanolic extracts of all plant parts against Fusarium sp., Candida sp. and Mucor using the agar diffusion disc exhibited moderate to appreciable antifungal activities compared to streptomycin used as positive control.     

Two class of synchronous matrix multisplitting schemes for solving linear complementarity problems

Many problems in the areas of scientific computing and engineering applications can lead to the solution of the linear complementarity problem LCP (M,q). It is well known that the matrix multisplitting methods have been found very useful for solving LCP (M,q). In this article, by applying the generalized accelerated overrelaxation (GAOR) and the symmetric successive overrelaxation (SSOR) techniques, we introduce two class of synchronous matrix multisplitting methods to solve LCP (M,q). Convergence results for these two methods are presented when M is an H-matrix (and also an M-matrix). Also the monotone convergence of the new methods is established. Finally, the numerical results show that the introduced methods are effective for solving the large and sparse linear complementary problems.     

A hybrid legendre tau method for the solution of a class of nonlinear wave equations with nonlinear dissipative terms

This article presents a technique based on the hybrid Legendre tau‐finite difference method to solve the fourth order wave equation which arises in the elasto‐plastic‐microstructure models for longitudinal motion of an elasto‐plastic bar. Illustrative examples and numerical results obtained using new technique demonstrate that the proposed approach is efficient in treating longitudinal equation of ealsto‐plastic bar. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1055–1071, 2011