Lucas Polynomials Research Articles

Forecasting Wind Speed Using the Proposed Wavelet Neural Network

Wind energy is one of the speedy processing technologies in the energy generation industry and the most economical methods of electrical power generation. For the reliability of system, it is wanted to improve highly appropriate wind speed forecasting methods. The wavelet transform is a powerful mathematical technique that converts an analyzed signal into a time-frequency representation. This technique has proven useful in a nonstationary time series forecasting. The aims of this study are to propose a wavelet function by derivation of a quotient from two different Lucas polynomials, as well as a comparison between an artificial neural network (ANN) and wavelet-artificial neural network (WNN). We used the proposed wavelet, Mexican hat, Morlet, Gaussian, Haar, Daubechies, and Coiflet to transform the wind speed data using the continuous wavelet transform (CWT). MATLAB software was used to implement the CWT and ANN. The proposed models were applied in the meteorological field to forecast the daily wind speed data that were collected from the meteorological directorate of Sulaymaniyah which is a city located in the Kurdistan region of Iraq for the period (Jan. 2011–Dec. 2020). Five different performance criteria during calibration and validation, the root mean square error ( R M S E ), mean square error ( M S E ), mean absolute percentage error M A P E , mean absolute error M A E , and coefficient of determination ( R 2 ), were evaluated. When studying, analyzing, and comparing these models, the results of the study concluded that the proposed wavelet-ANN is the best result ( M S E = 0.00072 , R M S E = 0.02683 , M A P E = 2.32400 , and R 2 = 0.99983 .

Lucas Polynomial Solution of the Single Degree of Freedom System

Free vibration of a single degree of freedom system is a fundamental topic in mechanical vibrations. The present study introduces a novel and simple numerical method for the solution of this system in terms of Lucas polynomials in the matrix form. Particular and general solutions of the differential equation can be determined by this method. The method is illustrated by a numerical application and the results obtained are compared with those of the exact solution.

A Pell–Lucas Collocation Approach for an SIR Model on the Spread of the Novel Coronavirus (SARS CoV-2) Pandemic: The Case of Turkey

In this article, we present a study about the evolution of the COVID-19 pandemic in Turkey. The modelling of a new virus named SARS-CoV-2 is considered by an SIR model consisting of a nonlinear system of differential equations. A collocation approach based on the Pell–Lucas polynomials is studied to get the approximate solutions of this model. First, the approximate solution in forms of the truncated Pell–Lucas polynomials are written in matrix forms. By utilizing the collocation points and the matrix relations, the considered model is converted to a system of the nonlinear algebraic equations. By solving this system, the unknown coefficients of the assumed Pell–Lucas polynomial solutions are determined, and so the approximate solutions are obtained. Secondly, two theorems about the error analysis are given and proved. The applications of the methods are made by using a code written in MATLAB. The parameters and the initial conditions of the model are determined according to the reported data from the Turkey Ministry of Health. Finally, the approximate solutions and the absolute error functions are visualized. To demonstrate the effectiveness of the method, our approximate solutions are compared with the approximate solutions obtained by the Runge–Kutta method. The reliable results are obtained from numerical results and comparisons. Thanks to this study, the tendencies of the pandemic can be estimated. In addition, the method can be applied to other countries after some necessary arrangements.

The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For p=0, the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for p=0 and by Komatsu and Ying for the general non-negative integer p.

Numerical Solution of High-Order Linear Fredholm Integro-Differential Equations by Lucas Collocation Method

In this paper, a useful matrix approach for high-order linear Fredholm integro-differential equations with initial boundary conditions expressed as Lucas polynomials is proposed. Using a matrix equationwhich is equivalent to a set of linear algebraic equations the method transforms to integro-differential equation. When compared to other methods that have been proposed in the literature, the numerical results from the suggested technique reveal that it is effective and promising. And also, error estimation of the scheme was derived. These results were compared with the exact solutions and the other numerical methods to the tested problems.

Some Novel Formulas of Lucas Polynomials via Different Approaches

Some new formulas related to the well-known symmetric Lucas polynomials are the primary focus of this article. Different approaches are used for establishing these formulas. A matrix approach to Lucas polynomials is followed in order to obtain some fundamental properties. Particularly, some recurrence relations and determinant forms are determined by suitable Hessenberg matrices. Conjugate Lucas polynomials and generating functions are derived and examined. Several connection problems between the Lucas polynomials and other celebrated symmetric and non-symmetric orthogonal polynomials such as the first and second kinds of Chebyshev polynomials and their shifted counterparts are solved. We prove that several argument-type hypergeometric functions are involved in the connection coefficients. In addition, we construct new formulas for high-order derivatives of Lucas polynomials in terms of their original polynomials, as well as formulas for repeated integrals of Lucas polynomials.

On a new subclass of biunivalent functions associated with the $(p,q)$-Lucas polynomials and bi-Bazilevic type functions of order $\rho+i\xi$

Using $ (p, q) $-Lucas polynomials and bi-Bazilevic type functions of order $\rho +i\xi,$ we defined a new subclass of biunivalent functions. We obtained coefficient inequalities for functions belonging to the new subclass. In addition to these results, the upper bound for the Fekete-Szegö functional was obtained. Finally, for some special values of parameters, several corollaries were presented.

IDENTITIES RELATED TO GENERALIZED FIBONACCI AND LUCAS POLYNOMIALS

The famous Fibonacci and Lucas polynomials possess various astonishing properties and identities. The Fibonacci polynomial has been generalized in many ways by preserving the recurrence relation and others by preserving the initial condition. In this paper, we dene generalized Fibonacci and Lucas polynomials and proved some famous identities in our settings.

Coefficient bounds and Fekete-Szegő inequality for a certain family of holomorphic and bi-univalent functions defined by (M,N)-Lucas polynomials

In the current work, we use the (M,N)-Lucas Polynomials to introduce a new family of holomorphic and bi-univalent functions which involve a linear combination between Bazilevic functions and ?-pseudo-starlike function defined in the unit disk D and establish upper bounds for the second and third coefficients of functions belongs to this new family. Also, we discuss Fekete-Szeg? problem in this new family.

FRACTIONAL VARIATIONAL PROBLEM INVOLVING INDEFINITE INTEGRALS AND NONSINGULAR KERNELS

This paper deals with a class of fractional variational problems involving Atangana–Baleanu (AB) derivative. The problem under consideration is a graceful combination of AB derivative with indefinite integral. A proposed numerical technique based on the shifted Vieta–Lucas polynomials is utilized for obtaining the solutions to the given problem. Our numerical process is principally based on the operational matrix (OM) of fractional integral in the AB sense in order to approximate solutions of the problem under study. By utilizing the OM along with the Lagrange multiplier method, the given problems are converted into a problem easier to solve. The resultant problem can then be solved using standard methods. Convergence of the presented method is investigated. In order to ensure and display the efficiency and accuracy of the proposed technique, some examples are given.

(U; V )-Lucas polynomial coefficient relations of the bi-univalent function class

In geometric function theory, Lucas polynomials and other special polynomials have recently gained importance. In this study, we develop a new family of bi-univalent functions. Also we examined coefficient inequalities and Fekete-Szegö problem for this new family via these polynomials.

A New Projection Method for a System of Fractional Cauchy Integro-Differential Equations via Vieta–Lucas Polynomials

This work presents a projection method based on Vieta–Lucas polynomials and an effective approach to solve a Cauchy-type fractional integro-differential equation system. The suggested established model overcomes two linear equation systems. We prove the existence of the problem’s approximate solution and conduct an error analysis in a weighted space. The theoretical results are numerically supported.

A computational method based on the generalized Lucas polynomials for fractional optimal control problems

Nonorthogonal polynomials have many useful properties like being used as a basis for spectral methods, being generated in an easy way, having exponential rates of convergence, having fewer terms and reducing computational errors in comparison with some others, and producing most important basic polynomials. In this regard, this paper deals with a new indirect numerical method to solve fractional optimal control problems based on the generalized Lucas polynomials. Through the way, the left and right Caputo fractional derivatives operational matrices for these polynomials are derived. Based on the Pontryagin maximum principle, the necessary optimality conditions for this problem reduce into a two-point boundary value problem. The main and efficient characteristic behind the proposed method is to convert the problem under consideration into a system of algebraic equations which reduces many computational costs and CPU time. To demonstrate the efficiency, applicability, and simplicity of the proposed method, several examples are solved, and the obtained results are compared with those obtained with other methods.

Approximate solution of Cauchy integral equations by using Lucas polynomials

Approximate solution of Cauchy integral equations by using Lucas polynomials

Lucas Collocation Method for the Solution of Differential Difference Equations

In order to solve mth-order linear differential difference equations with variable coefficients under mixed conditions, this research suggests a combined operational matrix approach based on Lucas polynomials. The simplicity of the proposed method’s application is a benefit. The technique simplifies the provided problem by turning it into a matrix equation. Absolute errors are used to validate illustrative examples. The solutions are enhanced by residual error estimates. The results, which are displayed in graphs and tables, are contrasted with the accepted approaches in the literature.

Vieta–Lucas wavelets method for fractional linear and nonlinear delay differential equations

PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

ON SUMS OF BIVARIATE FIBONACCI POLYNOMIALS AND BIVARIATE LUCAS POLYNOMIALS

In this paper, we present the sum of s+1 consecutive member of Bivariate Fibonacci Polynomials and Bivariate Lucas Polynomials and related identities consisting even and odd terms. We present its two cross two matrix and find interesting properties such as nth power of the matrix. Also, we present the identity which generalizes Catlan’s, Cassini’s and d’Ocagne’s identity. Binet’s formula will employ to obtain the identities.

On Generalized Fibonacci Polynomials: Horadam Polynomials

In this paper, we investigate the generalized Fibonacci (Horadam) polynomials and we deal with, in detail, two special cases which we call them $(r,s)$-Fibonacci and $(r,s)$-Lucas polynomials. We present Binet's formulas, generating functions, Simson's formulas, and the summation formulas for these polynomial sequences. Moreover, we give some identities and matrices associated with these sequences. Finally, we present several expressions and combinatorial results of the generalized Fibonacci polynomials.

Inverse tangent series involving pell and pell-lucas polynomials

AbstractBy means of the telescoping method, several summation formulae are established for the arctangent function with its argument being Pell and Pell–Lucas polynomials. Numerous infinite series identities involving Fibonacci and Lucas numbers are included as particular cases.

A new characterization of (P, Q)-Lucas polynomial coefficients of the bi-univalent function class associated with q-analogue of Noor integral operator

A new characterization of (P, Q)-Lucas polynomial coefficients of the bi-univalent function class associated with q-analogue of Noor integral operator