scholarly journals Comparison principles of fractional differential equations with non-local derivative and their applications

2021 ◽  
Vol 6 (2) ◽  
pp. 1443-1451
Author(s):  
Mohammed Al-Refai ◽  
◽  
Dumitru Baleanu ◽  
◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Comparison Principles ◽  
Non Local ◽  
Fractional Differential ◽  
Local Derivative

scholarly journals On Λ-Fractional Viscoelastic Models

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 22
Author(s):  
Anastassios K. Lazopoulos ◽  
Dimitrios Karaoulanis
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Present Article ◽  
Fractional Differential Equations ◽  
Zener Model ◽  
Viscoelastic Models ◽  
Non Local ◽  
Fractional Differential ◽  
Local Derivative

Λ-Fractional Derivative (Λ-FD) is a new groundbreaking Fractional Derivative (FD) introduced recently in mechanics. This derivative, along with Λ-Transform (Λ-T), provides a reliable alternative to fractional differential equations’ current solving. To put it straightforwardly, Λ-Fractional Derivative might be the only authentic non-local derivative that exists. In the present article, Λ-Fractional Derivative is used to describe the phenomenon of viscoelasticity, while the whole methodology is demonstrated meticulously. The fractional viscoelastic Zener model is studied, for relaxation as well as for creep. Interesting results are extracted and compared to other methodologies showing the value of the pre-mentioned method.

scholarly journals Monotone Iterative Technique for Conformable Fractional Differential Equations with Deviating Arguments

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Huiling Chen ◽  
Shuman Meng ◽  
Yujun Cui
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Deviating Arguments ◽  
Comparison Principles ◽  
Monotone Iterative ◽  
Fractional Differential

This paper is concerned with the existence of extremal solutions for periodic boundary value problems for conformable fractional differential equations with deviating arguments. We first build two comparison principles for the corresponding linear equation with deviating arguments. With the help of new comparison principles, some sufficient conditions for the existence of extremal solutions are established by combining the method of lower and upper solutions and the monotone iterative technique. As an application, an example is presented to enrich the main results of this article.

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.

Some comparison principles for fractional differential equations and systems

2019 ◽  
Vol 44 (3) ◽  
pp. 2405-2415
Author(s):  
Amal Alshabanat ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Comparison Principles ◽  
Fractional Differential

scholarly journals TWO COLLOCATION TYPE METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-LOCAL BOUNDARY CONDITIONS

2017 ◽  
Vol 22 (5) ◽  
pp. 654-670 ◽  
Author(s):  
Mikk Vikerpuur
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Differential Operators ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Spline Collocation ◽  
Local Boundary ◽  
Regularity Properties ◽  
Non Local ◽  
Fractional Differential

A class of non-local boundary value problems for linear fractional differential equations with Caputo-type differential operators is considered. By using integral equation reformulation of the boundary value problem, we study the existence and smoothness of the exact solution. Using the obtained regularity properties and spline collocation techniques, we construct two numerical methods (Method 1 and Method 2) for finding approximate solutions. By choosing suitable graded grids, we derive optimal global convergence estimates and obtain some super-convergence results for Method 2 by requiring additional assumptions on equation and collocation parameters. Some numerical illustrations for verification of theoretical results is also presented.

COMPARISON PRINCIPLES FOR HADAMARD-TYPE FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850056 ◽  
Author(s):  
CHUNTAO YIN ◽  
LI MA ◽  
CHANGPIN LI
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Fractional Derivatives ◽  
Comparison Principles ◽  
Right Hand ◽  
Fractional Differential ◽  
The Right ◽  
Continuous Dependence Of Solutions

The aim of this paper is to establish the comparison principles for differential equations involving Hadamard-type fractional derivatives. First, the continuous dependence of solutions on the right-hand side functions of Hadamard-type fractional differential equations (HTFDEs) is proposed. Then, we state and prove the first and second comparison principles for HTFDEs, respectively. The corresponding examples are provided as well.

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