scholarly journals A New Attractive Method in Solving Families of Fractional Differential Equations by a New Transform

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3039
Author(s):  
Ahmad Qazza ◽  
Aliaa Burqan ◽  
Rania Saadeh
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Integral Transform ◽  
Series Solutions ◽  
Numerical Examples ◽  
Fractional Differential ◽  
Expansion Coefficients

In this paper, we use the ARA transform to solve families of fractional differential equations. New formulas about the ARA transform are presented and implemented in solving some applications. New results related to the ARA integral transform of the Riemann-Liouville fractional integral and the Caputo fractional derivative are obtained and the last one is implemented to create series solutions for the target equations. The procedure proposed in this article is mainly based on some theorems of particular solutions and the expansion coefficients of binomial series. In order to achieve the accuracy and simplicity of the new method, some numerical examples are considered and solved. We obtain the solutions of some families of fractional differential equations in a series form and we show how these solutions lead to some important results that include generalizations of some classical methods.

scholarly journals Improvement on Conformable Fractional Derivative and Its Applications in Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Feng Gao ◽  
Chunmei Chi
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Physical Meaning ◽  
Fractional Integral ◽  
Conformable Fractional Derivative ◽  
Conformable Derivative ◽  
Fractional Differential ◽  
Definition Of

In this paper, we made improvement on the conformable fractional derivative. Compared to the original one, the improved conformable fractional derivative can be a better replacement of the classical Riemann-Liouville and Caputo fractional derivative in terms of physical meaning. We also gave the definition of the corresponding fractional integral and illustrated the applications of the improved conformable derivative to fractional differential equations by some examples.

scholarly journals Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Operational Matrix ◽  
Classical Solutions ◽  
Operational Matrices ◽  
Fractional Differential

We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs). In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD), and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI). The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.

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.

A Robust Computational Algorithm of Homotopy Asymptotic Method for Solving Systems of Fractional Differential Equations

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Zaid Odibat ◽  
Sunil Kumar
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Asymptotic Method ◽  
Computational Algorithm ◽  
Series Solutions ◽  
Numerical Examples ◽  
Convergence Of Series ◽  
New Ideas ◽  
Fractional Differential ◽  
Optimal Construction

In this paper, we present new ideas for the implementation of homotopy asymptotic method (HAM) to solve systems of nonlinear fractional differential equations (FDEs). An effective computational algorithm, which is based on Taylor series approximations of the nonlinear equations, is introduced to accelerate the convergence of series solutions. The proposed algorithm suggests a new optimal construction of the homotopy that reduces the computational complexity and improves the performance of the method. Some numerical examples are tested to validate and illustrate the efficiency of the proposed algorithm. The obtained results demonstrate the improvement of the accuracy by the new algorithm.

scholarly journals On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms

2020 ◽  
Vol 40 (2) ◽  
pp. 227-239
Author(s):  
John R. Graef ◽  
Said R. Grace ◽  
Ercan Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Nonoscillatory Solutions ◽  
Fractional Differential

This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

scholarly journals On a Fractional Operator Combining Proportional and Classical Differintegrals

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 360 ◽  
Author(s):  
Dumitru Baleanu ◽  
Arran Fernandez ◽  
Ali Akgül
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Inverse Operator ◽  
Fractional Operator ◽  
Special Cases ◽  
Non Local ◽  
Fractional Differential

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

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