Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations

AbstractIn this work, we propose a new operational method based on a Genocchi wavelet-like basis to obtain the numerical solutions of non-linear fractional order differential equations (NFDEs). To the best of our knowledge this is the first time a Genocchi wavelet-like basis is presented. The Genocchi wavelet-like operational matrix of a fractional derivative is derived through waveletpolynomial transformation. These operational matrices are used together with the collocation method to turn the NFDEs into a system of non-linear algebraic equations. Error estimates are shown and some illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed technique.

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Haar wavelet operational matrix method for system of fractional nonlinear differential equations

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691317500436 ◽

2017 ◽

Vol 15 (05) ◽

pp. 1750043 ◽

Cited By ~ 3

Author(s):

Umer Saeed

Keyword(s):

Differential Equations ◽

Reliable Method ◽

Nonlinear Differential Equations ◽

Operational Matrix ◽

Implementation Process ◽

Haar Wavelet ◽

Operational Matrices ◽

Algebraic Equations ◽

Operational Matrix Method ◽

Fractional Differential

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

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An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

Symmetry ◽

10.3390/sym12111755 ◽

2020 ◽

Vol 12 (11) ◽

pp. 1755

Author(s):

M. S. Al-Sharif ◽

A. I. Ahmed ◽

M. S. Salim

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Fractional Integral ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Science And Engineering ◽

Numerical Examples ◽

Fractional Differential

Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

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

Mathematics ◽

10.3390/math8010096 ◽

2020 ◽

Vol 8 (1) ◽

pp. 96 ◽

Cited By ~ 1

Author(s):

İbrahim Avcı ◽

Nazim I. Mahmudov

Keyword(s):

Differential Equations ◽

Numerical Solutions ◽

Fractional Integration ◽

Operational Matrix ◽

Main Idea ◽

System Of Equations ◽

Algebraic Equations ◽

Fractional Order Differential Equations ◽

Fractional Differential ◽

The Given

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.

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Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

Open Physics ◽

10.1515/phys-2016-0028 ◽

2016 ◽

Vol 14 (1) ◽

pp. 226-230 ◽

Cited By ~ 9

Author(s):

A. Bolandtalat ◽

E. Babolian ◽

H. Jafari

Keyword(s):

Differential Equations ◽

Numerical Method ◽

Fractional Differential Equations ◽

Numerical Solutions ◽

Fractional Integration ◽

Operational Matrix ◽

Algebraic Equations ◽

Boubaker Polynomials ◽

Fractional Differential ◽

The Given

AbstractIn this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

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Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2017/8630895 ◽

2017 ◽

Vol 2017 ◽

pp. 1-12

Author(s):

Qingxue Huang ◽

Fuqiang Zhao ◽

Jiaquan Xie ◽

Lifeng Ma ◽

Jianmei Wang ◽

...

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Fractional Differential Equations ◽

Operational Matrix ◽

Numerical Approach ◽

Principal Characteristic ◽

Legendre Functions ◽

Algebraic Equations ◽

Linear Algebraic Equations ◽

Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

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A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

Complexity ◽

10.1155/2021/9915551 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Haifa Bin Jebreen ◽

Fairouz Tchier

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Operational Matrix ◽

Pseudospectral Method ◽

Algebraic Equations ◽

Legendre Wavelets ◽

Derivative Operator ◽

Numerical Examples ◽

Linear Algebraic Equations ◽

Fractional Differential

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

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Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2021.0438 ◽

2021 ◽

Vol 477 (2253) ◽

Author(s):

Chandrali Baishya ◽

P. Veeresha

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Laguerre Polynomial ◽

Fractional Integration ◽

Laguerre Polynomials ◽

Operational Matrix ◽

Computational Technique ◽

Operational Matrices ◽

Algebraic Equations ◽

Fractional Differential

The Atangana–Baleanu derivative and the Laguerre polynomial are used in this analysis to define a new computational technique for solving fractional differential equations. To serve this purpose, we have derived the operational matrices of fractional integration and fractional integro-differentiation via Laguerre polynomials. Using the derived operational matrices and collocation points, we reduce the fractional differential equations to a system of linear or nonlinear algebraic equations. For the error of the operational matrix of the fractional integration, an error bound is derived. To illustrate the accuracy and the reliability of the projected algorithm, numerical simulation is presented, and the nature of attained results is captured in diverse order. Finally, the achieved consequences enlighten that the solutions obtained by the proposed scheme give better convergence to the actual solution than the results available in the literature.

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A numerical method based on Haar wavelets for the Hadamard-type fractional differential equations

Engineering Computations ◽

10.1108/ec-04-2021-0223 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Zain ul Abdeen ◽

Mujeeb ur Rehman

Keyword(s):

Differential Equations ◽

Numerical Method ◽

Fractional Differential Equations ◽

Numerical Scheme ◽

Numerical Solutions ◽

Haar Wavelets ◽

Operational Matrices ◽

Algebraic Equations ◽

Content Type ◽

Fractional Differential

PurposeThe purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.Design/methodology/approachThe aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.FindingsThe upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.Originality/valueThe numerical method is purposed for solving Hadamard-type fractional differential equations.

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Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691319500267 ◽

2019 ◽

Vol 17 (04) ◽

pp. 1950026 ◽

Cited By ~ 1

Author(s):

Umer Saeed

Keyword(s):

Differential Equations ◽

Nonlinear Differential Equations ◽

Operational Matrix ◽

Type Equation ◽

Operational Matrices ◽

Algebraic Equations ◽

Operational Matrix Method ◽

Fractional Differential ◽

Quasilinearization Technique ◽

Distribution Equation

In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

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A fast numerical method for fractional partial differential equations

Advances in Difference Equations ◽

10.1186/s13662-019-2390-z ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 2

Author(s):

S. Mockary ◽

E. Babolian ◽

A. R. Vahidi

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Chebyshev Polynomials ◽

Fractional Integration ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Linear Algebraic Equations

Abstract In this paper, we use operational matrices of Chebyshev polynomials to solve fractional partial differential equations (FPDEs). We approximate the second partial derivative of the solution of linear FPDEs by operational matrices of shifted Chebyshev polynomials. We apply the operational matrix of integration and fractional integration to obtain approximations of (fractional) partial derivatives of the solution and the approximation of the solution. Then we substitute the operational matrix approximations in the FPDEs to obtain a system of linear algebraic equations. Finally, solving this system, we obtain the approximate solution. Numerical experiments show an exponential rate of convergence and hence the efficiency and effectiveness of the method.

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