Showing papers on "Legendre polynomials published in 2022"

Certain New Models of the Multi-Space Fractal-Fractional Kuramoto-Sivashinsky and Korteweg-de Vries Equations

Abstract: 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.

An analytical integration Legendre polynomial series approach for Lamb waves in fractional order thermoelastic multilayered plates

Abstract: The Legendre polynomial series approach (LPSA) has been widely used to solve guided wave propagation in various structures since 1999. The LPSA directly introduces the boundary conditions into the control equations through the rectangular window function. However, in the solving process, the Legendre polynomial series, the rectangular window function, and their derivatives are introduced into the integral kernel functions, which results in a lot of CPU time on the abundant numerical integration calculations. To overcome this defect, an analytical integration Legendre polynomial series approach (AILPSA) is presented and is used to solve the Lamb waves in fractional order thermoelastic multilayered plates. Coupled wave equations and heat conduction equation are solved by the AILPSA and the LPSA, respectively. Comparison between two approaches indicates the computational efficiency of the AILPSA is improved by more than 90%. In addition, a backward error estimation is given in the convergency analysis to make up for the deficiency of the numerical experiments. Finally, the influence of fractional order on the thermoelastic Lamb wave is discussed.

Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order

Abstract: In this article, neural network method (NNM) is presented to solve the spatiotemporal variable-order fractional advection-diffusion equation with a nonlinear source term. The network is established by using shifted Legendre orthogonal polynomials with adjustable coefficients. According to the properties of variable fractional derivative, the loss function of neural network is deduced theoretically. Assume that the source function satisfies the Lipschitz hypothesis, the reasonable range for learning rate is discussed in details. Then neural networks are trained repeatedly on the training set to reduce the loss functions for two numerical examples. Numerical results show that the neural network method is better than the finite difference method in solving some nonlinear variable fractional order problems. Finally, several graphs and some numerical analysis are given to confirm our conclusions.

Numerical solution of distributed order integro-differential equations

Abstract: In this paper, a numerical algorithm is presented to obtain approximate solution of distributed order integro-differential equations. The approximate solution is expressed in the form of a polynomial with unknown coefficients and in place of differential and integral operators, we make use of matrices that we deduce from the shifted Legendre polynomials. To compute the numerical values of the polynomial coefficients, we set up a system of equations that tallies with the number of unknowns, we achieve this goal through the Legendre–Gauss quadrature formula and the collocation technique. The theoretical aspects of the error bound are discussed. Illustrative examples are included to demonstrate the validity and applicability of the method.

A generalized operational matrix of mixed partial derivative terms with applications to multi-order fractional partial differential equations

Abstract: In this paper, a computational approach based on the operational matrices in conjunction with orthogonal shifted Legendre polynomials (OSLPs) is designed to solve numerically the multi-order partial differential equations of fractional order consisting of mixed partial derivative terms. Our computational approach has ability to reduce the fractional problems into a system of Sylvester types matrix equations which can be solved by using MATLAB builtin function lyap(.). The solution is approximated as a basis vectors of OSLPs. The efficiency and the numerical stability is examined by taking various test examples.

On a new method for finding numerical solutions to integro-differential equations based on Legendre multi-wavelets collocation

Abstract: In this article, a wavelet collocation method based on linear Legendre multi-wavelets is proposed for the numerical solution of the first as well as higher orders Fredholm, Volterra and Volterra–Fredholm integro-differential equations. The presented numerical method has the capability to tackle the solutions of both linear and nonlinear problems of these model equations. In order to endorse accuracy and efficiency of the method, it is tested on various numerical problems from literature with the aid of maximum absolute errors and rates of convergence. L ∞ norms are used to compare the numerical results with other available methods such as Multi-Scale-Galerkin’s method, Haar wavelet collocation method and Meshless method from literature. The comparability of the presented method with other existing numerical methods demonstrates superior efficiency and accuracy.

A generalized operational matrix of mixed partial derivative terms with applications to multi-order fractional partial differential equations

Abstract: In this paper, a computational approach based on the operational matrices in conjunction with orthogonal shifted Legendre polynomials (OSLPs) is designed to solve numerically the multi-order partial differential equations of fractional order consisting of mixed partial derivative terms. Our computational approach has ability to reduce the fractional problems into a system of Sylvester types matrix equations which can be solved by using MATLAB builtin function lyap(.). The solution is approximated as a basis vectors of OSLPs. The efficiency and the numerical stability is examined by taking various test examples.

A meshfree approach for free vibration analysis of laminated sectorial and rectangular plates with varying fiber angle

Abstract: This paper focuses on the free vibration analysis of laminated sectorial and rectangular plates with varying fiber angle by means of the first-order shear deformation theory (FSDT) and the meshfree strong form method. A meshfree Legendre-radial point interpolation method (Legendre-RIPM) shape function using the combined basis of multi-quadrics (MQ) radial function and Legendre polynomials is presented. The equations of motion and boundary conditions are discretized by meshfree strong form method and the displacements are approximated using the Legendre-RIPM shape function. The accuracy and reliability of the current method are validated by comparing the results of the literature and the ABAQUS. The effects of some geometrical parameters on the natural frequencies of laminated sectorial and rectangular plates with varying fiber angles are investigated through numerical examples. • Free vibration analysis of sectorial and rectangular plates with varying fiber angle is conducted. • Various curved and linear fibers of laminated plates are represented by fiber angle function. • A meshfree Legendre-radial point interpolation shape function is constructed. • Governing equations and boundary conditions are discretized by meshfree strong form method. • Displacement components of laminated plate are approximated by proposed shape function.

The spectral collocation method for solving a fractional integro-differential equation

Abstract: In this paper, we propose a high-precision numerical algorithm for a fractional integro-differential equation based on the shifted Legendre polynomials and the idea of Gauss-Legendre quadrature rule and spectral collocation method. The error analysis of this method is also given in detail. Some numerical examples are give to illustrate the exponential convergence of our method.

Dynamics and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method

Abstract: The aim of this study is to investigate the dynamics of epidemic transmission of COVID-19 SEIR stochastic model with generalized saturated incidence rate. We assume that the random perturbations depends on white noises, which implies that it is directly proportional to the steady states. The existence and uniqueness of the positive solution along with the stability analysis is provided under disease-free and endemic equilibrium conditions for asymptotically stable transmission dynamics of the model. An epidemiological metric based on the ratio of basic reproduction is used to describe the transmission of an infectious disease using different parameters values involve in the proposed model. A higher order scheme based on Legendre spectral collocation method is used for the numerical simulations. For the better understanding of the proposed scheme, a comparison is made with the deterministic counterpart. In order to confirm the theoretical analysis, we provide a number of numerical examples.

Application of Legendre polynomials based neural networks for the analysis of heat and mass transfer of a non-Newtonian fluid in a porous channel

Abstract: Abstract In this paper, the mathematical models for flow and heat-transfer analysis of a non-Newtonian fluid with axisymmetric channels and porous walls are analyzed. The governing equations of the problem are derived by using the basic concepts of continuity and momentum equations. Furthermore, artificial intelligence-based feedforward neural networks (ANNs) are utilized with hybridization of a generalized normal-distribution optimization (GNDO) algorithm and sequential quadratic programming (SQP) to study the heat-transfer equations and calculate the approximate solutions for the momentum of a non-Newtonian fluid. Legendre polynomials based Legendre neural networks (LNN) are used to develop a mathematical model for the governing equations, which are further exploited by the global search ability of GNDO and SQP for rapid localization convergence. The proposed technique is applied to study the effect of variations in Reynolds number Re on the velocity profile $(f^{\prime })$ ( f ′ ) and the temperature profile $(q)$ ( q ) . The results obtained by the LeNN-GNDO-SQP algorithm are compared with the differential transformation method (DTM), which shows the stability of the results and the correctness of the technique. Extensive graphical and statistical analyses are conducted in terms of minimum, mean, and standard deviation based on fitness value, absolute errors, mean absolute deviation (MAD), error in the Nash–Sutcliffe efficiency (NSE), and root mean square error (RMSE).

A new operational vector approach for time‐fractional subdiffusion equations of distributed order based on hybrid functions

Abstract: This research study deals with the numerical solutions of linear and nonlinear time‐fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block‐pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.

A hybrid approach based on Legendre wavelet for numerical simulation of Helmholtz equation with complex solution

Abstract: In this article, with the help of Legendre wavelets a hybrid approach is presented for the numerical solution of the Helmholtz equation which has a complex solution. The present hybrid approach is based on the Legendre Wavelet Collocation Method (LWCM). Initially, the Helmholtz equation is converted to the coupled equation with suitable transformation, later all the derivatives in the equations including boundary conditions are approximated with the help of Legendre Wavelets. Later, with the help of the Legendre wavelet operational matrix of integration, a system of linear algebraic equations are formed for obtaining the numerical solution of Helmholtz equation. The aforementioned proposed algorithm is very simple and can easily be executed in any computer-oriented language efficiently. To demonstrate the efficiency and performance of the newly established technique, the method is tested on various well-known examples from previous literature including other Legendre wavelet-based methods. The obtained results produce better accuracy and widespread use of the newly developed technique for a range of benchmark problems which have complex solutions.

Applications of the q-Derivative Operator to New Families of Bi-Univalent Functions Related to the Legendre Polynomials

Abstract: By using the q-derivative operator and the Legendre polynomials, some new subclasses of q-starlike functions and bi-univalent functions are introduced. Several coefficient estimates and Fekete–Szegö-type inequalities for functions in each of these subclasses are obtained. The results derived in this article are shown to extend and generalize those in some earlier works.

Fractional Conformable Stochastic Integrodifferential Equations: Existence, Uniqueness, and Numerical Simulations Utilizing the Shifted Legendre Spectral Collocation Algorithm

Abstract: Theoretical and numerical studies of fractional conformable stochastic integrodifferential equations are introduced in this study. Herein, to emphasize the solution’s existence, we provide proof based on Picard iterations and Arzela−Ascoli’s theorem, whilst the proof of the uniqueness mainly depends on the famous Gronwall’s inequality. Also, we introduce the basic concepts related to shifted Legendre orthogonal polynomials which are utilized to be the basic functions of the spectral collocation algorithm to obtain approximate solutions for the mentioned equations that are not easy to be solved analytically. The substantial idea of the proposed algorithm is to transform such equations into a system containing a finite number of algebraic equations that can be treated using familiar numerical methods. For computational aims, we make a suitable discretization to evaluate the values of the Brownian motion, the noise term considered in our problem, at specific points. In addition, the feasibility and efficiency of the proposed algorithm are proved through convergence analysis and mathematical examples. To exhibit the mathematical simulation, graphs and tables are lucidly shown. Obviously, the physical interpretation of the displayed graphics accurately describes the behavior of the solutions. Despite the simplicity of the presented technique, it produces accurate and reasonable results as notarized in the conclusion section.

On Orthogonal Polynomials and Finite Moment Problem

Abstract: This paper is an improvement of a previous work on the problem recovering a function or probability density function from a finite number of its geometric moments, [1]. The previous worked solved the problem with the help of the B-Spline theory which is a great approach as long as the resulting linear system is not very large. In this work, two solution algorithms based on the approximate representation of the target probability distribution function via an orthogonal expansion are provided. One primary application of this theory is the reconstruction of the Particle Size Distribution (PSD), occurring in chemical engineering applications. Another application of this theory is the reconstruction of the Radon transform of an image at an unknown angle using the moments of the transform at known angles which leads to the reconstruction of the image form limited data. The aim is to recover a probability density function from a finite number of its geometric moments. The tool is the orthogonal expansion approach. The Shifted-Legendre Polynomials and the Chebyshev Polynomials as bases for the orthogonal expansion are used in this study. A high degree of accuracy has been obtained in recovering a function without facing a possible ill-conditioned linear system, which is the case with many typical approaches of solving the problem. In fact, for a normalized template function f on the interval [0, 1], and a reconstructed function ; the reconstruction accuracy is measured in two domains. One is the moment domain, in which the error (difference between the moments of f and the moments of ) is zero. The other measure is the standard difference in the norm -space ||f- || which can be ≈ 10-6 or less. This paper discusses the problem of recovering a function from a finite number of its geometric moments for the PSD application. Linear transformations were used, as needed, so that the function is supported on the unit interval [0, 1], or on [0, α] for some choice of α. This transformation forces the sequence of moments to vanish. Then, an orthogonal expansion of the Scaled Shifted Legendre Polynomials, as well as the Chebyshev Polynomials, are developed. The result shows good accuracy in recovering different types of synthetic functions. It is believed that up to fifteen moments, this approach is safe and reliable.

Fractional delay integrodifferential equations of nonsingular kernels: existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

Abstract: This paper considers linear and nonlinear fractional delay Volterra integrodifferential equation of order [Formula: see text] in the Atangana–Beleanu–Caputo (ABC) sense. We used continuous Laplace transform (CLT) to find equivalent Volterra integral equations that have been used together with the Arzela–Ascoli theorem and Schauder’s fixed point theorem to prove the local existence solution. Moreover, the obtained Volterra integral equations and the contraction mapping theorem have been successfully applied to construct and prove the global existence and uniqueness of the solution for the considered fractional delay integrodifferential equation (FDIDE). The Galerkin algorithm instituted within shifted Legendre polynomials (SLPs) is applied in the approximation procedure for the corresponding delay equation. Indeed, by this algorithm, we get algebraic system models and by solving this system we gained the approximated nodal solution. The reliability of the method and reduction in the size of the computational work give the algorithm wider applicability. Linear and nonlinear examples are included with some tables and figures to show the effectiveness of the method in comparison with the exact solutions. Finally, some valuable notes and details extracted from the presented results were presented in the last part, with the sign to some of our future works.

One‐parameter families of Legendre curves and plane line congruences

Abstract: Families of curves in the Euclidean plane naturally contain singular curves, where the frame of classical differential geometry does not work well. We introduce the notions of one‐parameter family of Legendre curves in the Euclidean plane, congruent equivalence and curvature. Especially, a one‐parameter family of Legendre curves can contain singular curves, and is determined by the curvature up to congruence. We also give properties of one‐parameter families of Legendre curves. As applications, we give a relation between one‐parameter families of Legendre curves and Legendre surfaces. Moreover, we study plane line congruences (one‐parameter families of lines in plane) in terms of the curvatures as one‐parameter families of Legendre curves.

Orthonormal shifted discrete Legendre polynomials for the variable-order fractional extended Fisher–Kolmogorov equation

Abstract: This paper presents a numerical technique for solving the variable-order fractional extended Fisher–Kolmogorov equation. The method suggested to solve this problem is based on the orthonormal shifted discrete Legendre polynomials and the collocation method. First, we expand the unknown solution of the problem using the these polynomialss. Also, we approximate the second- and fourth-order classical derivatives, as well as the variable-order fractional derivatives by these basis functions. Then, we substitute these approximations in the equation. Next, we utilize the classical and fractional derivative matrices together with the collocation method to convert the main equation into a system containing nonlinear algebraic equations. We show the correctness of the proposed scheme by providing several numerical examples.