scholarly journals A New Operational Matrix of Fractional Derivatives to Solve Systems of Fractional Differential Equations via Legendre Wavelets

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 238 ◽  
Author(s):  
Aydin Secer ◽  
Selvi Altun
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Matrix Method ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Operational Matrix Method ◽  
Fractional Differential

This paper introduces a new numerical approach to solving a system of fractional differential equations (FDEs) using the Legendre wavelet operational matrix method (LWOMM). We first formulated the operational matrix of fractional derivatives in some special conditions using some notable characteristics of Legendre wavelets and shifted Legendre polynomials. Then, the system of fractional differential equations was transformed into a system of algebraic equations by using these operational matrices. At the end of this paper, several examples are presented to illustrate the effectivity and correctness of the proposed approach. Comparing the methodology with several recognized methods demonstrates that the advantages of the Legendre wavelet operational matrix method are its accuracy and the understandability of the calculations.

scholarly journals Legendre wavelet operational matrix method for solving fractional differential equations in some special conditions

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 203-214
Author(s):  
Aydin Secer ◽  
Selvi Altun ◽  
Mustafa Bayram
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Mathematical Tool ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Operational Matrix Method ◽  
Fractional Differential ◽  
A New Technique

This paper proposes a new technique which rests upon Legendre wavelets for solving linear and non-linear forms of fractional order initial and boundary value problems. In some particular circumstances, a new operational matrix of fractional derivative is generated by utilizing some significant properties of wavelets and orthogonal polynomials. We approached the solution in a finite series with respect to Legendre wavelets and then by using these operational matrices, we reduced the fractional differential equations into a system of algebraic equations. Finally, the introduced technique is tested on several illustrative examples. The obtained results demonstrate that this technique is a very impressive and applicable mathematical tool for solving fractional differential equations.

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

2017 ◽  
Vol 15 (05) ◽  
pp. 1750043 ◽  
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.

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

Symmetry ◽  
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.

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

Complexity ◽  
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.

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

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.

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

2019 ◽  
Vol 17 (04) ◽  
pp. 1950026 ◽  
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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