Generalized Legendre Polynomials Research Articles

Generalized Legendre Polynomial Configuration Method for Solving Numerical Solutions of Fractional Pantograph Delay Differential Equations

This paper develops a numerical approach for solving fractional pantograph delay differential equations using generalized Legendre polynomials. These polynomials are derived from generalized Taylor bases, which facilitate the approximation of the underlying analytical solutions, leading to the formulation of numerical solutions. The fractional pantograph delay differential equation is then transformed into a finite set of nonlinear algebraic equations using collocation points. Following this step, Newton's iterative method is applied to the resultant set of nonlinear algebraic equations to compute their numerical solutions. An error analysis for this methodology is subsequently presented, accompanied by numerical examples demonstrating its accuracy and efficiency. Overall, this study contributes a more streamlined and productive tool for determining the numerical solution of fractional differential equations.

Some integrals involving squares of Bessel functions and generalized Legendre polynomials

This paper develops new integral formulas intended for detailed studies of electromagnetics normal modes in spherical and spherical annular cavities.

Numerical study for the fractional RL, RC, and RLC electrical circuits using Legendre pseudo‐spectral method

SummaryIn the presented work, we present an accurate procedure, which is the spectral method, to find a solution to a certain class of the very important fractional (described by the Liouville–Caputo sense) models of the electrical RL, RC, and RLC circuits. This method is collocated using some important advantages of the generalized Legendre polynomials to obtain the numerical solution for these models. Besides, we compare the approximate solutions that are obtained using the presented technique for the proposed models with their exact solution in the given numerical examples.

Solving Nonlinear Filtering Problems in Real Time by Legendre Galerkin Spectral Method

It is well known that the nonlinear filtering (NLF) problem has important applications in both military and civil industries. The central question is to solve the posterior conditional density function of the states, which satisfies the Kushner or the Duncan–Mortensen–Zakai (DMZ) equation after suitable change of probability measure. In this article, we shall follow the so-called Yau–Yau's algorithm to split the solution of the DMZ equation into on- and off-line part, where the off-line part is to solve the forward Kolmogorov equation (FKE) with the initial conditions to be the orthonormal bases in some suitable function space. Instead of the generalized Hermite function investigated by the second and the third author of this article, we shall explore the generalized Legendre polynomials. The Legendre spectral method (LSM) is used to numerically solve the FKE. Under certain conditions, the convergence rate of LSM is twice faster than that of the Hermite spectral method. Two two-dimensional numerical experiments of NLF problems (time-invariant and time-varying cases) have been numerically solved to illustrate the feasibility of our algorithm. Our algorithm outperforms the extended Kalman Filter and particle filter in both real-time manner and accuracy.

Generalized Legendre Polynomials for Support Vector Machines (SVMS) Classification

In this paper, we introduce a set of new kernel functions derived from the generalized Legendre polynomials to obtain more robust and higher support vector machine (SVM) classification accuracy. The generalized Legendre kernel functions are suggested to provide a value of how two given vectors are like each other by changing the inner product of these two vectors into a greater dimensional space. The proposed kernel functions satisfy the Mercer’s condition and orthogonality properties for reaching the optimal result with low number support vector (SV). For that, the new set of Legendre kernel functions could be utilized in classification applications as effective substitutes to those generally used like Gaussian, Polynomial and Wavelet kernel functions. The suggested kernel functions are calculated in compared to the current kernels such as Gaussian, Polynomial, Wavelets and Chebyshev kernels by application to various non-separable data sets with some attributes. It is seen that the suggested kernel functions could give competitive classification outcomes in comparison with other kernel functions. Thus, on the basis test outcomes, we show that the suggested kernel functions are more robust about the kernel parameter change and reach the minimal SV number for classification generally.

A generalized Legendre–Gauss collocation method for solving nonlinear fractional differential equations with time varying delays

We introduce a new numerical plan for nonlinear autonomous fractional differential equations with time varying delay. Generalized Legendre polynomials and their properties are developed for deriving a general procedure for solving a class of fractional differential equations. The application of the Laplace transform and generalized Legendre polynomials for solving the fractional differential equations with time varying delays is explained. Also, the properties and convergence of the proposed method are surveyed by some theorems. Finally, some numerical examples are implemented to show the efficiency and accuracy of the proposed method. The numerical result shows that the suggested method has high accuracy compared with the existing methods.

Generating Functions for Legendre Polynomials by using Group Theoretic Method

In this paper we obtain generating functions for the Legendre polynomials \(P_n (x)\) in its modified form by using Weisner's group theoretic method. Whereas, we deployed it to determine the new generating relations between the generalized Legendre polynomials and with easy way. The ideas in consent with proofs are originated from the book of McBride [8] and is used to determine new generating relations which involve modified Legendre polynomials.

GENERALIZED LEGENDRE POLYNOMIALS FOR SUPPORT VECTOR MACHINES (SVMS) CLASSIFICATION

In this paper, we introduce a set of new kernel functions derived from the generalized Legendre polynomials to obtain more robust and higher support vector machine (SVM) classification accuracy. The generalized Legendre kernel functions are suggested to provide a value of how two given vectors are like each other by changing the inner product of these two vectors into a greater dimensional space. The proposed kernel functions satisfy the Mercer’s condition and orthogonality properties for reaching the optimal result with low number support vector (SV). For that, the new set of Legendre kernel functions could be utilized in classification applications as effective substitutes to those generally used like Gaussian, Polynomial and Wavelet kernel functions. The suggested kernel functions are calculated in compared to the current kernels such as Gaussian, Polynomial, Wavelets and Chebyshev kernels by application to various non-separable data sets with some attributes. It is seen that the suggested kernel functions could give competitive classification outcomes in comparison with other kernel functions. Thus, on the basis test outcomes, we show that the suggested kernel functions are more robust about the kernel parameter change and reach the minimal SV number for classification generally.

Generalized Legendre polynomials and related supercongruences

For any positive integer n and variables a and x we define the generalized Legendre polynomial Pn(a,x) by Pn(a,x)=∑k=0n(ak)(−1−ak)(1−x2)k. Let p be an odd prime. In this paper we prove many congruences modulo p2 related to Pp−1(a,x). For example, we show that Pp−1(a,x)≡(−1)〈a〉pPp−1(a,−x)(modp2), where a is a rational p-adic integer and 〈a〉p is the least nonnegative residue of a modulo p. We also generalize some congruences of Zhi-Wei Sun, and establish congruences for ∑k=0p−1(2kk)(3kk)/54k and ∑k=0p−1(ak)(b−ak)(modp2).

A new basis set for molecular bending degrees of freedom

We present a new basis set as an alternative to Legendre polynomials for the variational treatment of bending vibrational degrees of freedom in order to highly reduce the number of basis functions. This basis set is inspired from the harmonic oscillator eigenfunctions but is defined for a bending angle in the range theta in [0:pi]. The aim is to bring the basis functions closer to the final (ro)vibronic wave functions nature. Our methodology is extended to complicated potential energy surfaces, such as quasilinearity or multiequilibrium geometries, by using several free parameters in the basis functions. These parameters allow several density maxima, linear or not, around which the basis functions will be mainly located. Divergences at linearity in integral computations are resolved as generalized Legendre polynomials. All integral computations required for the evaluation of molecular Hamiltonian matrix elements are given for both discrete variable representation and finite basis representation. Convergence tests for the low energy vibronic states of HCCH(++), HCCH(+), and HCCS are presented.

Generalized q-Legendre polynomials

This article gives a q-version of the generalized Legendre polynomials recently introduced by the author. The generalization makes use of the little q-Jacobi polynomials. In conclusion some open problems are posed.

Generalized Legendre polynomials.

Generalized Legendre polynomials.

Harmoniques sphériques Généralisées, coefficients de Fourier des polynômes de Legendre associés généralisés: exemple d'un programme Fortran

A computer program for numerical calculation has been written in Fortran which enables Fourier coefficients of associated generalized Legendre polynomials to be obtained in the general case. Thus the generalized spherical harmonics, involved in the quantitative analysis of the distribution function of the orientations of a texture, can be calculated.

Irreducible Cartesian tensor expansions of scalar fields

It is shown how a scalar function V (‖R + Σni=1 ai‖) of a sum of n + 1 vectors can be expanded as a multiple Cartesian tensor series in the vectors ai. This expansion is a rearrangement of the multiple Taylor series expansion of such a function. In order to prove the fundamental theorem, Eq. (3.1) below, generalized Cartesian Legendre polynomials are defined. The theorem is applied to the eigenfunctions of the Laplace operator and to inverse powers. The expansions of the latter type of function leads to forms involving generalized hypergeometric functions in several variables. As a special case, the Cartesian form of the multipole expansion of the electrostatic potential between two linear molecules is derived. A number of sum rules for hypergeometric functions and addition formulas for (standard and modified) spherical Bessel functions are proved using a reduction property of the generalized Legendre polynomials. The case of the expansion of a tensorial function is also briefly discussed.

On the theory of generalized legendre polynomials

On the theory of generalized legendre polynomials