scholarly journals A New Neumann Series Method for Solving a Family of Local Fractional Fredholm and Volterra Integral Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Xiao-Jing Ma ◽  
H. M. Srivastava ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Integral Operator ◽  
Integral Equations ◽  
Fractional Integral ◽  
Neumann Series ◽  
Volterra Integral Equations ◽  
Fractional Integral Operator ◽  
Series Method ◽  
Operator Type ◽  
Local Fractional Integral

We propose a new Neumann series method to solve a family of local fractional Fredholm and Volterra integral equations. The integral operator, which is used in our investigation, is of the local fractional integral operator type. Two illustrative examples show the accuracy and the reliability of the obtained results.

Local Fractional Fourier Series With Applications to Representations of Fractal Signals

2013 ◽  
Author(s):  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Fourier Series ◽  
Integral Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Series Representations ◽  
Local Fractional Integral ◽  
Fractal Signal

In this paper we discuss the local fractional Fourier series representations of fractal signals. Fractal signal processes are described within the local fractional integral operator. Four examples are presented in order to illustrate the developed technique.

scholarly journals On local fractional Volterra integral equations in fractal heat transfer

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 795-800 ◽  
Author(s):  
Zhong-Hua Wu ◽  
Amar Debbouche ◽  
Juan Guirao ◽  
Xiao-Jun Yang
Keyword(s):  
Heat Transfer ◽  
Integral Equations ◽  
Fractional Integral ◽  
Volterra Integral Equations ◽  
Fourier Law ◽  
Fractal Media ◽  
Local Fractional Integral ◽  
Linear And Nonlinear ◽  
Transfer Problems ◽  
Transfer Models

In the article, the fractal heat-transfer models are described by the local fractional integral equations. The local fractional linear and nonlinear Volterra integral equations are employed to present the heat transfer problems in fractal media. The local fractional integral equations are derived from the Fourier law in fractal media.

scholarly journals Integral equations involving generalized Mittag-Leffler function

2020 ◽  
Vol 72 (5) ◽  
Author(s):  
Rachana Desai ◽  
I. A. Salehbhai ◽  
A. K. Shukla
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Integral Equations ◽  
Fractional Integral ◽  
Necessary Conditions ◽  
Fractional Integral Operator

UDC 517.5 The paper deals with solving the integral equation with a generalized Mittag-Leffler function E α , β γ , q ( z ) that defines a kernel using a fractional integral operator. The existence of the solution is justified and necessary conditions on the integral equation admiting a solution are discussed. Also, the solution of the integral equation is derived.

A note on fractional integral operator associated with multiindex Mittag-Leffler functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1931-1939 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Fractional Integral Operator ◽  
Integration Formula ◽  
Special Cases

Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway type) involving the generalized multiindex Mittag-Leffler function E?,k[(?j,?j)m;z]. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

scholarly journals New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1753
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Omar Bazighifan ◽  
Georgia Irina Oros
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Convex Mappings ◽  
Special Cases ◽  
Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

scholarly journals FRACTIONAL INTEGRAL OPERATORS IN NONHOMOGENEOUS SPACES

2009 ◽  
Vol 80 (2) ◽  
pp. 324-334 ◽  
Author(s):  
H. GUNAWAN ◽  
Y. SAWANO ◽  
I. SIHWANINGRUM
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Type Inequality ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Fractional Integral Operator ◽  
Homogeneous Type ◽  
Fractional Integral Operators

AbstractWe discuss here the boundedness of the fractional integral operatorIαand its generalized version on generalized nonhomogeneous Morrey spaces. To prove the boundedness ofIα, we employ the boundedness of the so-called maximal fractional integral operatorIa,κ*. In addition, we prove an Olsen-type inequality, which is analogous to that in the case of homogeneous type.

scholarly journals From fractional order equations to integer order equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Vector Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Fractional Integral ◽  
State Of The Art ◽  
Constant Coefficients ◽  
Fractional Order Integral ◽  
Ordinary Integral

AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

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