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The primary objective of this study is to develop a new constitutive model by combining a fractional‐order Kelvin–Voigt model with an Abel dashpot element in parallel. Subsequently, this new model will be incorporated into the Euler–Bernoulli beam's governing equation, utilizing shifted Legendre polynomials as basis functions, a classical orthogonal polynomial system, to solve the fractional‐order partial differential equations. By comparing the numerical solutions with the analytical solutions, we aim to evaluate the applicability of shifted Legendre polynomials in solving such problems and the accuracy of the obtained numerical solutions. Furthermore, we will investigate the performance of viscoelastic HDPE beams under different loading conditions and conduct a comparative analysis of the displacements of HDPE beams under the new constitutive model and the traditional fractional‐order Kelvin–Voigt model. Through this research, we hope to gain a deeper understanding of the characteristics of fractional‐order phenomena and provide more accurate and efficient numerical simulation and analysis methods for the field of structural mechanics, promoting the development of related engineering applications. [ABSTRACT FROM AUTHOR]

Objective: The aim of this study was to identify suitable polynomial regression for modeling the average growth trajectory and to estimate the relative development of the rib eye area, scrotal circumference, and morphometric measurements of Guzerat young bulls. Methods: A total of 45 recently weaned males, aged 325.8±28.0 days and weighing 219.9±38.05 kg, were evaluated. The animals were kept on Brachiaria brizantha pastures, received multiple supplementations, and were managed under uniform conditions for 294 days, with evaluations conducted every 56 days. The average growth trajectory was adjusted using ordinary polynomials, Legendre polynomials, and quadratic B-splines. The coefficient of determination, mean absolute deviation, mean square error, the value of the restricted likelihood function, Akaike information criteria, and consistent Akaike information criteria were applied to assess the quality of the fits. For the study of allometric growth, the power model was applied. Results: Ordinary polynomial and Legendre polynomial models of the fifth order provided the best fits. B-splines yielded the best fits in comparing models with the same number of parameters. Based on the restricted likelihood function, Akaike's information criterion, and consistent Akaike's information criterion, the B-splines model with six intervals described the growth trajectory of evaluated animals more smoothly and consistently. In the study of allometric growth, the evaluated traits exhibited negative heterogeneity (b<1) relative to the animals' weight (p<0.01), indicating the precocity of Guzerat cattle for weight gain on pasture. Conclusion: Complementary studies of growth trajectory and allometry can help identify when an animal's weight changes and thus assist in decision-making regarding management practices, nutritional requirements, and genetic selection strategies to optimize growth and animal performance. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method based on an operational matrix of Legendre polynomials for resolving the class of time fractional diffusion (TFD) equations. The operational matrix of fractional order derivatives of the Legendre polynomials is derived as a product of matrices. The collocation method together with the operational matrix of Legendre polynomials are employed to transform the TFD equations into a set of algebraic equations. The perturbation method is applied to show the stability of the discussed method. The accuracy of the suggested method is validated using numerical experiments. The solution obtained by this method is in excellent agreement with the exact solution for the integer order of derivatives and is more precise than the solution obtained by the existing method in which Bernstein polynomials are taken as the basis polynomials. [ABSTRACT FROM AUTHOR]

To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method's effectiveness and accuracy. [ABSTRACT FROM AUTHOR]

This article presents an extension of the double Legendre polynomial method to model and evaluate the axisymmetric free vibration problem for the axially polarized piezoelectric resonator. The aim is to show the benefit of the employed method. The polynomial series was used to define the mechanical displacements and electrical potential at the same time. The proposed approach considerably improves computing efficiency by avoiding much of the numerical integration calculations. As a result, the electromechanical coupling coefficient shows that the fundamental mode is the most piezoactive mode in the structure. Furthermore, the electrical input impedance and field profiles are easily presented. [ABSTRACT FROM AUTHOR]

Legendre polynomials' first derivatives have been used as the basis function via the pseudo-Galerkin spectral method. Operational matrices for derivatives have been used and extended to deal with the system of ordinary differential-algebraic equations. An algorithm via those matrices has been designed. The accuracy and efficiency of the proposed algorithm had been shown by two techniques, theoretically, via the boundedness of the approximated expansion and numerically through numerical examples. [ABSTRACT FROM AUTHOR]

We provide novel sufficient conditions for the uniform recovery of sparse Legendre expansions using ℓ1 minimization, where the sampling points are drawn according to orthogonalization (uniform) measure. So far, conditions of the form m≳Θ2s×logfactors have been relied on to determine the minimum number of samples m that guarantees successful reconstruction of s‐sparse vectors when the measurement matrix is associated to an orthonormal system. However, in case of sparse Legendre expansions, the uniform bound Θ of Legendre systems is so high that these conditions are unable to provide meaningful guarantees. In this paper, we present an analysis which employs the envelop bound of all Legendre polynomials instead, and prove a new recovery guarantee for s‐sparse Legendre expansions, m≳s2×logfactors, which is independent of Θ. Arguably, this is the first recovery condition established for orthonormal systems without assuming the uniform boundedness of the sampling matrix. The key ingredient of our analysis is an extension of chaining arguments, recently developed in Bourgain and Chkifa et al., to handle the envelope bound. Furthermore, our recovery condition is proved via restricted eigenvalue property, a less demanding replacement of restricted isometry property which is perfectly suited to the considered scenario. Along the way, we derive simple criteria to detect good sample sets. Our numerical tests show that sets of uniformly sampled points that meet these criteria will perform better recovery on average. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for the system of prey-predator with immigrant prey has been analyzed to find an approximate solution for immigrant prey population density, local prey population density, and predator population density. Furthermore, we present a novel soft computing technique named LeNN-WOA-NM algorithm for solving the mathematical model of the prey-predator system with immigrant prey. The proposed algorithm uses a function approximating ability of Legendre polynomials based on Legendre neural networks (LeNNs), global search ability of the whale optimization algorithm (WOA), and a local search mechanism of the Nelder–Mead algorithm. The LeNN-WOA-NM algorithm is applied to study the effect of variations on the growth rate, the force of interaction, and the catching rate of local prey and immigrant prey. The statistical data obtained by the proposed technique establish the effectiveness of the proposed algorithm when compared with techniques in the latest literature. The efficiency of solutions obtained by LeNN-WOA-NM is validated through performance measures including absolute errors, MAD, TIC, and ENSE. [ABSTRACT FROM AUTHOR]

Simple Summary: This study aimed to compare different order Legendre polynomials for Arabian racing horses. The present work's results indicated that the L(2,2) Legendre polynomial (that suggests generalizations and connections to the various mathematical structures and numerical applications) model could be reliably used to estimate heritability values for the racing speed of Arabian horses in the presence of repeated observations. This work aimed to compare the fitting performance of the random regression models applied to the different order orthogonal Legendre polynomials on the race completion speed (m/s) of Arabian racing horses. Legendre polynomial function for additive genetic, permanent environmental variances and heritability values with the L(2,2), L(2,3), L(3,2) and L(3,3) models (where L(i,j) means L(order of fit for additive genetic effects, order of fit for permanent environmental effects)) was estimated. A total of 233,491 race speed records (m/s) of Arabian horses were taken from the Jockey Club of Turkey between 2005 and 2016. The mean and standard deviation of heritability values were estimated as 0.294 ± 0.0746, 0.285 ± 0.0620, 0.302 ± 0.0767 and 0.290 ± 0.1018 for L(2,2), L(2,3), L(3,2), and L(3,3), respectively. The steady decreasing trend of permanent environmental variances for L(2,2) provided stationery for heritability values. According to Akaike information criterion (AIC) and Bayesian information criterion (BIC) values, the L(2,2) model could be reliably used to estimate heritability values for the racing speed of Arabian horses in the presence of repeated observations. [ABSTRACT FROM AUTHOR]

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–Baleanu derivative sense. To solve the problem, operational matrices of AB-fractional integration and multiplication, together with the Lagrange multiplier method for the constrained extremum, are considered. The method reduces the main problem to a system of nonlinear algebraic equations. In this framework by solving the obtained system, the approximate solution is calculated. An error estimate of the numerical solution is also proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are presented to demonstrate the accuracy and validity of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, operational matrices of two-dimensional hybrid of block-pulse functions and Legendre polynomials are employed to approximate the solutions of two-dimensional nonlinear Volterra integral equations of the first kind (2DNVIEF). To this aim, we first convert 2DNVIEF to the two-dimensional linear Volterra integral equations of the second kind (2DLVIES). Then the operational matrices of integration, together with the product operational matrix, can be applied to reduce the 2DLVIES to a system of algebraic equations. The convergence of the proposed method is studied, and an error estimation under the L 2 -norm is provided. Finally, some numerical examples are prepared to clarify the accuracy and efficiency of the numerical method especially in dealing with functions that are not continuous in the whole interval. [ABSTRACT FROM AUTHOR]

An alternative form of ad hoc collision model to the Fokker–Planck field collision operator is proposed. The model is so constructed that the results of exact Fokker–Planck field collision, applied to the low-order Legendre and associated Laguerre polynomials, are used to impose self-adjointness first and then collisional conservation laws. The model is shown to be actually identical to the conventional ad hoc model in the l = 0 and 1 parts in Legendre polynomial expansion in an isothermal case, while it is not limited to extend to the higher order, in contrast to the conventional ad hoc model, in which the collisional conservation laws are imposed primarily, using the exact test collision form of the Fokker–Planck operator. Furthermore, it is discussed that the proposed model is identified to the lowest order in an expansion of formal Hirshman–Sigmar operator, while there is a difference between the formal and conventional Hirshman–Sigmar operators. Numerical examples demonstrate that the proposed model can yield comparable results to those by the exact Fokker–Planck and conventional Hirshman–Sigmar operators. [ABSTRACT FROM AUTHOR]

In this work, our motivation is to design a new collocation method based on Müntz–Legendre polynomial involving operational matrices to solve variable‐order stochastic fractional integro‐differential equation. We first prove the existence and uniqueness result for the solution of considered problem. The operational matrices are used to convert the variable‐order stochastic fractional integro‐differential equation into non‐linear system of algebraic equations. Further, the accuracy and efficiency of the proposed method are investigated through numerical experiments. Finally, we show the advantage of the variable‐order model over the constant order model through a real‐world example of mathematical finance. [ABSTRACT FROM AUTHOR]

Fractional order differential equations accurately model dynamic systems and processes. In some of the fractional optimal control problems (FOCPs), due to the ambiguity in the initial conditions and the transfer of ambiguity to the solution, it is necessary to use fuzzy mathematics. In this paper, a numerical method is presented to approximate the solution for a class of Fuzzy Fractional Optimal Control Problems (FFOCPs) using the Legendre basis functions. The fuzzy fractional derivative is described in the Caputo sense. The performance index of an FFOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of Fuzzy Fractional Differential Equations (FFDEs). After obtaining Euler–Lagrange equations for FFOCPs and the necessary and sufficient conditions for the existence of solutions, using the definition of generalized Hukuhara differentiability (types I, II), the problem is considered in two cases. Then the distance function and an approach similar to the variational type along with the Lagrange multiplier method are used to formulate and solve the equations in a system. Time-invariant and time-varying examples are provided to assess the presented method. Numerical results show a similar trend for the state and control variables for various numbers of Legendre polynomials. Also, the convergence of state and control variables for the time-invariant system can be seen, and the same is true for control variables for the time-varying system. [ABSTRACT FROM AUTHOR]

The solutions of integral-algebraic equations with non-vanishing delays (DIAEs) may contain so-called primary discontinuity points. At these points the solution of the DIAEs, regardless of how regular the given functions are, will in general exhibit a low degree of regularity. In this paper, we study the existence, uniqueness and regularity properties of solutions for the DIAEs and consider a spectral collocation method based on Legendre polynomials as approximate solution of this equation. Numerical solution is described by a constructive technique and dividing the definition domain into several subintervals according to the primary discontinuous points associated with the delay function. The error analysis is derived and numerical experiments confirm the theoretical expectations. [ABSTRACT FROM AUTHOR]

Compared to the traditional integer-order viscoelastic model, a fractional-order derivative viscoelastic model is shown to be advantageous. A quadrature-free Legendre polynomial method combining the Weyl definition of fractional order derivative is for the first time employed to solve guided waves in functionally graded fractional viscoelastic plates. The presented method avoids a lot of numerical integration calculation in traditional polynomial method and greatly improves the computational efficiency. The effects of different graded fields and fractional orders on the dispersion and attenuation curves are illustrated. The displacement distributions, Poynting vectors and power flows are analyzed to reveal guided wave characteristics. [ABSTRACT FROM AUTHOR]

This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equations. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests provide theoretical results. [ABSTRACT FROM AUTHOR]

A closed form expression of a quadruple integral involving the Legerndre polynomial Pn(x) is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new. [ABSTRACT FROM AUTHOR]

In this paper, the Legendre polynomials approximations are studied for the numerical solutions of differential-algebraic equations (DAEs). An algorithm of the method constructed using Legendre polynomials approximations is developed to solve DAE systems. Two test problems are answered to illustrate the Legendre polynomials approximations, and then the obtained solutions are confronted with the exact solutions of the problems. As a first step, the power series of an allowed equation system is determined, in second step it is converted into Legendre polynomials approximations structure which allows an arbitrary order. Furthermore, to get numerical solution of DAEs with Legendre polynomials approximations, a Maple algorithm is implemented. Graphs of obtained solutions are sketched, and are created tables to compare the exact solutions by using Maple programming. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce and study the numerical solutions of the multi-space fractal-fractional Kuramoto-Sivashinsky equation (MSFFKS) and the multi-space fractal-fractional Korteweg-de Vries equation (MSFFKDV). These models are obtained by replacing the classical derivative by the fractal-fractional derivative based upon the generalized Mittag-Leffler kernel. In our investigation, we use the spectral collocation method (SCM) involving the shifted Legendre polynomials (SLPs) in order to reduce the new models to a system of algebraic equations. We then use one of the known numerical methods, the Newton-Raphson method (NRM), for solving the resulting system of the nonlinear algebraic equations. The efficiency and accuracy of the numerical results are validated by calculating the absolute error as well as the residual error. We also present several illustrative examples and graphical representations for the various results which we have derived in this paper. [ABSTRACT FROM AUTHOR]

The numerical solutions are obtained for rotating beams; the inclusion of centrifugal force term makes it difficult to get the analytical solutions. In this paper, we solve the free vibration problem of rotating Rayleigh beam using Chebyshev and Legendre polynomials where weak form of meshless local Petrov-Galerkin method is used. The equations which are derived for rotating beams result in stiffness matrices and the mass matrix. The orthogonal polynomials are used and results obtained with Chebyshev polynomials and Legendre polynomials are exactly the same. The results are compared with the literature and the conventional finite element method where only first seven terms of both the polynomials are considered. The first five natural frequencies and respective mode shapes are calculated. The results are accurate when compared to literature. [ABSTRACT FROM AUTHOR]

In this work, the interactions between the axial translational motions and aspherical oscillations of two gas bubbles in an incompressible liquid are considered. Representing the surface function by the Legendre polynomial of first order, we derive a dynamic model to describe the motions of two aspherical bubbles in Lagrangian mechanics. An apple-shaped bubble from simulations based on the model can be well consistent with known experimental observation. The bubble appears as the shape of a sphere at maximum expansion. The maximum asymmetry of the bubbles occurs during collapse. The surface tension is a key factor to stable oscillatory deformation. It is also found that the aspherical amplitudes of two bubbles decrease with increasing distance or decreasing driving pressure. [ABSTRACT FROM AUTHOR]

A new differentiation technique, fractional pseudospectral shifted Legendre differentiation matrices (FSL D-matrices), was introduced. It depends on shifted Legendre polynomials (SLPs) as a base function. We take into consideration its extreme points and inner product. The technique was used to solve fractional ordinary differential equations (FODEs). Moreover, it extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The novel FSL D-matrices transformed these fractional differential problems (FDPs) into an algebraic system of equations. Also, an error and a convergence analysis for that technique were investigated. Finally, the correctness and efficiency of this technique were examined with test functions and several examples. All the results were compared with the results of other methods to ensure the investigated error analysis. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a numerical method to approximate the solution of non-linear fractal-fractional Burgers equation. In this model, differential operators are defined in the Atangana-RiemannLiouville sense with Mittag-Leffler kernel. We first expand the spatial derivatives using barycentric interpolation method, and then we derive an operational matrix (OM) of the fractal-fractional derivative for the Legendre polynomials. To be more precise, two approximation tools are coupled to convert the fractal-fractional Burgers equation into a system of algebraic equations which is technically uncomplicated and can be solved using available mathematical software such as MATLAB. To investigate the agreement between exact and approximate solutions, several examples are examined. [ABSTRACT FROM AUTHOR]

In 2012, Lima introduced a remarkable two-term dilogarithm identity, based on a proof for the Basel problem due to Beukers et al. Using a series transform obtained very recently via Legendre polynomial expansions, we nontrivially extend Lima's identity, and offer a new proof of this same identity. [ABSTRACT FROM AUTHOR]

A new inversion based Fourier transformation technique named as Legendre-Polynomials Least-Squares Fourier Transformation (L-LSQ-FT) and Legendre-Polynomials Iteratively Reweighted Least-Squares Fourier Transformation (L-IRLS-FT) are presented. The introduced L-LSQ-FT algorithm establishes an overdetermined inverse problem from the Fourier transform. The spectrum was approximated by a series expansion limited to a finite number of terms, and the solution of inverse problem, which gives the values of series expansion coefficients, was obtained by the LSQ method. Practically, results from the least square method are responsive to data outliers, thus scattered large errors and the estimated model values may be far from reality. A definitely better option is attained by introducing Steiner's Most Frequent Value method. By combining the IRLS algorithm with Cauchy-Steiner weights, the Fourier transformation process was robustified to give the L-IRLS-FT method. In both cases Legendre polynomials were applied as basis functions. Thus the approximation of the continuous Fourier spectra is given by a finite series of Legendre polynomials and their coefficients. The series expansion coefficients were obtained as a solution to an overdetermined non-linear inverse problem. The traditional DFT and the L-IRLS-FT were tested numerically using synthetic datasets as well as field magnetic data. The resulting images clearly show the reduced sensitivity of the newly developed L-IRLS-FT methods to outliers and scattered noise compared to the traditional DFT. Conclusively, the newly developed L-IRLS-FT can be considered to be a better alternative to the traditional DFT. [ABSTRACT FROM AUTHOR]

• The integrations in the Legendre polynomial approach is analytically solved to improve the computational efficiency. • The solutions according to the global matrix method are derived to verify the effectiveness of the presented approach. • Longitudinal mode attenuation curves rapidly decrease when torsional mode curves is approaching them. • The temperature amplitude of flexural torsional mode is not negligible, especially for a small radius-thickness ratio pipe. • Attenuation differences of various flexural orders are large at cut-off frequencies and mutation frequencies. A new solving procedure based on the Legendre polynomial series is developed to investigate the longitudinal guided waves in fractional order thermoelastic inhomogeneous hollow cylinders. Different from the available Legendre polynomial approach, integrals are calculated by the explicit and simple expressions to replace the numerical solutions. The solutions according to the global matrix method (GMM) are also derived for the first time to verify the effectiveness of the presented approach. Comparison of the CPU time indicates that the computational efficiency of the new solving procedure has a huge improvement. Then, the guided wave phase velocity dispersion curves, attenuation curves, displacement and temperature distributions for functionally graded hollow cylinders with different fractional orders, flexural orders, radius-thickness ratios and gradient fields are analyzed. It is found that the attenuation coefficients of flexural longitudinal modes rapidly decrease when the flexural torsional mode curve is close to the flexural longitudinal mode curve. Furthermore, notable influence of the flexural order on attenuation curves mainly occurs at the cut-off frequencies and mutation frequencies. In addition, a smaller radius-thickness ratio implies a larger attenuation, and the temperature amplitudes of flexural torsional mode are more and more close to those of longitudinal mode when the radius-thickness ratio decreases. [ABSTRACT FROM AUTHOR]

This research aims to assemble two methodical spectral Legendre's derivative algorithms to numerically attack the Lane-Emden, Bratu's, and singularly perturbed type equations. We discretize the exact unknown solution as a truncated series of Legendre's derivative polynomials. Then, via tau and collocation methods for linear and nonlinear problems, respectively, we obtain linear/nonlinear systems of algebraic equations in the unknown expansion coefficients. Finally, with the aid of the Gaussian elimination technique in the linear case and Newton's iterative method for the non-linear case - with vanishing initial guess- we solve these systems to obtain the desired solutions. The stability and convergence analyses of the numerical schemes were studied in-depth. The schemes are convergent and accurate. Some numerical test problems are performed to verify the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this article, we study the discrete version of Legendre spectral and iterated Legendre spectral techniques to solve the second kind Hammerstein type weakly singular integral equations. To obtain the convergence analysis, we use the appropriate numerical quadrature rule and obtain the order O (n − r + 1) in discrete Legendre spectral method. If the quadrature rule is minimal, i.e. the number of quadrature nodes and the dimension of the approximating subspace are same, then the optimal rate is obtained O (n − r) in iterated form of discrete Legendre spectral collocation method in L 2 norm and uniform norm. Numerical aspects are given to verify the hypothetical results. [ABSTRACT FROM AUTHOR]

• Considered two major interface models on an inhomogeneity interface in a half-space. • Interface effects affect stress distribution through five dimensionless parameters. • Interface flexural rigidities lead to obvious changes to Gurtin-Murdoch solutions. • Soft nanoinhomogeneities are affected most by interface stress and bending moments. • Deviatoric interface curvature change tensor is more significant than the mean one. This paper examines the interface elasticity between an elastic half-space and a spherical nanoinhomogeneity subjected to a unidirectional far-field tension that is parallel to the half-space plane boundary. The spherical interface is modeled by the full version of Steigmann–Ogden interface theory. Due to their marginal significance, the effects of half-space plane boundary are neglected. The problem is solved by decomposing the unidirectional load into an all-around and an equal-but-opposite traction field. Benefitting from the property of axial symmetry, the former subproblem is solved by the sole use of Boussinesq displacement potentials. The latter condition is addressed by six harmonic functions extracting from Boussinesq, Papkovich–Neuber and Dougall displacement potentials. The relations between cylindrical and spherical harmonic functions are employed to clear the tractions along the half-space plane boundary and to satisfy the nonclassical traction equilibrium conditions across the matrix/nanoinhomogeneity interface. Truncating the resultant infinite series and equating the coefficients preceding Legendre polynomials, associated Legendre functions as well as their derivatives lead to a system of algebraic equations with respect to the dimensionless unknown parameters in displacement potentials. Extensive parametric studies are further performed in order to analyze the importance of interface tension, interface Lamé constants, interface flexural rigidities, shear moduli ratio and nanoinhomogeneity radius-to-depth ratio on stress distributions and stress concentrations. Comparisons against both the classical and the Gurtin–Murdoch solutions help to clarify the separate effects of individual interface material properties. Stresses inside a soft nanoinhomogeneity is most affected by the additional incorporation of interface flexural rigidities. In addition, the deviatoric component of the interface curvature change tensor is found to be more important than the mean part for the proposed nanoinhomogeneity problem. [ABSTRACT FROM AUTHOR]

In this paper, the properties of the primitives of Legendre polynomials on the interval are studied. It is proved that the Legendre polynomials form an "almost" orthogonal system. Namely, for a fixed order of the primitive, only finitely many of these polynomials can be nonorthogonal. These properties underly the relationship between the spectral problems for differential operators in and the spectral properties of generalized Jacobi matrices. [ABSTRACT FROM AUTHOR]

In this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body's natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique. [ABSTRACT FROM AUTHOR]

The scattering term within the Boltzmann equation was for decades approximated either through the Legendre polynomial for deterministic solvers or by S (α , β) /free gas model approach for Monte Carlo solvers. This, to some extent, inaccurate approach led to the assumption in several cases that the scattering term can be further "tuned" to simplify complex mathematical solver of the transport equation, mainly by reducing considerably the computation time, assuming no consequences on the physical results. The introduction of the resonant dependent scattering kernel to MONTE CARLO code and in particular the experimental validation within the resolved resonance range for Uranium and Thorium, in RPI (Renselear Polytechnic Institute), pointed out that the scattering term cannot be taken as a second order negligible term, but rather should be accurately regarded for any solution of the transport equation. Corollary to the considerable high impact of that "physical" scattering kernel, this study extends its importance beyond the epithermal resolved resonance range and aims to proof that the scattering kernel can and should be accurately dealt also at higher energies at least up to about several tens of keV. Moreover, in the debate between the purely quantum mechanics treatment known as the Optical Model -OM and the Doppler broadening based classical approach, the latter seems to be correct for this extended energy range. Strictly speaking, Newton's Laws for the scattering kernel evaluation remain intact, yet in accordance to their quantum mechanics based integrated scattering cross section values as was shown for example by the well-known fundamental Breit Wigner formula. [ABSTRACT FROM AUTHOR]

This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family of basis functions, are employed to generate an operational matrix method for this equation. A new fractional operational matrix related to these polynomials is extracted and employed to construct the presented method. Using this approach, an algebraic system of equations is obtained instead of the original variable-order equation. The numerical solution of this system can be found easily. Some numerical examples are provided for verifying the accuracy of the generated approach. [ABSTRACT FROM AUTHOR]

This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family of basis functions, are employed to generate an operational matrix method for this equation. A new fractional operational matrix related to these polynomials is extracted and employed to construct the presented method. Using this approach, an algebraic system of equations is obtained instead of the original variable-order equation. The numerical solution of this system can be found easily. Some numerical examples are provided for verifying the accuracy of the generated approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose an artificial neural network model (ANN) to solve a partial differential equation (PDE) constrained optimization problem. Here, the discretize then optimize approach is used. At first, the Legendre polynomials are used to discretize the optimization problem and transform it into a quadratic optimization problem with linear constraint. Then an ANN model is proposed to solve the obtained quadratic optimization problem. Finally, several examples are presented to illustrate the abilities and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

In this work, we study a space–time Petrov–Galerkin method for third‐ and fifth‐order time fractional Korteweg–de Vries–Burgers equations. The method is based on the framework of Legendre and Jacobi polynomials. The basis functions of the fractional part are constructed by the generalized Jacobi functions, which contained the singularity of weak solutions. The numerical schemes of the problems are transformed into the nonlinear schemes constructed by matrices. Based on the orthogonality of ideal basis functions, we get the optimal estimation under the specific weighted Sobolev spaces. Numerical experiments confirm the expected convergence. [ABSTRACT FROM AUTHOR]

This article tackles the decentralized near-optimal control problem for the class of nonlinear polynomial interconnected system based on a shifted Legendre polynomials direct approach. The proposed method converts the interconnected optimal control problems into a nonlinear programming one with multiple constraints. In light of the formulated NLP optimization, state and control coefficients are used to design a nonlinear decentralized state feedback controller. Overall closed-loop system stability sufficient conditions are investigated with the help of Grönwall lemma. The triple inverted pendulum case is considered for simulation. Satisfactory results are obtained in both open-loop and closed-loop schemes with comparison to collocation and state-dependent Riccati equation techniques. [ABSTRACT FROM AUTHOR]

In this paper, an intelligence method based on single layer legendre neural network is proposed to solve fractional optimal control problems where the dynamic control system depends on Caputo fractional derivatives. First, with the help of an approximation, the Caputo derivative is replaced to integer order derivative. According to the Pontryagin minimum principle for optimal control problems and by constructing an error function, an unconstrained minimization problem is then defined. In the optimization problem, trial solutions are used for state, costate and control functions, where these trial solutions are constructed by using Legendre polynomial based functional link artificial neural network. In the following, error back propagation algorithm is used for updating the network parameters (weights). At the end, some illustrative examples are included to demonstrate the validity and capability of the proposed method. Three applicable examples about chaos control of Malkus waterwheel, finance fractional chaotic models and fractional-order geomagnetic field models are also considered. [ABSTRACT FROM AUTHOR]

This article is devoted to developing an accurate operational matrix (OM) method for the solution of a new category of nonlinear optimal control problems (OCPs) explained by fractal‐fractional dynamical systems. The fractal‐fractional differentiation is considered in the sense of Atangana‐Riemann‐Liouville. The method is based on the shifted Legendre polynomials (LPs). Through the way, a new OM of fractal‐fractional differentiation is extracted for the shifted LPs. The optimality conditions are converted to an algebraic system of equations by using the shifted LP expansions of the state and control variables and applying the method of constrained extrema. More precisely, these variables are approximated by the shifted LPs with undetermined coefficients. Then, these expansions are replaced in the performance index, and the Gauss‐Legendre integration method is used to approximate the performance index for extracting a nonlinear algebraic equation. In the sequel, the mentioned OM and the collocation method are used to generate some algebraic equations from the dynamical system. Finally, the method of the constrained extrema is used by coupling the algebraic constraints generated from the dynamical system and the initial condition with the algebraic equation elicited from the performance index by a set of unknown Lagrange multipliers. The accuracy of the method is studied by solving some numerical examples. [ABSTRACT FROM AUTHOR]

The paper deals with sharp estimates of derivatives of intermediate order in the Sobolev space , . The functions under study are the smallest possible quantities in inequalities of the form The properties of the primitives of shifted Legendre polynomials on the interval are used to obtain an explicit description of these functions in terms of hypergeometric functions. In the paper, a new relation connecting the derivatives and primitives of Legendre polynomials is also proved. [ABSTRACT FROM AUTHOR]

• New numerical schemes are proposed for solving the Riesz-space fractional partial differential equation. • The matrix transform method in space and operational matrix method in the time domain is applied. • Optimal error bound and stability analysis are investigated for the schemes. • Numerical stability of the schemes is verified by applying some noise. • The effectiveness and accuracy of the numerical schemes are discussed. In this paper, we construct two efficient numerical schemes by combining the finite difference method and operational matrix method (OMM) to solve Riesz-space fractional diffusion equation (RFDE) and Riesz-space fractional advection-dispersion equation (RFADE) with initial and Dirichlet boundary conditions. We applied matrix transform method (MTM) for discretization of Riesz-space fractional derivative and OMM based on shifted Legendre polynomials (SLP) and shifted Chebyshev polynomial (SCP) of second kind for approximating the time derivatives. The proposed schemes transform the RFDE and RFADE into the system of linear algebraic equations. For a better understanding of the methods, numerical algorithms are also provided for the considered problems. Furthermore, optimal error bound for the numerical solution is derived, and theoretical unconditional stability has been proved with respect to L 2 -norm. The stability of the schemes is also verified numerically. The schemes are observed to be of second-order accurate in space. The effectiveness and accuracy of the schemes are tested by taking two numerical examples of RFDE and RFADE and found to be in good agreement with the exact solutions. It is observed that the numerical schemes are simple, easy to implement, yield high accurate results with both the basis functions. Moreover, the CPU time taken by the schemes with SLP basis is very less as compared to schemes with SCP basis. [ABSTRACT FROM AUTHOR]

In this study, a numerical method based on Hermite polynomial approximation for solving a class of fractional optimal control problems is presented. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Operational matrices of integration by using such known formulas as Caputo and Riemann-Liouville operators for computing fractional derivatives and integration of polynomials is introduced and used to reduce the problem of a system of algebraic equations. The convergence of the proposed method is analyzed, and the error upper bound for the operational matrix of the fractional integration is obtained. To confirm the validity and accuracy of the proposed numerical method, three numerical examples are presented along with a comparison between our numerical results and those obtained using Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the new technique. [ABSTRACT FROM AUTHOR]

COMPARED TO THE TRADITIONAL INTEGER ORDER VISCOELASTIC MODEL, a fractional order derivative viscoelastic model is shown to be advantageous. The characteristics of guided circumferential waves in an anisotropic fractional order Kelvin-- Voigt viscoelastic hollow cylinder are investigated by a quadrature-free Legendre polynomial approach combining the Weyl definition of fractional order derivatives. The presented approach can obtain dispersion solutions in a stable manner from an eigenvalue/eigenvector problem for the calculation of wavenumbers and displacement profiles of viscoelastic guided wave, which avoids a lot of numerical integration calculation in a traditional polynomial method and greatly improves the computational efficiency. Comparisons with the related studies are conducted to validate the correctness of the presented approach. The full three dimensional spectrum of an anisotropic fractional Kelvin--Voigt hollow cylinder is plotted. The influence of fractional order and material parameters on the phase velocity dispersion and attenuation curves of guided circumferential wave is discussed in detail. Moreover, the difference of the phase velocity dispersion and attenuation characteristics between the Kelvin--Voigt and hysteretic viscoelastic models is also illustrated. The presented approach along with the observed wave features should be particularly useful in non-destructive evaluations using waves in viscoelastic waveguides. [ABSTRACT FROM AUTHOR]