scholarly journals On Caputo modification of Hadamard-type fractional derivative and fractional Taylor series

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Rashida Zafar ◽  
Mujeeb ur Rehman ◽  
Moniba Shams
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Taylor Series ◽  
General Framework ◽  
Taylor Expansion ◽  
Fractional Integral ◽  
Haar Wavelet ◽  
Wavelet Method ◽  
Fractional Differential Operator ◽  
Fractional Differential

Abstract In this paper a general framework is presented on some operational properties of Caputo modification of Hadamard-type fractional differential operator along with a new algorithm proposed for approximation of Hadamard-type fractional integral using Haar wavelet method. Moreover, a generalized Taylor expansion based on Caputo–Hadamard-type fractional differential operator is also established, and an example is presented for illustration.

scholarly journals Study of new class of q-fractional integral operator

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5713-5721
Author(s):  
M. Momenzadeh ◽  
N.I. Mahmudov
Keyword(s):  
Differential Operator ◽  
Integral Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Integral Approach ◽  
Cauchy Integral ◽  
New Class ◽  
Fractional Differential Operator ◽  
Fractional Differential

In this paper, we study on the new class of q-fractional integral operator. In the aid of iterated Cauchy integral approach to fractional integral operator, we applied tp f(t) in these integrals and a new class of q-fractional integral operator with parameter p, is introduced. Recently, the q-analogue of fractional differential integral operator is studied and all of the operators defined in these studies are q-analogue of Riemann fractional differential operator. We show that our new class of operator generalize all the operators in use, and additionally, it can cover the q-analogue of Hadamard fractional differential operator, as well. Some properties of this operator are investigated.

scholarly journals Second Hankel Determinant for a Class of Analytic Functions Defined by Fractional Derivative

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
A. K. Mishra ◽  
P. Gochhayat
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Analytic Functions ◽  
Hadamard Product ◽  
Integral Transform ◽  
Hankel Determinant ◽  
Nonlinear Functional ◽  
Fractional Differential Operator ◽  
Second Hankel Determinant ◽  
Fractional Differential

By making use of the fractional differential operatorΩzλdue to Owa and Srivastava, a class of analytic functionsℛλ(α,ρ)    (0≤ρ≤1,  0≤λ<1,    |α|<π/2)is introduced. The sharp bound for the nonlinear functional|a2a4−a32|is found. Several basic properties such as inclusion, subordination, integral transform, Hadamard product are also studied.

scholarly journals -Dimensional Fractional Lagrange's Inversion Theorem

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
F. A. Abd El-Salam
Keyword(s):  
Differential Operator ◽  
Fractional Calculus ◽  
Unknown Function ◽  
Taylor Expansion ◽  
Higher Dimensions ◽  
Lagrange Inversion ◽  
Inversion Theorem ◽  
Fractional Differential Operator ◽  
Independent Variable ◽  
Fractional Differential

Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized. For extending the treatment in higher dimensions, some relevant vectors and tensors definitions and notations are presented. A fractional Taylor expansion of a function of -dimensional polyadics is derived. A fractional -dimensional Lagrange inversion theorem is proved.

On the kinetics of Hadamard-type fractional differential systems

2020 ◽  
Vol 23 (2) ◽  
pp. 553-570 ◽  
Author(s):  
Li Ma
Keyword(s):  
Differential Operator ◽  
Numerical Simulations ◽  
Type Inequality ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Kinetics Of ◽  
Theoretical Results

AbstractThis paper is devoted to the investigation of the kinetics of Hadamard-type fractional differential systems (HTFDSs) in two aspects. On one hand, the nonexistence of non-trivial periodic solutions for general HTFDSs, which are considered in some functional spaces, is proved and the corresponding eigenfunction of Hadamard-type fractional differential operator is also discussed. On the other hand, by the generalized Gronwall-type inequality, we estimate the bound of the Lyapunov exponents for HTFDSs. In addition, numerical simulations are addressed to verify the obtained theoretical results.

scholarly journals On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. F. Imaga ◽  
S. A. Iyase
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Half Line ◽  
At Resonance ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Problem At Resonance

AbstractIn this work, we consider the solvability of a fractional-order p-Laplacian boundary value problem on the half-line where the fractional differential operator is nonlinear and has a kernel dimension equal to two. Due to the nonlinearity of the fractional differential operator, the Ge and Ren extension of Mawhin’s coincidence degree theory is applied to obtain existence results for the boundary value problem at resonance. Two examples are used to validate the established results.

scholarly journals On Multivalent Analytic Functions Considered by a Multi-Arbitrary Differential Operator in a Complex Domain

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 315
Author(s):  
Najla M. Alarifi ◽  
Rabha W. Ibrahim
Keyword(s):  
Differential Operator ◽  
Analytic Functions ◽  
Integral Operators ◽  
Upper And Lower Bounds ◽  
Fractional Operator ◽  
Convolution Operators ◽  
Linear Convolution ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Special Case

(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have been undertaken. (2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator. This class is defined by utilizing the subordination and superordination theory. (3) Results: We illustrate different fractional inequalities of starlike and convex formulas. Moreover, we discuss the main conditions to obtain sandwich inequalities involving the fractional operator. (4) Conclusion: We indicate that the suggested class is a generalization of recent works and can be applied to discuss the upper and lower bounds of a special case of fractional differential equations.

Symmetry analysis of the generalized space and time fractional Korteweg–de Vries equation

2021 ◽  
Author(s):  
Jian-Gen Liu ◽  
Xiao-Jun Yang ◽  
Yi-Ying Feng ◽  
Lu-Lu Geng
Keyword(s):  
Fractional Derivative ◽  
Kdv Equation ◽  
Symmetry Analysis ◽  
Analysis Method ◽  
Space And Time ◽  
Fractional Differential Operator ◽  
Time Differential ◽  
De Vries ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

In this paper, we studied the generalized space and time fractional Korteweg–de Vries (KdV) equation in the sense of the Riemann–Liouville fractional derivative. Initially, the symmetry of this considered equation through the symmetry analysis method was obtained. Next, a one-parameter Lie group of point transformation was yielded. Then, this considered fractional model can be translated into an ordinary differential equation of fractional order via the Erdélyi–Kober fractional differential operator and the Erdélyi–Kober fractional integral operator. Finally, with the help of the nonlinear self-adjointness, conservation laws of the generalized space and time fractional KdV equation can be found. This approach can provide us with a new scheme for studying space and time differential equations of fractional derivative.

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