A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

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Spectral Tau method for solving general fractional order differential equations with linear functional argument

Journal of the Egyptian Mathematical Society ◽

10.1186/s42787-019-0039-4 ◽

2019 ◽

Vol 27 (1) ◽

Cited By ~ 1

Author(s):

Kamal R. Raslan ◽

Mohamed A. Abd El salam ◽

Khalid K. Ali ◽

Emad M. Mohamed

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Linear Functional ◽

Numerical Technique ◽

Tau Method ◽

Fractional Order Differential Equations ◽

Functional Argument

Abstract In this paper, a numerical technique for solving new generalized fractional order differential equations with linear functional argument is presented. The spectral Tau method is extended to study this problem, where the derivatives are defined in the Caputo fractional sense. The proposed equation with its functional argument represents a general form of delay and advanced differential equations with fractional order derivatives. The obtained results show that the proposed method is very effective and convenient.

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Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series

Advances in Difference Equations ◽

10.1186/s13662-020-02951-z ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 2

Author(s):

Khalid K. Ali ◽

Mohamed A. Abd El Salam ◽

Emad M. H. Mohamed ◽

Bessem Samet ◽

Sunil Kumar ◽

...

Keyword(s):

Differential Equations ◽

Collocation Method ◽

Linear Functional ◽

Arbitrary Order ◽

Numerical Technique ◽

Operational Matrix ◽

Numerical Test ◽

Proportional Delay ◽

Chebyshev Series ◽

The Given

Abstract In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.

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Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations

Mathematics ◽

10.3390/math7010040 ◽

2019 ◽

Vol 7 (1) ◽

pp. 40 ◽

Cited By ~ 15

Author(s):

Shumaila Javeed ◽

Dumitru Baleanu ◽

Asif Waheed ◽

Mansoor Shaukat Khan ◽

Hira Affan

Keyword(s):

Differential Equations ◽

Perturbation Method ◽

Fractional Order ◽

Homotopy Perturbation Method ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Fractional Order Differential Equations ◽

Homotopy Perturbation ◽

The Given ◽

Ordinary Derivative

The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

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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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