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

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

Mathematics ◽

10.3390/math6110238 ◽

2018 ◽

Vol 6 (11) ◽

pp. 238 ◽

Cited By ~ 4

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.

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

Thermal Science ◽

10.2298/tsci180920034s ◽

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.

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Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices

Advances in Mathematical Physics ◽

10.1155/2013/954015 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 10

Author(s):

Mohsen Alipour ◽

Dumitru Baleanu

Keyword(s):

Nonlinear System ◽

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Caputo Fractional Derivative ◽

Fractional Integral ◽

Operational Matrix ◽

Classical Solutions ◽

Operational Matrices ◽

Fractional Differential

We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.

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Computational method based on Bernstein operational matrices for multi-order fractional differential equations

Filomat ◽

10.2298/fil1403591r ◽

2014 ◽

Vol 28 (3) ◽

pp. 591-601 ◽

Cited By ~ 8

Author(s):

Davood Rostamy ◽

Hossein Jafari ◽

Mohsen Alipour ◽

Chaudry Khalique

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Operational Matrix ◽

Adomian Decomposition Method ◽

Computational Method ◽

Operational Matrices ◽

Algebraic Equations ◽

Transform Method ◽

Adomian Decomposition ◽

Fractional Differential

In this paper, the Bernstein operational matrices are used to obtain solutions of multi-order fractional differential equations. In this regard we present a theorem which can reduce the nonlinear fractional differential equations to a system of algebraic equations. The fractional derivative considered here is in the Caputo sense. Finally, we give several examples by using the proposed method. These results are then compared with the results obtained by using Adomian decomposition method, differential transform method and the generalized block pulse operational matrix method. We conclude that our results compare well with the results of other methods and the efficiency and accuracy of the proposed method is very good.

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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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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 wavelet method for solving Caputo–Hadamard fractional differential equation

Engineering Computations ◽

10.1108/ec-03-2021-0165 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Umer Saeed

Keyword(s):

Differential Equations ◽

Boundary Value Problems ◽

Fractional Differential Equations ◽

Boundary Value ◽

Fractional Integration ◽

Operational Matrix ◽

Operational Matrices ◽

Content Type ◽

Hadamard Fractional Differential Equations ◽

Fractional Differential

PurposeThe purpose of the present work is to propose a wavelet method for the numerical solutions of Caputo–Hadamard fractional differential equations on any arbitrary interval.Design/methodology/approachThe author has modified the CAS wavelets (mCAS) and utilized it for the solution of Caputo–Hadamard fractional linear/nonlinear initial and boundary value problems. The author has derived and constructed the new operational matrices for the mCAS wavelets. Furthermore, The author has also proposed a method which is the combination of mCAS wavelets and quasilinearization technique for the solution of nonlinear Caputo–Hadamard fractional differential equations.FindingsThe author has proved the orthonormality of the mCAS wavelets. The author has constructed the mCAS wavelets matrix, mCAS wavelets operational matrix of Hadamard fractional integration of arbitrary order and mCAS wavelets operational matrix of Hadamard fractional integration for Caputo–Hadamard fractional boundary value problems. These operational matrices are used to make the calculations fast. Furthermore, the author works out on the error analysis for the method. The author presented the procedure of implementation for both Caputo–Hadamard fractional initial and boundary value problems. Numerical simulation is provided to illustrate the reliability and accuracy of the method.Originality/valueMany scientist, physician and engineers can take the benefit of the presented method for the simulation of their linear/nonlinear Caputo–Hadamard fractional differential models. To the best of the author’s knowledge, the present work has never been proposed and implemented for linear/nonlinear Caputo–Hadamard fractional differential equations.

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