Existence of mild solution of Atangana–Baleanu fractional differential equations with non-instantaneous impulses and with non-local conditions

2020 ◽  
Vol 132 ◽  
pp. 109551 ◽  
Author(s):  
Ashish Kumar ◽  
Dwijendra N. Pandey
Keyword(s):  
Differential Equations ◽  
Mild Solution ◽  
Fractional Differential Equations ◽  
Local Conditions ◽  
Non Local ◽  
Fractional Differential

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.

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.

scholarly journals Results for Mild solution of fractional coupled hybrid boundary value problems

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Dumitru Baleanu ◽  
Hossein Jafari ◽  
Hasib Khan ◽  
Sarah Jane Johnston
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Mild Solution ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Coupled System ◽  
Banach Contraction ◽  
Fractional Differential

AbstractThe study of coupled system of hybrid fractional differential equations (HFDEs) needs the attention of scientists for the exploration of its different important aspects. Our aim in this paper is to study the existence and uniqueness of mild solution (EUMS) of a coupled system of HFDEs. The novelty of this work is the study of a coupled system of fractional order hybrid boundary value problems (HBVP) with n initial and boundary hybrid conditions. For this purpose, we are utilizing some classical results, Leray–Schauder Alternative (LSA) and Banach Contraction Principle (BCP). Some examples are given for the illustration of applications of our results.

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.

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