The operational matrix of Caputo fractional derivatives of modified generalized Laguerre polynomials and its applications

This paper deals with modified generalized Laguerre spectral tau and collocation methods for solving linear and nonlinear multiterm fractional differential equations (FDEs) on the half line. A new formula expressing the Caputo fractional derivatives of modified generalized Laguerre polynomials of any degree and for any fractional order in terms of the modified generalized Laguerre polynomials themselves is derived. An efficient direct solver technique is proposed for solving the linear multiterm FDEs with constant coefficients on the half line using a modified generalized Laguerre tau method. The spatial approximation with its Caputo fractional derivatives is based on modified generalized Laguerre polynomialsLi(α,β)(x)withx∈Λ=(0,∞),α>−1, andβ>0, andiis the polynomial degree. We implement and develop the modified generalized Laguerre collocation method based on the modified generalized Laguerre-Gauss points which is used as collocation nodes for solving nonlinear multiterm FDEs on the half line.

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A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Entropy ◽

10.3390/e18100345 ◽

2016 ◽

Vol 18 (10) ◽

pp. 345 ◽

Cited By ~ 12

Author(s):

Waleed Abd-Elhameed ◽

Youssri Youssri

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Fractional Derivatives ◽

Operational Matrix ◽

Fibonacci Polynomials ◽

Caputo Fractional Derivatives ◽

Fractional Differential ◽

Derivatives Of

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A new operational matrix of Caputo fractional derivatives of Fermat polynomials: an application for solving the Bagley-Torvik equation

Advances in Difference Equations ◽

10.1186/s13662-017-1123-4 ◽

2017 ◽

Vol 2017 (1) ◽

Cited By ~ 16

Author(s):

Youssri H Youssri

Keyword(s):

Fractional Derivatives ◽

Operational Matrix ◽

Caputo Fractional Derivatives ◽

Derivatives Of

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New Numerical Algorithm to Solve Variable-Order Fractional Integrodifferential Equations in the Sense of Hilfer-Prabhakar Derivative

Abstract and Applied Analysis ◽

10.1155/2021/8817794 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

MohammadHossein Derakhshan

Keyword(s):

Fractional Derivatives ◽

Numerical Approximation ◽

Numerical Technique ◽

Operational Matrix ◽

Integrodifferential Equations ◽

Variable Order ◽

Algebraic Equations ◽

Lagrange Multiplier Technique ◽

Derivatives Of ◽

Variable Order Fractional Derivative

In this article, a numerical technique based on the Chebyshev cardinal functions (CCFs) and the Lagrange multiplier technique for the numerical approximation of the variable-order fractional integrodifferential equations are shown. The variable-order fractional derivative is considered in the sense of regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives. To solve the problem, first, we obtain the operational matrix of the regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives of CCFs. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional integrodifferential equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate its accuracy and efficiency.

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On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions

Advances in Difference Equations ◽

10.1186/s13662-020-03196-6 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Mohammed S. Abdo ◽

Thabet Abdeljawad ◽

Saeed M. Ali ◽

Kamal Shah

Keyword(s):

Boundary Value Problems ◽

Existence And Uniqueness ◽

Fractional Derivatives ◽

Fractional Integral ◽

Fixed Point Theorems ◽

Boundary Value ◽

Integral Conditions ◽

Caputo Fractional Derivatives ◽

Derivatives Of ◽

Fractional Boundary Value Problems

AbstractIn this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders $0<\vartheta \leq 1$ 0 < ϑ ≤ 1 and $1<\vartheta \leq 2$ 1 < ϑ ≤ 2 . We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.

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On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

Fractal and Fractional ◽

10.3390/fractalfract5030117 ◽

2021 ◽

Vol 5 (3) ◽

pp. 117

Author(s):

Briceyda B. Delgado ◽

Jorge E. Macías-Díaz

Keyword(s):

Fractional Order ◽

Fractional Derivatives ◽

Decomposition Theorem ◽

Helmholtz Decomposition ◽

Fractional Operators ◽

Gradient Vector ◽

Caputo Fractional Derivatives ◽

General Solutions ◽

Derivatives Of ◽

Fractional Space

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

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On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain

Nonlinear Engineering ◽

10.1515/nleng-2018-0073 ◽

2019 ◽

Vol 8 (1) ◽

pp. 318-327 ◽

Cited By ~ 5

Author(s):

E.H. Doha ◽

Y.H. Youssri

Keyword(s):

Recurrence Relations ◽

Laguerre Polynomials ◽

Infinite Domain ◽

Laguerre Expansion ◽

Varying Coefficients ◽

Generalized Laguerre Polynomials ◽

General Order ◽

In Series ◽

Expansion Coefficients ◽

Derivatives Of

Abstract Herein, three important theorems were stated and proved. The first relates the modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients; and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced. The second theorem gives new modified generalized Laguerre coefficients of the moments of one single modified generalized Laguerre polynomials of any degree. Finally, the third theorem expresses explicitly the modified generalized Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its modified generalized Laguerre coefficients. Some spectral applications of these theorems for solving ordinary differential equations with varying coefficients and some specific applied differential problems, by reducing them to recurrence relations in their expansion coefficients of the solution are considered.

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A Note on the Generalized Relativistic Diffusion Equation

Mathematics ◽

10.3390/math7111009 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1009

Author(s):

Luisa Beghin ◽

Roberto Garra

Keyword(s):

Fourier Transform ◽

Fundamental Solution ◽

Diffusion Equation ◽

Fractional Derivatives ◽

Stable Process ◽

Caputo Fractional Derivatives ◽

The Fourier Transform ◽

Derivatives Of

We study here a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function. We find the Fourier transform of the fundamental solution and discuss the probabilistic meaning of the results obtained in relation to the time-scaled fractional relativistic stable process. We briefly consider also the application of fractional derivatives of a function with respect to another function in order to generalize fractional Riesz-Bessel equations, suggesting their stochastic meaning.

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On the absence of global solutions to two-times-fractional differential inequalities involving Hadamard-Caputo and Caputo fractional derivatives

AIMS Mathematics ◽

10.3934/math.2022323 ◽

2022 ◽

Vol 7 (4) ◽

pp. 5830-5843

Author(s):

Ibtehal Alazman ◽

◽

Mohamed Jleli ◽

Bessem Samet ◽

Keyword(s):

Global Solution ◽

Differential Inequality ◽

Fractional Derivatives ◽

Singular Potential ◽

Global Solutions ◽

Differential Inequalities ◽

Caputo Fractional Derivatives ◽

Potential Term ◽

Fractional Differential ◽

Derivatives Of

<abstract><p>In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.</p></abstract>

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Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

Abstract and Applied Analysis ◽

10.1155/2013/546502 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 37

Author(s):

D. Baleanu ◽

A. H. Bhrawy ◽

T. M. Taha

Keyword(s):

Initial Conditions ◽

Laguerre Polynomials ◽

Operational Matrix ◽

Infinite Interval ◽

Solution Technique ◽

Generalized Laguerre Polynomials ◽

Spectral Solution ◽

Spectral Algorithms ◽

And Performance ◽

Pseudo Spectral

We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.

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