A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation

In this paper, we employ the boundary-only meshfree method to find out numerical solution of the classical Boussinesq equation in one dimension. The proposed method in the current paper is a combination of boundary knot method and meshless analog equation method. The boundary knot technique is an integration free, boundary-only, meshless method which is used to avoid the known disadvantages of the method of fundamental solution. Also, we use the meshless analog equation method to replace the nonlinear governing equation with an equivalent nonhomogeneous linear equation. A predictor-corrector scheme is proposed to solve the resulted differential equation of the collocation. The numerical results and conclusions are obtained for both the ‘good’ and the ‘bad’ Boussinesq equations.     

Phorbol Esters from Jatropha Meal Triggered Apoptosis, Activated PKC-δ, Caspase-3 Proteins and Down-Regulated the Proto-Oncogenes in MCF-7 and HeLa Cancer Cell Lines

Jatropha meal produced from the kernel of Jatropha curcas Linn. grown in Malaysia contains phorbol esters (PEs). The potential benefits of PEs present in the meal as anticancer agent are still not well understood. Hence, this study was conducted to evaluate the cytotoxic effects and mode of actions of PEs isolated from Jatropha meal against breast (MCF-7) and cervical (HeLa) cancer cell lines. Isolated PEs inhibited cells proliferation in a dose-dependent manner of both MCF-7 and HeLa cell lines with the IC₅₀ of 128.6 ± 2.51 and 133.0 ± 1.96 μg PMA equivalents/mL respectively, while the values for the phorbol 12-myristate 13-acetate (PMA) as positive control were 114.7 ± 1.73 and 119.6 ± 3.73 μg/mL, respectively. Microscopic examination showed significant morphological changes that resemble apoptosis in both cell lines when treated with PEs and PMA at IC₅₀ concentration after 24 h. Flow cytometry analysis and DNA fragmentation results confirmed the apoptosis induction of PEs and PMA in both cell lines. The PEs isolated from Jatropha meal activated the PKC-δ and down-regulated the proto-oncogenes (c-Myc, c-Fos and c-Jun). These changes probably led to the activation of Caspase-3 protein and apoptosis cell death occurred in MCF-7 and HeLa cell lines upon 24 h treatment with PEs and PMA. Phorbol esters of Jatropha meal were found to be promising as an alternative to replace the chemotherapeutic drugs for cancer therapy.     

Phenolics and flavonoids compounds, phenylanine ammonia lyase and antioxidant activity responses to elevated CO₂ in Labisia pumila (Myrisinaceae)

A split plot 3 × 3 experiment was designed to examine the impact of three concentrations of CO? (400, 800 and 1,200 μmol·mol?1) on the phenolic and flavonoid compound profiles, phenylalanine ammonia lyase (PAL) and antioxidant activity in three varieties of Labisia pumila Benth. (var. alata, pumila and lanceolata) after 15 weeks of exposure. HPLC analysis revealed a strong influence of increased CO? concentration on the modification of phenolic and flavonoid profiles, whose intensity depended on the interaction between CO? levels and L. pumila varieties. Gallic acid and quercetin were the most abundant phenolics and flavonoids commonly present in all the varieties. With elevated CO? (1,200 μmol·mol?1) exposure, gallic acid increased tremendously, especially in var. alata and pumila (101-111%), whilst a large quercetin increase was noted in var. lanceolata (260%), followed closely by alata (201%). Kaempferol, although detected under ambient CO? conditions, was undetected in all varieties after exposure. Instead, caffeic acid was enhanced tremendously in var. alata (338~1,100%) and pumila (298~433%). Meanwhile, pyragallol and rutin were only seen in var. alata (810 μg·g?1 DW) and pumila (25 μg·g?1 DW), respectively, under ambient conditions; but the former compound went undetected in all varieties while rutin continued to increase by 262% after CO? enrichment. Interestingly, naringenin that was present in all varieties under ambient conditions went undetected under enrichment, except for var. pumila where it was enhanced by 1,100%. PAL activity, DPPH and FRAP also increased with increasing CO? levels implying the possible improvement of health-promoting quality of Malaysian L. pumila under high CO? enrichment conditions.     

The numerical solution of the non-linear integro-differential equations based on the meshless method

This article investigates the numerical solution of the nonlinear integro-differential equations. The numerical scheme developed in the current paper is based on the moving least square method. The moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. It consists of a local weighted least square fitting, valid on a small neighborhood of a point and only based on the information provided by its n closet points. Hence the method is a meshless method and does not need any background mesh or cell structures. The error analysis of the proposed method is provided. The validity and efficiency of the new method are demonstrated through several tests.     

Numerical solution of nonlinear Jaulent–Miodek and Whitham–Broer–Kaup equations

In this work we investigate the numerical solution of Jaulent–Miodek (JM) and Whitham–Broer–Kaup (WBK) equations. The proposed numerical schemes are based on the fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations (PDEs) with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in JM equation is diagonal but in WBK equation is not diagonal. However for WBK equation we can also implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrates that those methods are accurate and readily implemented.     

The Sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients

This paper deals with the numerical solution of classes of fractional convection–diffusion equations with variable coefficients. The fractional derivatives are described based on the Caputo sense. Our approach is based on the collocation techniques. The method consists of reducing the problem to the solution of linear algebraic equations by expanding the required approximate solution as the elements of shifted Legendre polynomials in time and the Sinc functions in space with unknown coefficients. The properties of Sinc functions and shifted Legendre polynomials are then utilized to evaluate the unknown coefficients. Several examples are given and the numerical results are shown to demonstrate the efficiency of the newly proposed method.     

Modeling and numerical simulation of linear and nonlinear spacecraft attitude dynamics and gravity gradient moments: A comparative study

In this paper linear and nonlinear models of spacecraft attitude dynamics equations and gravity gradient moments are investigated. In addition, effects of gravity gradient moments on attitude dynamics of the satellite are studied. The purpose of this paper is to present a comparison between nonlinear and linear models of spacecraft attitude dynamics and gravity gradient moments in order to determine divergence of linear approximation from the nonlinear model. Simulation results indicate that designer of spacecraft attitude control subsystem should be meticulous in applying linear approximation of equations especially in low earth orbits. Consequently, finding an upper bound for small angle to keep the linear model valid and precise enough would be a vital part of using linear approximation. Results supported by numerical examples demonstrate various features of this study.     

Numerical solution of the Klein–Gordon equation via He’s variational iteration method

In this paper, we present the solution of the Klein--Gordon equation. Klein--Gordon equation is the relativistic version of the Schrödinger equation, which is used to describe spinless particles. The He’s variational iteration method (VIM) is implemented to give approximate and analytical solutions for this equation. The variational iteration method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Application of variational iteration technique to this problem shows rapid convergence of the sequence constructed by this method to the exact solution. Moreover, this technique reduces the volume of calculations by avoiding discretization of the variables, linearization or small perturbations.