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.

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.

Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives

2021 ◽  
Vol 24 (2) ◽  
pp. 483-508
Author(s):  
Mohammed D. Kassim ◽  
Nasser-eddine Tatar
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Logarithmic Function ◽  
Fractional Differential ◽  
Derivatives Of ◽  
L’Hôpital’S Rule

Abstract The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

Fractional calculus and Sinc methods

2011 ◽  
Vol 14 (4) ◽  
Author(s):  
Gerd Baumann ◽  
Frank Stenger
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Integral Equations ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Singular Integral Equations ◽  
Fractional Integrals ◽  
Sinc Methods ◽  
Weakly Singular Integral Equations ◽  
Fractional Differential

AbstractFractional integrals, fractional derivatives, fractional integral equations, and fractional differential equations are numerically solved by Sinc methods. Sinc methods are able to deal with singularities of the weakly singular integral equations of Riemann-Liouville and Caputo type. The convergence of the numerical method is numerically examined and shows exponential behavior. Different examples are used to demonstrate the effective derivation of numerical solutions for different types of fractional differential and integral equations, linear and non-linear ones. Equations of mixed ordinary and fractional derivatives, integro-differential equations are solved using Sinc methods. We demonstrate that the numerical calculation needed in fractional calculus can be gained with high accuracy using Sinc methods.

scholarly journals Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1675
Author(s):  
Nur Amirah Zabidi ◽  
Zanariah Abdul Majid ◽  
Adem Kilicman ◽  
Faranak Rabiei
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lagrange Interpolation ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Third Order ◽  
Fractional Differential ◽  
Derivatives Of

Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept of a third-order Adam–Bashforth numerical scheme by implementing Lagrange interpolation for fractional case, where the fractional derivatives are defined in the Caputo sense. Furthermore, the study includes a discussion on stability and convergence analysis of the method. Several numerical examples are also provided in order to validate the reliability and efficiency of the proposed method. The examples in this study cover solving linear and nonlinear fractional differential equations for the case of both single order as α∈(0,1) and higher order, α∈1,2, where α denotes the order of fractional derivatives of Dαy(t). The comparison in terms of accuracy between the proposed method and other existing methods demonstrate that the proposed method gives competitive performance as the existing methods.

scholarly journals On existence–uniqueness results for proportional fractional differential equations and incomplete gamma functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zaid Laadjal ◽  
Thabet Abdeljawad ◽  
Fahd Jarad
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Uniqueness Of Solutions ◽  
Gamma Functions ◽  
Uniqueness Results ◽  
Incomplete Gamma Functions ◽  
Fractional Differential ◽  
Derivatives Of

AbstractIn this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form $$ D^{\rho }= (1-\rho )+ \rho D, \quad \rho \in [0,1], $$ D ρ = ( 1 − ρ ) + ρ D , ρ ∈ [ 0 , 1 ] , where D is the ordinary differential operator.

scholarly journals Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Atimad Harir ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Higher Order ◽  
Conformable Derivatives ◽  
Fuzzy Solution ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, the generalized concept of conformable fractional derivatives of order q ∈ n , n + 1 for fuzzy functions is introduced. We presented the definition and proved properties and theorems of these derivatives. The fuzzy conformable fractional differential equations and the properties of the fuzzy solution are investigated, developed, and proved. Some examples are provided for both the new solutions.

scholarly journals On a system of Riemann–Liouville fractional differential equations with coupled nonlocal boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fixed Point Theory ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Point Theory ◽  
Fractional Integrals ◽  
Fractional Differential

AbstractWe investigate the existence of solutions for a system of Riemann–Liouville fractional differential equations with nonlinearities dependent on fractional integrals, subject to coupled nonlocal boundary conditions which contain various fractional derivatives and Riemann–Stieltjes integrals. In the proof of our main results, we use some theorems from the fixed point theory.

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