On the Use of Recursive Evaluation of Derivatives and Padé Approximation to Solve the Blasius Problem

The Blasius problem is one of the well-known problems in fluid mechanics in the study of boundary layers. It is described by a third-order ordinary differential equation derived from the Navier-Stokes equation by a similarity transformation. Crocco and Wang independently transformed this third-order problem further into a second-order differential equation. Classical series solutions and their Padé approximants have been computed. These solutions however require extensive algebraic manipulations and significant computational effort. In this paper, we present a computational approach using algorithmic differentiation to obtain these series solutions. Our work produces results superior to those reported previously. Additionally, using increased precision in our calculations, we have been able to extend the usefulness of the method beyond limits where previous methods have failed.

A New Flux Splitting Scheme Based on Toro-Vazquez and HLL Schemes for the Euler Equations

This paper presents a new flux splitting scheme for the Euler equations. The proposed scheme termed TV-HLL is obtained by following the Toro-Vazquez splitting (Toro and Vázquez-Cendón, 2012) and using the HLL algorithm with modified wave speeds for the pressure system. Here, the intercell velocity for the advection system is taken as the arithmetic mean. The resulting scheme is more accurate when compared to the Toro-Vazquez schemes and also enjoys the property of recognition of contact discontinuities and shear waves. Accuracy, efficiency, and other essential features of the proposed scheme are evaluated by analyzing shock propagation behaviours for both the steady and unsteady compressible flows. The accuracy of the scheme is shown in 1D test cases designed by Toro.

Localized Pulsating Solutions of the Generalized Complex Cubic-Quintic Ginzburg-Landau Equation

We study the dynamics of the localized pulsating solutions of generalized complex cubic-quintic Ginzburg-Landau equation (CCQGLE) in the presence of intrapulse Raman scattering (IRS). We present an approach for identification of periodic attractors of the generalized CCQGLE. Using ansatz of the travelling wave and fixing some relations between the material parameters, we derive the strongly nonlinear Lienard-Van der Pol equation for the amplitude of the nonlinear wave. Next, we apply the Melnikov method to this equation to analyze the possibility of existence of limit cycles. For a set of fixed parameters we show the existence of limit cycle that arises around a closed phase trajectory of the unperturbed system and prove its stability. We apply the Melnikov method also to the equation of Duffing-Van der Pol oscillator used for the investigation of the influence of the IRS on the bandwidth limited amplification. We prove the existence and stability of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the corresponding unperturbed system. The condition of existence of the limit cycle derived here coincides with the relation between the critical value of velocity and the amplitude of the solitary wave solution (Uzunov, 2011).

Fourier Splitting Method for Kawahara Type Equations

In this work, we integrate numerically the Kawahara and generalized Kawahara equation by using an algorithm based on Strang’s splitting method. The linear part is solved using the Fourier transform and the nonlinear part is solved with the aid of the exponential operator method. To assess the accuracy of the solution, we compare known analytical solutions with the numerical solution. Further, we show that as t increases the conserved quantities remain constant.

Comparative DFT Study of Phytochemical Constituents of the Fruits of Cucumis trigonus Roxb. and Cucumis sativus Linn.

The hepatoprotective active phytochemical constituents from the ethanolic extracts of the fruits of Cucumis trigonus Roxb. and Cucumis sativus Linn. were identified by GC-MS analysis. The density functional theory (DFT) of these molecules was calculated by density functional B3LYP methods using B3LYP/6-311++G(d,p) basis set. The optimized geometries of phytochemical constituents were evaluated. Physicochemical properties such as HOMO, LUMO, ionization potential, electron affinity, electronegativity, electrochemical potential, hardness, softness, electrophilicity, total energy, and dipole moment have also been recorded. These are very important parameters to understand the chemical reactivity and biological activity of the phytochemical constituents. Glycodeoxycholic acid and 2-(2-methylcyclohexylidene)-hydrazinecarboxamide were found to be effective drugs selected on the basis of their HOMO and LUMO energy gap and softness. The effective properties of these compounds may be due to the presence of amino, carbonyl, and alcohol as a functional group.

Solving Fractional Diffusion Equation via the Collocation Method Based on Fractional Legendre Functions

A formulation of the fractional Legendre functions is constructed to solve the generalized time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre and path following methods. Analysis for the presented method is given and numerical results are presented.

Sinc Collocation Method for Solving the Benjamin-Ono Equation

We propose a simple, though powerful, technique for numerical solutions of the Benjamin-Ono equation. This approach is based on a global collocation method using Sinc basis functions. Some properties of the Sinc collocation method required for our subsequent development are given and utilized to reduce the computation of the Benjamin-Ono equation to a system of ordinary differential equations. The propagation of one soliton and the interaction of two solitons are used to validate our numerical method. The method is easy to implement and yields accurate results.

Spectral-Homotopy Perturbation Method for Solving Governing MHD Jeffery-Hamel Problem

We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problems. The technique is based on the standard homotopy perturbation method and blending of the Chebyshev pseudospectral methods. The implementation of the new approach is demonstrated by solving the MHD Jeffery-Hamel flow and the effect of MHD on the flow has been discussed. Comparisons are made between the proposed technique, the previous studies, the standard homotopy perturbation method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the presented approach. The results demonstrate that the new modification is more efficient and converges faster than the standard homotopy perturbation method at small orders. The MATLAB software has been used to solve all the equations in this study.

Prediction of Materials Density according to Number of Scattered Gamma Photons Using Optimum Artificial Neural Network

Through the study of scattered gamma beam intensity, material density could be obtained. Most important factor in this densitometry method is determining a relation between recorded intensity by detector and target material density. Such situation needs many experiments over materials with different densities. In this paper, using two different artificial neural networks, intensity of scattered gamma is obtained for whole densities. Mean relative error percentage for test data using best method is 1.27% that shows good agreement between the proposed artificial neural network model and experimental results.

Finite-Difference Simulation of Elastic Wave with Separation in Pure P- and S-Modes

Elastic wave equation simulation offers a way to study the wave propagation when creating seismic data. We implement an equivalent dual elastic wave separation equation to simulate the velocity, pressure, divergence, and curl fields in pure P- and S-modes, and apply it in full elastic wave numerical simulation. We give the complete derivations of explicit high-order staggered-grid finite-difference operators, stability condition, dispersion relation, and perfectly matched layer (PML) absorbing boundary condition, and present the resulting discretized formulas for the proposed elastic wave equation. The final numerical results of pure P- and S-modes are completely separated. Storage and computing time requirements are strongly reduced compared to the previous works. Numerical testing is used further to demonstrate the performance of the presented method.