scholarly journals A numerical study of a coupled system of fractional differential equations

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2585-2600
Author(s):  
Amal Alshabanat ◽  
Bessem Samet
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Study ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Coupled System ◽  
Numerical Approach ◽  
Coupled Systems ◽  
Caputo Fractional Derivatives ◽  
Fractional Differential

We consider a certain class of coupled systems of fractional differential equations involving ?-Caputo fractional derivatives. A numerical approach is provided for solving this class of systems. The method is based on operational matrix of fractional integration of an arbitrary ?-polynomial basis. A theoretical study related to the numerical scheme and the convergence of the method is presented. Next, several numerical examples are given using different types of polynomials aiming to confirm the efficiency of our approach.

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.

scholarly journals Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

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.

scholarly journals Analysis of Nonlinear Coupled Systems of Impulsive Fractional Differential Equations with Hadamard Derivatives

2019 ◽  
Vol 2019 ◽  
pp. 1-20 ◽  
Author(s):  
Usman Riaz ◽  
Akbar Zada ◽  
Zeeshan Ali ◽  
Manzoor Ahmad ◽  
Jiafa Xu ◽  
...  
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Coupled System ◽  
Coupled Systems ◽  
Ulam Stability ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Impulsive Fractional Differential Equations ◽  
Uniqueness Results ◽  
Fractional Differential

This work is committed to establishing the assumptions essential for at least one and unique solution of a switched coupled system of impulsive fractional differential equations having derivative of Hadamard type. Using Krasnoselskii’s fixed point theorem, the existence, as well as uniqueness results, is obtained. Along with this, different kinds of Hyers–Ulam stability are discussed. For supporting the theory, example is provided.

Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel

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.

A Study of Coupled Systems of ψ-Hilfer Type Sequential Fractional Differential Equations with Integro-Multipoint Boundary Conditions

2021 ◽  
Vol 5 (4) ◽  
pp. 162
Author(s):  
Ayub Samadi ◽  
Cholticha Nuchpong ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Coupled System ◽  
Coupled Systems ◽  
Uniqueness Of Solutions ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

In this paper, the existence and uniqueness of solutions for a coupled system of ψ-Hilfer type sequential fractional differential equations supplemented with nonlocal integro-multi-point boundary conditions is investigated. The presented results are obtained via the classical Banach and Krasnosel’skiĭ’s fixed point theorems and the Leray–Schauder alternative. Examples are included to illustrate the effectiveness of the obtained results.

A wavelet method for solving Caputo–Hadamard fractional differential equation

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.

scholarly journals Investigation of a Mild Solution to Coupled Systems of Impulsive Hybrid Fractional Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Mohamed Hannabou ◽  
Hilal Khalid
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Fractional Differential Equations ◽  
Coupled System ◽  
Coupled Systems ◽  
Banach Fixed Point Theorem ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

The study of coupled systems of hybrid fractional differential equations requires the attention of scientists for the exploration of their different important aspects. Our aim in this paper is to study the existence and uniqueness of the solution for impulsive hybrid fractional differential equations. The novelty of this work is the study of a coupled system of impulsive hybrid fractional differential equations with initial and boundary hybrid conditions. We used the classical fixed-point theorems such as the Banach fixed-point theorem and Leray–Schauder alternative fixed-point theorem for existence results. We also give an example of the main results.

scholarly journals Solving partial fractional differential equations by using the Laguerre wavelet-Adomian method

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nasser Aghazadeh ◽  
Amir Mohammadi ◽  
Ghader Ahmadnezhad ◽  
Shahram Rezapour
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Adomian Decomposition Method ◽  
Nonlinear Method ◽  
Adomian Decomposition ◽  
Laguerre Wavelets ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

AbstractBy using a nonlinear method, we try to solve partial fractional differential equations. In this way, we construct the Laguerre wavelets operational matrix of fractional integration. The method is proposed by utilizing Laguerre wavelets in conjunction with the Adomian decomposition method. We present the procedure of implementation and convergence analysis for the method. This method is tested on fractional Fisher’s equation and the singular fractional Emden–Fowler equation. We compare the results produced by the present method with some well-known results.

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