scholarly journals A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 747 ◽  
Author(s):  
Jehad Alzabut ◽  
Weerawat Sudsutad ◽  
Zeynep Kayar ◽  
Hamid Baghani
Keyword(s):  
Initial Data ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Initial Value ◽  
Weighted Function ◽  
Fractional Derivatives And Integrals ◽  
Right Hand ◽  
The Right

New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall–Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side.

scholarly journals A verified method for solving piecewise smooth initial value problems

2013 ◽  
Vol 23 (4) ◽  
pp. 731-747 ◽  
Author(s):  
Ekaterina Auer ◽  
Stefan Kiel ◽  
Andreas Rauh
Keyword(s):  
Initial Value Problems ◽  
System Model ◽  
Future Research ◽  
Initial Value ◽  
Smooth Functions ◽  
Solution Processes ◽  
Right Hand ◽  
Future Research Directions ◽  
The Right ◽  
Derivatives Of

Abstract In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

scholarly journals Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
H. M. Srivastava ◽  
Alireza Khalili Golmankhaneh ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Coupling Method ◽  
Initial Value ◽  
Local Fractional Calculus ◽  
Sumudu Transform

Local fractional derivatives were investigated intensively during the last few years. The coupling method of Sumudu transform and local fractional calculus (called as the local fractional Sumudu transform) was suggested in this paper. The presented method is applied to find the nondifferentiable analytical solutions for initial value problems with local fractional derivative. The obtained results are given to show the advantages.

scholarly journals Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2023
Author(s):  
Christopher Nicholas Angstmann ◽  
Byron Alexander Jacobs ◽  
Bruce Ian Henry ◽  
Zhuang Xu
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Integral Transform ◽  
Initial Value ◽  
Intrinsic Nature ◽  
Recent Interest ◽  
Special Cases ◽  
Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals Resolvent of nonautonomous linear delay functional differential equations

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Joël Blot ◽  
Mamadou I. Koné
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Differentiable Function ◽  
Functional Differential Equations ◽  
Initial Value ◽  
Complete Proof ◽  
Right Hand ◽  
Linear Delay ◽  
Functional Differential ◽  
The Right

AbstractThe aim of this paper is to give a complete proof of the formula for the resolvent of a nonautonomous linear delay functional differential equations given in the book of Hale and Verduyn Lunel [9] under the assumption alone of the continuity of the right-hand side with respect to the time,when the notion of solution is a differentiable function at each point, which satisfies the equation at each point, and when the initial value is a continuous function.

Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives

2020 ◽  
Vol 23 (1) ◽  
pp. 103-125 ◽  
Author(s):  
Latif A-M. Hanna ◽  
Maryam Al-Kandari ◽  
Yuri Luchko
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Operational Method ◽  
Initial Value ◽  
Right Hand ◽  
Fractional Differential ◽  
Left And Right ◽  
Fractional Integrals And Derivatives

AbstractIn this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the four-parameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

scholarly journals Desiderata for Fractional Derivatives and Integrals

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 149 ◽  
Author(s):  
Rudolf Hilfer ◽  
Yuri Luchko
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Special Issue ◽  
Fractional Derivatives And Integrals

The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria.

scholarly journals Response of Fractionally Damped Beams with General Boundary Conditions Subjected to Moving Loads

2012 ◽  
Vol 19 (3) ◽  
pp. 333-347 ◽  
Author(s):  
R. Abu-Mallouh ◽  
I. Abu-Alshaikh ◽  
H.S. Zibdeh ◽  
Khaled Ramadan
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Damping Characteristics ◽  
The Laplace Transform ◽  
The Right

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.

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