Operational Matrix Of Fractional Integration Research Articles

Fractional-Order Legendre Functions for Solving Fractional Delay Differential Equations

This research propounds a fractional-order Legendre functions operational matrix of fractional integration in order to solve the fractional delay differential equations. The fractional derivative is regarded in the Caputo sense. The suggested method is implemented to reduce the problem to the solution of a system of algebraic equations. Using the presented method, some examples are solved and numerical stability of the proposed method is examined. Compared to other published methods, the presented technique proved to be more accurate.

Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

THE ALTERNATIVE LEGENDRE TAU METHOD FOR SOLVING NONLINEAR MULTI-ORDER FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, the alternative Legendre polynomials (ALPs) are used to approximate the solution of a class of nonlinear multi-order fractional differential equations (FDEs). First, the operational matrix of fractional integration of an arbitrary order and the product operational matrix are derived for ALPs. These matrices together with the spectral Tau method are then utilized to reduce the solution of the mentioned equations into the one of solving a system of nonlinear algebraic equations with unknown ALP coefficients of the exact solution. The fractional derivatives are considered in the Caputo sense and the fractional integration is described in the Riemann-Liouville sense. Numerical examples illustrate that the present method is very effective for linear and nonlinear multi-order FDEs and high accuracy solutions can be obtained only using a small number of ALPs.

Numerical Solution for A Class of Fractional Variational Problem via Second Order B-Spline Function

In this paper, we solve a class of fractional variational problems (FVPs) by using operational matrix of fractional integration which derived from second order spline (B-spline) basis function. The fractional derivative is defined in the Caputo and Riemann-Liouville fractional integral operator. The B-spline function with unknown coefficients and B-spline operational matrix of integration are used to replace the fractional derivative which is in the performance index. Next, we applied the method of constrained extremum which involved a set of Lagrange multipliers. As a result, we get a system of algebraic equations which can be solve easily. Hence, the value for unknown coefficients of B-spline function is obtained as well as the solution for the FVPs. Finally, the illustrative examples shown the validity and applicability of this method to solve FVPs.

Haar wavelet collocation method for solving singular and nonlinear fractional time-dependent Emden\u2013Fowler equations with initial and boundary conditions

In this paper, we have applied an iterative method to the singular and nonlinear fractional partial differential of Emden–Fowler equations types. Haar wavelets operational matrix of fractional integration will be used to solve the problem with the Picard technique. The singular equations turn to Sylvester equations that will be solved so that numerically solvable is very cost- effective. Moreover, the proposed technique is reliable enough to overcome the difficulty of the singular point at x = 0. Numerical examples are providing to illustrate the efficiency and accuracy of the technique.

Solution of delay fractional optimal control problems using a hybrid of block-pulse functions and orthonormal Taylor polynomials

This paper introduces an efficient direct approach for solving delay fractional optimal control problems. The concepts of the fractional integral and the fractional derivative are considered in the Riemann–Liouville sense and the Caputo sense, respectively. The suggested framework is based on a hybrid of block-pulse functions and orthonormal Taylor polynomials. The convergence of the proposed hybrid functions with respect to the L2-norm is demonstrated. The operational matrix of fractional integration associated with the hybrid functions is constructed by using the Laplace transform method. The problem under consideration is transformed into a mathematical programming one. The method of Lagrange multipliers is then implemented for solving the resulting optimization problem. The performance and computational efficiency of the developed numerical scheme are assessed through various types of delay fractional optimal control problems. Our numerical findings are compared with either exact solutions or the existing results in the literature.

Legendre wavelets for fractional partial integro-differential viscoelastic equations with weakly singular kernels⋆

This study deals with a new class of fractional partial integro-differential equations (FPI-DEs) characterized by the presence of weakly singular kernel and a Newtonian viscoelasticity factor. To numerically solve such equations, a hybrid method is established by combining the Legendre wavelets (LWs), the collocation method, and a new operational matrix of fractional integration (OMFI). More precisely, the unknown solution is expanded by the LWs with unknown coefficients. Then, the OMFI and the collocation method are utilized to extract a system of algebraic equations whose solution is an approximation for the problem’s solution. Convergence and error estimation of the LWs expansion in two dimensions are investigated. Moreover, the efficiency and accuracy of the proposed method are demonstrated by solving some concrete examples. The obtained results confirm the presented approach is very accurate to provide satisfactory solutions.

The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative

In this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers’ equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation method to obtain a numerical solution of the coupled system of Burgers’ equations with a time-fractional derivative. The properties of Gegenbauer wavelets were used to transform this system to a system of nonlinear algebraic equations in the unknown expansion coefficients. The Galerkin method and collocation method were used to find these coefficients. The main aim of this study was to indicate that the Gegenbauer wavelets-based methods is suitable and efficient for the coupled system of Burgers’ equations with time-fractional derivative. The obtained results show the applicability and efficiency of the presented Gegenbaur wavelets-based methods.

A numerical study of fractional relaxation–oscillation equations involving $$\psi $$ ψ -Caputo fractional derivative

We provide a numerical method to solve a certain class of fractional differential equations involving $$\psi $$ -Caputo fractional derivative. The considered class includes as particular case fractional relaxation–oscillation equations. Our approach is based on operational matrix of fractional integration of a new type of orthogonal polynomials. More precisely, we introduce $$\psi $$ -shifted Legendre polynomial basis, and we derive an explicit formula for the $$\psi $$ -fractional integral of $$\psi $$ -shifted Legendre polynomials. Next, via an orthogonal projection on this polynomial basis, the problem is reduced to an algebraic equation that can be easily solved. The convergence of the method is justified rigorously and confirmed by some numerical experiments.

A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

AbstractThis paper aims to demonstrate the superiority of the discrete Chebyshev polynomials over the classical Chebyshev polynomials for solving time‐delay fractional optimal control problems (TDFOCPs). The discrete Chebyshev polynomials have been introduced and their properties are investigated thoroughly. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by these polynomials with unknown coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algabric system. A comparison has been made between the required CPU time and accuracy of the discrete and continuous Chebyshev polynomials methods. The obtained numerical results reveal that utilizing discrete Chebyshev polynomials is more efficient and less time‐consuming in comparison to the continuous Chebyshev polynomials.

Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel

In this paper, a new set of functions called fractional-order Euler functions (FEFs) is constructed to obtain the solution of fractional integro-differential equations. The properties of the fractional-order Euler functions are utilized to construct the operational matrix of fractional integration. By using the matrix and the functions approximation, the fractional integro-differential equations are reduced to systems of algebraic equations. The convergence analysis of fractional-order Euler functions approximation is given. Illustrative examples are included to demonstrate the high precision and good performance of the new scheme.

A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets

The aim of the present study is to present a numerical algorithm for solving time-delay fractional optimal control problems (TDFOCPs). First, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet functions and their properties are implemented to derive some operational matrices. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by means of the Chelyshkov wavelets. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algebraic system. Moreover, some illustrative examples are considered and the obtained numerical results were compared with those previously published in the literature.

Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials

In this paper, numerical techniques are presented for solving initial value problems of nonlinear fractional differential equations. The method is implemented by applying alternative Legendre polynomials. The operational matrix of fractional integration and the product for the alternative Legendre polynomials are derived in order to transform the nonlinear equations into a system of algebraic equations. The study of the error analysis of the obtained method is also considered. Furthermore, numerical examples demonstrate that this method is applicable and accurate.

An effective numerical method for solving fractional pantograph differential equations using modification of hat functions

In this work, a spectral method based on a modification of hat functions (MHFs) is proposed to solve the fractional pantograph differential equations. Some basic properties of fractional calculus and the operational matrices of MHFs are utilized to reduce the considered problem to a system of linear algebraic equations. The greatest advantage of using MHFs is the large number of zeros in their operational matrix of fractional integration, product operational matrix and also pantograph operational matrix. This property makes these functions computationally attractive. Some illustrative examples are included to show the high performance and applicability of the proposed method and a comparison is made with the existing results. These examples confirm that the method leads to the results of convergence order O(h3).

A wavelet approach for the multi-term time fractional diffusion-wave equation

ABSTRACTIn this paper, a Galerkin method based on the second kind Chebyshev wavelets (SKCWs) is established for solving the multi-term time fractional diffusion-wave equation. To do this, a new operational matrix of fractional integration for the SKCWs must be derived and in order to improve the computational efficiency, the hat functions are proposed to create a general procedure for constructing this matrix. Implementation of these wavelet basis functions and their operational matrix of fractional integration simplifies the problem under consideration to a system of linear algebraic equations, which greatly decreases the computational cost for finding an approximate solution. The main privilege of the proposed method is adjusting the initial and boundary conditions in the final system automatically. Theoretical error and convergence analysis of the SKCWs expansion approve the reliability of the approach. Also, numerical investigation reveals the applicability and accuracy of the presented method.

A new class of operational matrices method for solving fractional neutral pantograph differential equations

This paper uses new fractional integration operational matrices to solve a class of fractional neutral pantograph delay differential equations. A fractional-order function space is constructed where the exact solution lies in, and a set of orthogonal bases are given. Using them, we reduce the fractional delay differential equation to algebraic equations and get the approximate solution. Finally, we give the Legendre operational matrix of fractional integration to solve the equation as an example and show the efficiency of the method.

Application of Müntz–Legendre polynomials for solving the Bagley–Torvik equation in a large interval

In this paper, a new numerical method for the approximate solution of the Bagley–Torvik equation which belongs to a class of fractional differential equations is proposed. The basic idea of this method is to obtain the approximate solution in a generalized form of the Muntz–Legendre polynomials. For this purpose, first, we derive an operational matrix of fractional integration based on Muntz–Legendre polynomials. Then, by using this matrix and collocation method, the Bagley–Torvik equation is reduced into a system of algebraic equations. Hence, by solving this system, the unknown Muntz–Legendre coefficients are computed. The accuracy and performance of the proposed method are demonstrated by some numerical examples.

Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling

In the present paper, we employ a wavelets optimization method is employed for the elucidations of fractional partial differential equations of pricing European option accompanied by a Lévy model. We apply the Legendre wavelets optimization method (LWOM) to optimize the governing problem. The novelty of the proposed method is the inclusion of differential evolution algorithm (DE) in the Legendre wavelets method for the optimized approximations of the unknown terms of the Legendre wavelets. Sequentially, the functions and components of the pricing models are discretized by utilizing the operational matrix of fractional integration of Legendre wavelets. Illustratively, the implementation of the LWOM is exemplified on a pricing European option Lévy model and successfully depicted the stock paths. Moreover, comparison analysis of the Black-Scholes model with a class of Lévy model and LWOM with q-homotopy analysis transform method (q-HATM) is also deliberated out. Accordingly, the technique is found to be appropriate for financial models that can be expressed as partial differential equations of integer and fractional orders, subjected to initial or boundary conditions.

Numerical solution of systems of fractional delay differential equations using a new kind of wavelet basis

In the present paper, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet basis and their properties are utilized to obtain their operational matrix of fractional integration in the Riemann–Liouville sense and delay operational matrix. Convergence and error bound of the expansion by this kind of wavelet functions are investigated. Then, these operational matrices along with the Galerkin approach have been implemented to solve systems of fractional delay differential equations (SFDDEs). The main superiority of the proposed technique is that it reduces SFDDEs to a system of algebraic equations. Moreover, accuracy and efficiency of the suggested Chelyshkov wavelet approach are verified through some linear and nonlinear SFDDEs. Finally, the obtained numerical results are compared with those previously reported in the literature.

Modified methods for solving two classes of distributed order linear fractional differential equations

This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the αth Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picard's method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied.