A new Definition of Fractional Derivative and Fractional Integral

In this paper, the fuzzy fractional evolution equations of order q (FFEE) with fuzzy Caputo fractional derivative are introduced. We study the existence and uniqueness of mild solutions for FFEE under some conditions. Also, we generalize the definition of the fuzzy fractional integral and derivative order q. The fuzzy Laplace transform is presented and proved. The solvability of the problem (FFEE) and the properties of the fuzzy solution operator and its generator are investigated and developed.

Download Full-text

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

Journal of Function Spaces ◽

10.1155/2020/5852414 ◽

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.

Download Full-text

Hadamard’s inequality and its extensions for conformable fractional integrals of any order α>0

Creative Mathematics and Informatics ◽

10.37193/cmi.2018.02.12 ◽

2018 ◽

Vol 27 (2) ◽

pp. 197-206

Author(s):

ERHAN SET ◽

◽

AHMET OCAK AKDEMIR ◽

I. MUMCU ◽

◽

...

Keyword(s):

Fractional Calculus ◽

Fractional Derivative ◽

Fractional Integral ◽

Fractional Integrals ◽

Hadamard’S Inequality ◽

Conformable Fractional Integral ◽

Definition Of ◽

Appl Math

Recently the authors Abdeljawad [Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66] and Khalil et al. [Khalil, R., Horani, M. Al., Yousef, A. and Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70] introduced a new and simple well-behaved concept of fractional integral called conformable fractional integral. In this article, we establish Hermite-Hadamard’s inequalities for conformable fractional integral. We also give extensions of Hermite-Hadamard type inequalities for conformable fractional integrals.

Download Full-text

A remark on local fractional calculus and ordinary derivatives

Open Mathematics ◽

10.1515/math-2016-0104 ◽

2016 ◽

Vol 14 (1) ◽

pp. 1122-1124 ◽

Cited By ~ 12

Author(s):

Ricardo Almeida ◽

Małgorzata Guzowska ◽

Tatiana Odzijewicz

Keyword(s):

Fractional Calculus ◽

Fractional Derivative ◽

Short Note ◽

General Definition ◽

Local Fractional Derivative ◽

Fundamental Properties ◽

Local Fractional Calculus ◽

Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

Download Full-text

Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽

10.3390/axioms10020047 ◽

2021 ◽

Vol 10 (2) ◽

pp. 47

Author(s):

A. Torres-Hernandez ◽

F. Brambila-Paz ◽

U. Iturrarán-Viveros ◽

R. Caballero-Cruz

Keyword(s):

Fractional Derivative ◽

Iterative Methods ◽

Fractional Integral ◽

Order Of Convergence ◽

Fractional Operators ◽

Speed Of Convergence ◽

Newton Raphson ◽

Raphson Method

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

Download Full-text

On Caputo modification of Hadamard-type fractional derivative and fractional Taylor series

Advances in Difference Equations ◽

10.1186/s13662-020-02658-1 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Rashida Zafar ◽

Mujeeb ur Rehman ◽

Moniba Shams

Keyword(s):

Differential Operator ◽

Fractional Derivative ◽

Taylor Series ◽

General Framework ◽

Taylor Expansion ◽

Fractional Integral ◽

Haar Wavelet ◽

Wavelet Method ◽

Fractional Differential Operator ◽

Fractional Differential

Abstract In this paper a general framework is presented on some operational properties of Caputo modification of Hadamard-type fractional differential operator along with a new algorithm proposed for approximation of Hadamard-type fractional integral using Haar wavelet method. Moreover, a generalized Taylor expansion based on Caputo–Hadamard-type fractional differential operator is also established, and an example is presented for illustration.

Download Full-text

Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

Axioms ◽

10.3390/axioms10030130 ◽

2021 ◽

Vol 10 (3) ◽

pp. 130

Author(s):

Suphawat Asawasamrit ◽

Yasintorn Thadang ◽

Sotiris K. Ntouyas ◽

Jessada Tariboon

Keyword(s):

Boundary Conditions ◽

Fractional Derivative ◽

Boundary Value Problems ◽

Caputo Fractional Derivative ◽

Fractional Integral ◽

Boundary Value ◽

Integral Boundary Conditions ◽

Integral Boundary ◽

Uniqueness Results ◽

Nonlinear Alternative

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence of a unique solution is obtained via Banach’s contraction mapping principle, while an existence result is established by using Leray–Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

Download Full-text

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2021-0043 ◽

2021 ◽

Vol 24 (4) ◽

pp. 1003-1014

Author(s):

J. A. Tenreiro Machado

Keyword(s):

Fractional Calculus ◽

Fractional Derivative ◽

Fractional Order ◽

Restitution Coefficient ◽

Classical Physics ◽

Bouncing Ball ◽

Mechanical Experiment ◽

Fractional Models ◽

Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

Download Full-text

Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu

Numerical Methods for Partial Differential Equations ◽

10.1002/num.22195 ◽

2017 ◽

Vol 34 (5) ◽

pp. 1502-1523 ◽

Cited By ~ 109

Author(s):

Abdon Atangana ◽

J. F. Gómez-Aguilar

Keyword(s):

Fractional Derivative ◽

Numerical Approximation ◽

Definition Of

Download Full-text

Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative

Mathematics ◽

10.3390/math7070630 ◽

2019 ◽

Vol 7 (7) ◽

pp. 630

Author(s):

Dandan Yang ◽

Chuanzhi Bai

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Differential Inclusions ◽

Existence Of Solutions ◽

Sufficient Conditions ◽

Fractional Differential Inclusions ◽

Nonlinear Alternative ◽

Contractive Map ◽

Fractional Differential ◽

Definition Of

In this paper, we investigate the existence of solutions for a class of anti-periodic fractional differential inclusions with ψ -Riesz-Caputo fractional derivative. A new definition of ψ -Riesz-Caputo fractional derivative of order α is proposed. By means of Contractive map theorem and nonlinear alternative for Kakutani maps, sufficient conditions for the existence of solutions to the fractional differential inclusions are given. We present two examples to illustrate our main results.

Download Full-text