scholarly journals An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Ricardo Almeida ◽  
Delfim F. M. Torres
Keyword(s):  
Differential Equations ◽  
Approximation Method ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Variable Order ◽  
Fractional Operators ◽  
Expansion Formula ◽  
Approximation Formulas ◽  
Fractional Integrals And Derivatives ◽  
Derivatives Of

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.

Initialization of Fractional Differential Equations: Background and Theory

2007 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Companion Paper ◽  
Laplace Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Higher Order Derivative ◽  
Fractional Differential ◽  
Derivatives Of

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. This paper provides background on past work in the area and determines the Laplace transforms for initialized fractional integrals of any order and fractional derivatives of order less than one. A companion paper in this conference extends the theory to higher order derivative operators and provides application insight.

Initialization of Fractional Differential Equations: Theory and Application

2007 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Companion Paper ◽  
Laplace Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Derivatives Of

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. A companion paper in this conference determines the Laplace transforms for initialized fractional integrals of any order and fractional derivatives of order less than one. This paper extends the theory for the Laplace transform of the derivative to higher order and provides applications.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals Finding the fractional value of 2pi

2020 ◽  
Author(s):  
Anurag Vaidya
Keyword(s):  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Explorative Study ◽  
Detection Technology ◽  
Fractional Integrals And Derivatives ◽  
Definition Of ◽  
Rational Order ◽  
Derivatives Of ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Rational order integral and derivative of a myriad of functions—ln(x); e^(ax) and t^(ax)—are known. Nevertheless, the investigation focuses on rational order integrals and derivatives of sine and cosine as these functions follow a cyclic order and determiningwhether properties of sine and cosine extend to their fractional integrals and derivativescould expand current applications of sine and cosine, which are extensively in engineeringand economics. For example, Mehdi and Majid outline in, "Applications of FractionalCalculus" how fractional integrals and derivatives come handy while modeling ultrasonicwave propagation in human cancellous bones and bettering edge detection technology. Thus, this explorative study investigates the properties of fractional derivatives and fractional integrals of standard sine and cosine functions. THe study also looks at how to generalize the definition of the 2pi using fractional calculus.

An expansion formula for fractional derivatives of variable order

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Teodor Atanackovic ◽  
Marko Janev ◽  
Stevan Pilipovic ◽  
Dusan Zorica
Keyword(s):  
Stress Relaxation ◽  
Constitutive Equation ◽  
Relaxation Function ◽  
Fractional Derivatives ◽  
Variable Order ◽  
Elementary Functions ◽  
Expansion Formula ◽  
Derivatives Of

AbstractIn this work we extend our previous results and derive an expansion formula for fractional derivatives of variable order. The formula is used to determine fractional derivatives of variable order of two elementary functions. Also we propose a constitutive equation describing a solidifying material and determine the corresponding stress relaxation function.

scholarly journals General Fractional Integrals and Derivatives of Arbitrary Order

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 755
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Arbitrary Order ◽  
Fractional Integrals ◽  
Basic Properties ◽  
Suitable Generalization ◽  
Fractional Integrals And Derivatives ◽  
Fundamental Theorems ◽  
Derivatives Of

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

Initialization of Fractional-Order Operators and Fractional Differential Equations

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Laplace Transforms ◽  
Time Varying ◽  
Fractional Operators ◽  
Fractional Differential ◽  
Derivatives Of

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. The new transforms unify the initialization of systems of fractional and ordinary differential equations. The paper provides background on past work in the area and determines the Laplace transforms for the initialized fractional integral and fractional derivatives of any (real) order. An application provides insight and demonstrates the theory.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

scholarly journals On local fractional operators View of computational complexity: Diffusion and relaxation defined on cantor sets

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 755-767 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Zhi-Zhen Zhang ◽  
Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Computational Complexity ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Systems ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Fractional Partial Differential Equations ◽  
Comparative Results

This paper treats the description of non-differentiable dynamics occurring in complex systems governed by local fractional partial differential equations. The exact solutions of diffusion and relaxation equations with Mittag-Leffler and exponential decay defined on Cantor sets are calculated. Comparative results with other versions of the local fractional derivatives are discussed.

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.

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