Legendre Orthogonal Functions Research Articles

A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations

<abstract><p>A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-differential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.</p></abstract>

Image representation based on fractional order Legendre and Laguerre orthogonal moments

In this paper, we have introduced new sets of fractional order orthogonal basis moments based on Fractional order Legendre orthogonal Functions (FLeFs) and Fractional order Laguerre orthogonal Functions (FLaFs) for image representation. We have generated a novel set of Fractional order Legendre orthogonal Moments (FLeMs) from fractional order Legendre orthogonal functions and a new set of Fractional order Laguerre orthogonal Moments (FLaMs) from the fractional order Laguerre orthogonal functions. The new presented sets of (FLeMs) and (FLaMs) are tested with the recently introduced Fractional order Chebyshev orthogonal Moments (FCMs). This edge detection filter can be used successfully in the gray level image and color images. The new sets of fractional moments are used to reconstruct the gray level image. The numerical results show FLeMs and FLaMs are promised techniques for image representation. The computational time of the proposed techniques is compared with the computational time of Chebyshev orthogonal Moments techniques and gives better results. Also, the fractional parameters give the flexibility of studying global features of the image at different positions of moments.

Shifted fractional Legendre spectral collocation technique for solving fractional stochastic Volterra integro-differential equations

This paper presents a spectral collocation technique to solve fractional stochastic Volterra integro-differential equations (FSV-IDEs). The algorithm is based on shifted fractional order Legendre orthogonal functions generated by Legendre polynomials. The shifted fractional order Legendre–Gauss–Radau collocation (SFL-GR-C) method is developed for approximating the FSV-IDEs, with the objective of obtaining a system of algebraic equations. For computational purposes, the Brownian motion function W(x) is discretized by Lagrange interpolation, while the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Numerical examples demonstrate the accuracy and applicability of the proposed technique, even when dealing with non-smooth solutions.

Numerical simulation for fractional-order differential system of a Glioblastoma Multiforme and Immune system

In this paper, we present a numerical simulation to study a fractional-order differential system of a glioblastoma multiforme and immune system. This numerical simulation is based on spectral collocation method for tackling the fractional-order differential system of a glioblastoma multiforme and immune system. We introduce new shifted fractional-order Legendre orthogonal functions outputted by Legendre polynomials. Also, we state and derive some corollaries and theorems related to the new shifted fractional order Legendre orthogonal functions. The shifted fractional-order Legendre–Gauss–Radau collocation method is developed to approximate the fractional-order differential system of a glioblastoma multiforme and immune system. The basis of the shifted fractional-order Legendre orthogonal functions is adapted for temporal discretization. The solution of such an equation is approximated as a truncated series of shifted fractional-order Legendre orthogonal functions for temporal variable, and then we evaluate the residuals of the mentioned problem at the shifted fractionalorder Legendre–Gauss–Radau quadrature points. The accuracy of the novel method is demonstrated with several test problems.

Identification of a Non-Linear System Using Volterra Series Model with Calculated Kernels by Legendre Orthogonal Function

Modeling and identification of non-linear systems have gained lots of attentions especially in industrial processes. Most of the actual systems have non-linear behavior and the first and simplest solution in modeling such systems is to linearize them which in most cases the result of linearization is not satisfactory. In this paper, modeling of non-linear systems is investigated using Volterra series model based on Legendre orthogonal function. Expansion of Volterra series kernels by Legendre orthogonal functions causes a reduction in the number of model parameters; hence, complexity of calculations would be decreased. Besides, if the free parameter is selected properly in these orthogonal functions, error is reduced and convergence speed of parameters is increased which leads to an increase in identification accuracy. In this paper, identification of non-linear system is presented with Volterra series expanded by Legendre function and PSO algorithm is used to calculate the optimum free parameters of Legendre function. Finally, in order to validate the efficacy and accuracy, the proposed algorithm is implemented on a non-linear system i.e. heat exchanger with actual data

Proportional integral derivative controller design using Legendre orthogonal functions

The Legendre orthogonal functions are employed to design the family of PID controllers for a variety of plants. In the proposed method, the PID controller and the plant model are represented with their corresponding Legendre series. Matching the first three terms of the Legendre series of the loop gain with the desired one gives the PID controller parameters. The closed loop system stability conditions in terms of the Legendre basis function pole (λ) for a wide range of systems including the first order, second order, double integrator, first order plus dead time, and first order unstable plants are obtained. For first order and double integrator plants, the closed loop system stability is preserved for all values of λ and for the other plants, an appropriate range in terms of λ is obtained. The optimum value of λ to attain a minimum integral square error performance index in the presence of the control signal constraints is achieved. The numerical simulations demonstrate the benefits of the Legendre based PID controller.

The Study of 3D Digital Watermarking Algorithm Which is Based on a Set of Complete System of Legendre Orthogonal Function

The Study of 3D Digital Watermarking Algorithm Which is Based on a Set of Complete System of Legendre Orthogonal Function

Control of Robot Manipulators Based on Legendre Orthogonal Neural Network

In this paper, Legendre orthogonal functions neural network is used to achieve the control of nonlinear systems. The adaptive controller is constructed by using Legendre orthogonal functions neural network. The adaptive learning law of orthogonal neural network is derived to guarantee that the adaptive weight errors and tracking errors are bound by using Lyapunov stability theory. Simulation results are given for a two-link robot, and the control scheme is validated.

Tracking control of robot manipulators based on orthogonal neural network

A robust adaptive control based on orthogonal neural network (ONN) is presented for robot manipulators in this paper. The adaptive controller constructed by Legendre orthogonal function neural network has some advantages such as simple structure and fast convergence speed. The adaptive learning law of orthogonal neural network is derived to guarantee that the adaptive weight errors and tracking errors are bounded by using Lyapunov stability theory. Simulation results on a two-link robot manipulator validate the control scheme.

A system identification algorithm using orthogonal functions

An adaptive filter (ADF) structure is proposed for applications in which large-order ADFs are required. It is based on modeling the impulse response of the system to be identified as a linear combination of a set of discrete Legendre orthogonal functions. The proposed adaptive filter structure has desirable stability features and a unimodal mean-square error surface as well as a modular structure that permits an easy increase of the filter order without changing the previous stages. Computer simulations in which the proposed structure is used to identify actual acoustic echo path impulse responses show that the Legendre ADF has better convergence performance than the transversal ADF when identifying systems with long impulse response.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>