scholarly journals Generalized fractional derivatives and Laplace transform

2020 ◽  
Vol 13 (3) ◽  
pp. 709-722 ◽  
Author(s):  
Fahd Jarad ◽  
◽  
Thabet Abdeljawad ◽  
Keyword(s):  
Laplace Transform ◽  
Fractional Derivatives ◽  
Generalized Fractional Derivatives

scholarly journals Reachability and Observability of Positive Linear Electrical Circuits Systems Described by Generalized Fractional Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2856
Author(s):  
Tong Yuan ◽  
Hongli Yang ◽  
Ivan Ganchev Ivanov
Keyword(s):  
Laplace Transform ◽  
Fractional Derivatives ◽  
Electrical Circuits ◽  
Preliminary Results ◽  
Generalized Fractional Derivatives

Positive linear electrical circuits systems described by generalized fractional derivatives are studied in this paper. We mainly focus on the reachability and observability of linear electrical circuits systems. Firstly, generalized fractional derivatives and ρ-Laplace transform of f is presented and some preliminary results are provided. Secondly, the positivity of linear electrical circuits systems described by generalized fractional derivatives is investigated and conditions for checking positivity of the systems are derived. Thirdly, reachability and observability of the generalized fractional derivatives systems are studied, in which the ρ-Laplace transform of a Mittag-Leffler function plays an important role. At the end of the paper, illustrative electrical circuits systems are presented, and conclusions of the paper are presented.

scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

scholarly journals Opial typeLp-inequalities for fractional derivatives

2002 ◽  
Vol 31 (2) ◽  
pp. 85-95 ◽  
Author(s):  
G. A. Anastassiou ◽  
J. J. Koliha ◽  
J. Pecaric
Keyword(s):  
Fractional Derivatives ◽  
Continuous Functions ◽  
Taylor’S Formula ◽  
Integrable Functions ◽  
Generalized Fractional Derivatives ◽  
Taylor's Formula

This paper presents a class ofLp-type Opial inequalities for generalized fractional derivatives for integrable functions based on the results obtained earlier by the first author for continuous functions (1998). The novelty of our approach is the use of the index law for fractional derivatives in lieu of Taylor's formula, which enables us to relax restrictions on the orders of fractional derivatives.

POTENTIALS OF ARBITRARY FORCES WITH FRACTIONAL DERIVATIVES

2004 ◽  
Vol 19 (17n18) ◽  
pp. 3083-3092 ◽  
Author(s):  
EQAB M. RABEI ◽  
TAREQ S. ALHALHOLY ◽  
AKRAM ROUSAN
Keyword(s):  
Laplace Transform ◽  
General Formula ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Hamiltonian Mechanics ◽  
Dissipative Effects ◽  
Special Cases ◽  
Exact Agreement ◽  
The Laplace Transform ◽  
Lagrangian And Hamiltonian Mechanics

The Laplace transform of fractional integrals and fractional derivatives is used to develop a general formula for determining the potentials of arbitrary forces: conservative and nonconservative in order to introduce dissipative effects (such as friction) into Lagrangian and Hamiltonian mechanics. The results are found to be in exact agreement with Riewe's results of special cases. Illustrative examples are given.

Couette Flow of a Generalized Oldroyd-B Fluid due to Constantly Accelerated Shear Stress Coupled with the Pressure Gradient

2011 ◽  
Vol 354-355 ◽  
pp. 179-182
Author(s):  
Chun Rui Li ◽  
Lian Cun Zheng ◽  
Xin Xin Zhang ◽  
Jia Jia Niu
Keyword(s):  
Shear Stress ◽  
Laplace Transform ◽  
Pressure Gradient ◽  
Exact Solutions ◽  
Couette Flow ◽  
Fractional Derivatives ◽  
Hankel Transform ◽  
Time Dependent ◽  
Infinite Cylinder ◽  
Finite Hankel Transform

This paper presented an analysis for the couette flow of a generalized Oldroyd-B fluid within an infinite cylinder subject to a time-dependent shear stress with the influence of the internal constantly decelerated pressure gradient. The exact solutions are established by means of the combine of the sequential fractional derivatives Laplace transform and finite Hankel transform and presented by integral and series form in terms of the Mittag-Leffler function. Moreover, the effects of various parameters are analyzed in detail by graphical illustrations.

Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Myong-Ha Kim ◽  
Guk-Chol Ri ◽  
Hyong-Chol O
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
Operational Method ◽  
Initial Value ◽  
Constant Coefficients ◽  
Generalized Fractional Derivatives ◽  
Fractional Differential

AbstractThis paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski’s type. We prove that the initial value problem has the solution if and only if some initial values are zero.

scholarly journals ON THE WEIGHTED FRACTIONAL OPERATORS OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040011 ◽  
Author(s):  
F. JARAD ◽  
T. ABDELJAWAD ◽  
K. SHAH
Keyword(s):  
Laplace Transform ◽  
Fractional Derivatives ◽  
Fractional Operators ◽  
Weighted Integrals

The primary goal of this study is to define the weighted fractional operators on some spaces. We first prove that the weighted integrals are bounded in certain spaces. Afterwards, we discuss the weighted fractional derivatives defined on absolute continuous-like spaces. At the end, we present a modified Laplace transform that can be applied perfectly to such operators.

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