Showing papers on "Legendre polynomials published in 2021"

Neuroevolution machine learning potentials: Combining high accuracy and low cost in atomistic simulations and application to heat transport

Abstract: We develop a neuroevolution-potential (NEP) framework for generating neural network-based machine-learning potentials. They are trained using an evolutionary strategy for performing large-scale molecular dynamics (MD) simulations. A descriptor of the atomic environment is constructed based on Chebyshev and Legendre polynomials. The method is implemented in graphic processing units within the open-source gpumd package, which can attain a computational speed over ${10}^{7}$ atom-step per second using one Nvidia Tesla V100. Furthermore, per-atom heat current is available in NEP, which paves the way for efficient and accurate MD simulations of heat transport in materials with strong phonon anharmonicity or spatial disorder, which usually cannot be accurately treated either with traditional empirical potentials or with perturbative methods.

A fuzzy fractional model of coronavirus (COVID-19) and its study with Legendre spectral method.

Abstract: The virus which belongs to the family of the coronavirus was seen first in Wuhan city of China. As it spreads so quickly and fastly, now all over countries in the world are suffering from this. The world health organization has considered and declared it a pandemic. In this presented research, we have picked up the existing mathematical model of corona virus which has six ordinary differential equations involving fractional derivative with non-singular kernel and Mittag-Leffler law. Another new thing is that we study this model in a fuzzy environment. We will discuss why we need a fuzzy environment for this model. First of all, we find out the approximate value of ABC fractional derivative of simple polynomial function ( t - a ) n . By using this approximation we will derive and developed the Legendre operational matrix of fractional differentiation for the Mittag-Leffler kernel fractional derivative on a larger interval [ 0 , b ] , b ⩾ 1 , b ∈ N . For the numerical investigation of the fuzzy mathematical model, we use the collocation method with the addition of this newly developed operational matrix. For the feasibility and validity of our method we pick up a particular case of our model and plot the graph between the exact and numerical solutions. We see that our results have good accuracy and our method is valid for the fuzzy system of fractional ODEs. We depict the dynamics of infected, susceptible, exposed, and asymptotically infected people for the different integer and fractional orders in a fuzzy environment. We show the effect of fractional order on the suspected, exposed, infected, and asymptotic carrier by plotting graphs.

A numerical approach for solving fractional optimal control problems with mittag-leffler kernel:

Abstract: In this work, we present a numerical approach based on the shifted Legendre polynomials for solving a class of fractional optimal control problems. The derivative is described in the Atangana–Balea...

Dispersion characteristics of guided waves in functionally graded anisotropic micro/nano-plates based on the modified couple stress theory

Abstract: In this paper, the acoustic wave motion characteristics of Lamb and SH waves in functionally graded (FG) anisotropic micro/nano-plates are studied based on the modified couple stress theory. A higher efficient computational approach, the extended Legendre orthogonal polynomial method (LOPM) is utilized to deduce solving process. This polynomial method does not need to solve the FG micro/nano-plates hierarchically, which provides a more realistic analysis model for FG micro/nano-plates and has high computational efficiency. Simultaneously, the solutions based on the global matrix method (GMM) are also deduced to verify the correctness of the polynomial method. Furthermore, the effects of size and material gradient are studied in detail. Numerical results show that the size effect causes wrinkles in Lamb wave dispersion curves, and the material gradient characteristic changes the amplitude and range of wrinkles. For SH waves, the length scale parameter Lx increases the cut-off frequency but does not change the overall trend of the dispersion curve; on the contrary, Lz does not change the cut-off frequency but causes the dispersion curve to show an upward trend.

A new soft computing approach for studying the wire coating dynamics with Oldroyd 8-constant fluid

Abstract: In this paper, a mathematical model for wire coating in the presence of pressure type die along with the bath of Oldroyd 8-constant fluid is presented. The model is governed by a partial differential equation, transformed into a nonlinear ordinary differential equation in dimensionless form through similarity transformations. We have designed a novel soft computing paradigm to analyze the governing mathematical model of wire coating by defining weighted Legendre polynomials based on Legendre neural networks (LeNN). Training of design neurons in the network is carried out globally by using the whale optimization algorithm (WOA) hybrid with the Nelder–Mead (NM) algorithm for rapid local convergence. Designed scheme (LeNN-WOA-NM algorithm) is applied to study the effect of variations in dilating constant (α), pressure gradient (Ω), and pseudoplastic constant β on velocity profile w(r) of fluid. To validate the proposed technique's efficiency, solutions and absolute errors are compared with the particle swarm optimization algorithm. Graphical and statistical performance of fitness value, absolute errors, and performance measures in terms of minimum, mean, median, and standard deviations further establishes the worth of the designed scheme for variants of the wire coating process.

Numerical approximations to the nonlinear fractional-order Logistic population model with fractional-order Bessel and Legendre bases

Abstract: The main aim of this manuscript is to obtain the approximate solutions of the nonlinear Logistic equation of fractional order by developing a collocation approach based on the fractional-order Bessel and Legendre functions. The main characteristic of these polynomial approximation techniques is that they transform the governing differential equation into a system of algebraic equations, thus the computational efforts will be greatly reduced. Our secondary aim is to show a comparative investigation on the use of these fractional-order polynomials and to examine their utilities to solve the model problem. Numerical experiments are carried out to demonstrate the validity and applicability of the presented techniques and comparisons are made with methods available in the standard literature. The methods perform very well in terms of efficiency and simplicity to solve this population model especially when the Legendre bases are utilized.

Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions

Abstract: In this article, a new and efficient operational matrix method based on the amalgamation of Fibonacci wavelets and block pulse functions is proposed for the solutions of time-fractional telegraph equations with Dirichlet boundary conditions. The Fibonacci polynomials and the corresponding wavelets along with their fundamental properties are briefly studied at first. These functions along with their nice characteristics are then utilized to formulate the Fibonacci wavelet operational matrices of fractional integrals. The proposed method reduces the fractional model into a system of algebraic equations, which can be solved using the classical Newton iteration method. Approximate solutions of the time-fractional telegraph equation are compared with the recently appeared Legendre and Sinc-Legendre wavelet collocation methods. The numerical outcomes show that the Fibonacci technique yields precise outcomes and is computationally more effective than the current ones.

Analysis of Third-Order Nonlinear Multi-Singular Emden–Fowler Equation by Using the LeNN-WOA-NM Algorithm

Abstract: In this paper, a novel soft computing algorithm is designed for the numerical solution of third-order nonlinear multi-singular Emden–Fowler equation (TONMS-EFE) using the strength of universal approximation capabilities of Legendre polynomials based Legendre neural networks supported with optimization power of the Whale Optimization Algorithm (WOA) and Nelder-Mead (NM) algorithm. Unsupervised error functions are constructed in terms of mean square error for governing TONMS-EF equations of first and second order. Unknown designed parameters in LeNN structure are optimized initially by WOA for global search while NM algorithm further enhances the rapid local search convergence. The proposed algorithm’s objective is to show the accuracy and robustness in solving challenging problems like TONMS-EFE. To study our designed scheme’s performance and effectiveness, LeNN-WOA-NM is implemented on four cases of TONMS-EFE. The results obtained by the proposed algorithm are compared with the Particle Swarm Optimization (PSO) algorithm, Cuckoo search algorithm (CSA), and WOA. Extensive graphical and statistical analysis for fitness value, absolute errors, and performance indicators in terms of mean, median, and standard deviations show the proposed algorithm’s efficiency and accuracy.

How much faster does the best polynomial approximation converge than Legendre projection

Abstract: We compare the convergence behavior of best polynomial approximations and Legendre and Chebyshev projections and derive optimal rates of convergence of Legendre projections for analytic and differentiable functions in the maximum norm For analytic functions, we show that the best polynomial approximation of degree n is better than the Legendre projection of the same degree by a factor of $$n^{1/2}$$ For differentiable functions such as piecewise analytic functions and functions of fractional smoothness, however, we show that the best approximation is better than the Legendre projection by only some constant factors Our results provide some new insights into the approximation power of Legendre projections

Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

Abstract: Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.

Shifted Legendre Collocation Method for the Solution of Unsteady Viscous-Ohmic Dissipative Hybrid Ferrofluid Flow over a Cylinder

Abstract: A numerical treatment for the unsteady viscous-Ohmic dissipative flow of hybrid ferrofluid over a contracting cylinder is provided in this study. The hybrid ferrofluid was prepared by mixing a 50% water (H2O) + 50% ethylene glycol (EG) base fluid with a hybrid combination of magnetite (Fe3O4) and cobalt ferrite (CoFe2O4) ferroparticles. Suitable parameters were considered for the conversion of partial differential equations (PDEs) into ordinary differential equations (ODEs). The numerical solutions were established by expanding the unknowns and employing the truncated series of shifted Legendre polynomials. We begin by collocating the transformed ODEs by setting the collocation points. These collocated equations yield a system of algebraic equations containing shifted Legendre coefficients, which can be obtained by solving this system of equations. The effect of the various influencing parameters on the velocity and temperature flow profiles were plotted graphically and discussed in detail. The effects of the parameters on the skin friction coefficient and heat transfer rates were further presented. From the discussion, we come to the understanding that Eckert number considerably decreases both the skin friction coefficient and the heat transfer rate.

Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system

Abstract: A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.

New Color Image Zero-Watermarking Using Orthogonal Multi-Channel Fractional-Order Legendre-Fourier Moments

Abstract: Zero-watermarking methods provide promising solutions and impressive performance for copyright protection of images without changing the original images. In this paper, a novel zero-watermarking method for color images is envisioned. Our envisioned approach is based on multi-channel orthogonal Legendre Fourier moments of fractional orders, referred to as MFrLFMs. In this method, a highly precise Gaussian integration method is utilized to calculate MFrLFMs. Then, based on the selected accurate MFrLFMs moments, a zero-watermark is constructed. Due to their accuracy, geometric invariances, and numerical stability, the proposed MFrLFMs-based zero-watermarking method shows excellent resistance against various attacks. Performed experiments using the proposed watermarking method show the outperformance over existing watermarking algorithms.