Fractional Laplace transform and fractional calculus

2017 ◽  
Vol 12 (20) ◽  
pp. 991-1000
Author(s):  
Gustavo D. Medina ◽  
Nelson R. Ojeda ◽  
Jose H. Pereira ◽  
Luis G. Romero
Keyword(s):  
Fractional Calculus ◽  
Laplace Transform

scholarly journals Further Developments of Bessel Functions via Conformable Calculus with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Mahmoud Abul-Ez ◽  
Mohra Zayed ◽  
Ali Youssef
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Bessel Functions ◽  
Kinetic Equations ◽  
Test Problems ◽  
Physical Problems ◽  
Starting Point ◽  
Fractional Differential ◽  
Fractional Kinetic Equations

The theory of Bessel functions is a rich subject due to its essential role in providing solutions for differential equations associated with many applications. As fractional calculus has become an efficient and successful tool for analyzing various mathematical and physical problems, the so-called fractional Bessel functions were introduced and studied from different viewpoints. This paper is primarily devoted to the study of developing two aspects. The starting point is to present a fractional Laplace transform via conformable fractional-order Bessel functions (CFBFs). We establish several important formulas of the fractional Laplace Integral operator acting on the CFBFs of the first kind. With this in hand, we discuss the solutions of a generalized class of fractional kinetic equations associated with the CFBFs in view of our proposed fractional Laplace transform. Next, we derive an orthogonality relation of the CFBFs, which enables us to study an expansion of any analytic functions by means of CFBFs and to propose truncated CFBFs. A new approximate formula of conformable fractional derivative based on CFBFs is provided. Furthermore, we describe a useful scheme involving the collocation method to solve some conformable fractional linear (nonlinear) multiorder differential equations. Accordingly, several practical test problems are treated to illustrate the validity and utility of the proposed techniques and examine their approximate and exact solutions. The obtained solutions of some fractional differential equations improve the analog ones provided by various authors using different techniques. The provided algorithm may be beneficial to enrich the Bessel function theory via fractional calculus.

scholarly journals Integral representations of Mittag-Leffler function on the positive real axis

2019 ◽  
Vol 20 (2) ◽  
pp. 217
Author(s):  
Eliana Contharteze Grigoletto ◽  
Edmundo Capelas Oliveira ◽  
Rubens Figueiredo Camargo
Keyword(s):  
Fractional Calculus ◽  
Laplace Transform ◽  
Real Axis ◽  
Integral Representations ◽  
Positive Real Axis ◽  
Inverse Laplace Transform ◽  
Positive Real

The Mittag-Leffler functions appear in many problems associated with fractional calculus. In this paper, we use the methodology for evaluation of the inverse Laplace transform, proposed by M. N. Berberan-Santos, to show that the three-parameter Mittag-Leffler function has similar integral representations on the positive real axis. Some of the integrals are also presented.

scholarly journals Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2117
Author(s):  
Tohru Morita ◽  
Ken-ichi Sato
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Nonstandard Analysis ◽  
Distribution Theory ◽  
Nonregular Distributions

For Euler’s differential equation of order n, a theorem is presented to give n solutions, by modifying a theorem given in a recent paper of the present authors in J. Adv. Math. Comput. Sci. 2018; 28(3): 1–15, and then the corresponding theorem in distribution theory is given. The latter theorem is compared with recent studies on Euler’s differential equation in distribution theory. A supplementary argument is provided on the solutions expressed by nonregular distributions, on the basis of nonstandard analysis and Laplace transform.

On the generating function e xt+yϕ(t) and its fractional calculus

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Alireza Ansari ◽  
Amirhossein Sheikhani ◽  
Sohrab Kordrostami
Keyword(s):  
Fractional Calculus ◽  
Laplace Transform ◽  
Generating Function ◽  
Mellin Transform ◽  
Complex Function ◽  
Inverse Laplace Transform ◽  
Explicit Solutions ◽  
Associated Functions ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

AbstractIn this article, we derive the coefficient set {H m(x,y)}m=1∞ using the generating function ext+yϕ(t). When the complex function ϕ(t) is entire, using the inverse Mellin transform, and when ϕ(t) has singular points, using the inverse Laplace transform, the coefficient set is obtained. Also, bi-orthogonality of this set with its associated functions and its applications in the explicit solutions of partial fractional differential equations is discussed.

scholarly journals Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

2021 ◽  
Vol 5 (2) ◽  
pp. 43
Author(s):  
Gerd Baumann
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Laplace Transforms ◽  
Inverse Laplace Transform ◽  
Exact Methods ◽  
Sinc Methods ◽  
Fractional Differential

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

Axial MHD Flow of a Generalized Oldroyd-B Fluid Due to Two Oscillating Cylinders

2011 ◽  
Vol 354-355 ◽  
pp. 83-86
Author(s):  
Ya Qing Liu ◽  
Lian Cun Zheng ◽  
Jun Tie
Keyword(s):  
Shear Stress ◽  
Velocity Field ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fluid Velocity ◽  
Mhd Flow ◽  
Hankel Transform ◽  
Constitutive Relationship ◽  
Relationship Of ◽  
Fluid Velocity Field

Axial magnetohydrodynamic (MHD) flows for Oldroyd-B fluid are investigated between two cylinders. The motion of the fluid is produced by the two oscillating cylinders. The fractional calculus approach is introduced to establish the constitutive relationship of a viscolastic fluid. Velocity field and shear stress of the motion are determined in terms of Bessel function and generalized Mittag-Leffler function by using Laplace transform and Hankel transform. The influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn.

scholarly journals Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and their Approximation for Fractional Calculus

2021 ◽  
Author(s):  
Gerd Baumann
Keyword(s):  
Fractional Calculus ◽  
Laplace Transform ◽  
Numerical Approximation ◽  
Laplace Transforms ◽  
Inverse Laplace Transform ◽  
Exact Methods ◽  
Sinc Methods ◽  
Fractional Differential

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approxi- mation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

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