scholarly journals A decomposition algorithm coupled with operational matrices approach with applications to fractional differential equations

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 449-455
Author(s):  
Imran Talib ◽  
Nur Alam ◽  
Dumitru Baleanu ◽  
Danish Zaidi
Keyword(s):  
Numerical Method ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Legendre Function ◽  
Operational Matrix ◽  
Decomposition Algorithm ◽  
Operational Matrices ◽  
Initial Value ◽  
Multiple Orders ◽  
Fractional Differential

In this article, we solve numerically the linear and non-linear fractional initial value problems of multiple orders by developing a numerical method that is based on the decomposition algorithm coupled with the operational matrices approach. By means of this, the fractional initial value problems of multiple orders are decomposed into a system of fractional initial value problems which are then solved by using the operational matrices approach. The efficiency and advantage of the developed numerical method are highlighted by comparing the results obtained otherwise in the literature. The construction of the new derivative operational matrix of fractional legendre function vectors in the Caputo sense is also a part of this research. As applications, we solve several fractional initial value problems of multiple orders. The numerical results are displayed in tables and plots.

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 Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 226-230 ◽  
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.

On the initial value problems for the Caputo–Fabrizio impulsive fractional differential equations

2020 ◽  
pp. 2150073
Author(s):  
Mohamed I. Abbas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Initial Value ◽  
Impulsive Fractional Differential Equations ◽  
Uniqueness Results ◽  
Fractional Differential

This paper is devoted to initial value problems for impulsive fractional differential equations of Caputo–Fabrizio type fractional derivative. By means of Banach’s fixed point theorem and Schaefer’s fixed point theorem, the existence and uniqueness results are obtained. Finally, an example is given to illustrate one of the main results.

Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function

2021 ◽  
Vol 24 (4) ◽  
pp. 1220-1230
Author(s):  
Mohammed Al-Refai
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Maximum Principles ◽  
Initial Value ◽  
Derivative Operator ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Two Parameters

Abstract In this paper, we formulate and prove two maximum principles to nonlinear fractional differential equations. We consider a fractional derivative operator with Mittag-Leffler function of two parameters in the kernel. These maximum principles are used to establish a pre-norm estimate of solutions, and to derive certain uniqueness and positivity results to related linear and nonlinear fractional initial value problems.

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