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

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

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0281 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Khalid K. Ali ◽

Mohamed A. Abd El salam ◽

Emad M. H. Mohamed

Keyword(s):

Differential Equations ◽

Collocation Method ◽

Fractional Order ◽

Linear Functional ◽

Numerical Technique ◽

Operational Matrix ◽

Fractional Order Differential Equations ◽

The Given ◽

Functional Argument ◽

Special Case

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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A new exponential Chebyshev operational matrix of derivatives for solving high-order ordinary differential equations in unbounded domains

Journal of Modern Methods in Numerical Mathematics ◽

10.20454/jmmnm.2016.1068 ◽

2016 ◽

Vol 7 (1) ◽

pp. 19 ◽

Cited By ~ 2

Author(s):

Mohamed Ramadan ◽

Kamal Raslan ◽

Talaat El Danaf ◽

Mohamed A. Abd Elsalam

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Collocation Method ◽

Operational Matrix ◽

Variable Coefficients ◽

High Order ◽

Algebraic Equations ◽

Linear Ordinary Differential Equations ◽

The Given ◽

Derivatives Of

The purpose of this paper is to investigate a new exponential Chebyshev (EC) operational matrix of derivatives. The new operational matrix of derivatives of the EC functions is derived and introduced for solving high-order linear ordinary differential equations with variable coefficients in unbounded domain using the collocation method. This method transforms the given differential equation and conditions to matrix equation with unknown EC coefficients. These matrices together with the collocation method are utilized to reduce the solution of high-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of EC functions. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive with good accuracy.

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Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

Journal of Function Spaces ◽

10.1155/2021/3084110 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

F. M. Alharbi ◽

A. M. Zidan ◽

Muhammad Naeem ◽

Rasool Shah ◽

Kamsing Nonlaopon

Keyword(s):

Differential Equations ◽

Fractional Integral ◽

Numerical Technique ◽

Fractional Integration ◽

Operational Matrix ◽

Nonlinear Problems ◽

Haar Wavelet ◽

Mathematical Tool ◽

Suggested Technique ◽

Operational Matrix Of Integration

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

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Applications of Legendre spectral collocation method for solving system of time delay differential equations

Advances in Mechanical Engineering ◽

10.1177/1687814020922113 ◽

2020 ◽

Vol 12 (6) ◽

pp. 168781402092211

Author(s):

Sami Ullah Khan ◽

Ishtiaq Ali

Keyword(s):

Time Delay ◽

Differential Equations ◽

Collocation Method ◽

Differential System ◽

Numerical Test ◽

Spectral Collocation Method ◽

Numerical Techniques ◽

Spectral Collocation ◽

Delay Differential System ◽

Delay Differential

The numerical techniques are regarded as the backbone of modern research. In literature, the exact solution of time delay differential models are hardly achievable or impossible. Therefore, numerical techniques are the only way to find their solution. In this article, a novel numerical technique known as Legendre spectral collocation method is used for the approximate solution of time delay differential system. Legendre spectral collocation method and their properties are applied to determined the general procedure for solving time delay differential system with detail error and convergence analysis. The method first convert the proposed system to a system of ordinary differential equations and then apply the Legendre polynomials to solve the resultant system efficiently. Finally, some numerical test problems are given to confirm the efficiency of the method and were compared with other available numerical schemes in the literature.

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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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Highly accurate technique for studying some chaotic models described by ABC-fractional differential equations of variable-order

International Journal of Modern Physics C ◽

10.1142/s0129183121500182 ◽

2020 ◽

pp. 2150018

Author(s):

M. M. Khader ◽

Ibrahim Al-Dayel

Keyword(s):

Exact Solution ◽

Differential Equations ◽

Numerical Solutions ◽

Polynomial Interpolation ◽

Numerical Technique ◽

Variable Order ◽

Lagrange Polynomial ◽

Efficient Tool ◽

Fractional Differential ◽

The Given

The propose of this paper is to introduce and investigate a highly accurate technique for solving the fractional Logistic and Ricatti differential equations of variable-order. We consider these models with the most common nonsingular Atangana–Baleanu–Caputo (ABC) fractional derivative which depends on the Mittag–Leffler kernel. The proposed numerical technique is based upon the fundamental theorem of the fractional calculus as well as the Lagrange polynomial interpolation. We satisfy the efficiency and the accuracy of the given procedure; and study the effect of the variation of the fractional-order [Formula: see text] on the behavior of the solutions due to the presence of ABC-operator by evaluating the solution with different values of [Formula: see text]. The results show that the given procedure is an easy and efficient tool to investigate the solution for such models. We compare the numerical solutions with the exact solution, thereby showing excellent agreement which we have found by applying the ABC-derivatives. We observe the chaotic solutions with some fractional-variable-order functions.

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Systems of nonlinear Volterra integro-differential equations of arbitrary order

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v36i4.31478 ◽

2018 ◽

Vol 36 (4) ◽

pp. 33-54 ◽

Cited By ~ 6

Author(s):

Kourosh Parand ◽

Mehdi Delkhosh

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Approximate Method ◽

Fractional Order ◽

Chebyshev Polynomials ◽

Arbitrary Order ◽

Operational Matrix ◽

Algebraic Equations ◽

Arbitrary Integer ◽

Orthogonal Functions

In this paper, a new approximate method for solving of systems of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional) order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs). Also, we construct the fractional derivative operational matrix of order $\alpha$ in the Caputo's definition for GFCFs. This method reduced a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.

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Solving Nonlinear Fractional Integro-Differential Equations of Volterra Type Using Novel Mathematical Matrices

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4029281 ◽

2015 ◽

Vol 10 (6) ◽

Cited By ~ 3

Author(s):

Farshid Mirzaee ◽

Saeed Bimesl ◽

Emran Tohidi

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Collocation Method ◽

Operational Matrix ◽

Volterra Type

In this paper, the operational matrix of Euler functions for fractional derivative of order β in the Caputo sense is derived. Via this matrix, we develop an efficient collocation method for solving nonlinear fractional Volterra integro-differential equations. Illustrative examples are given to demonstrate the validity and applicability of the proposed method, and the comparisons are made with the existing results.

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