A REVIEW PAPER ON FRACTIONAL CALCULUS WITH THEIR APPLICATIONS

Review article

Vol. 8 Issue 1 Page No. 1-7

Dr. Govind Sharma*, Himani Aggrawal

School of Applied Sciences, Suresh Gyan Vihar University, Jaipur, India

*Corresponding Author’s email address – Govind.sharma@mygyanvihar.com

Paper Received – 22/10/2021                Paper Published – 10/02/2022

Keywords

Fractional Calculus

Operators

Differential Operator

Application

Abstract

The main object of this paper is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional “differ integral” equations and used several important definitions for fractional-order derivatives, including: the Riemann-Liouville, the Gr¨unwald-Letnikov, the Liouville-Caputo, the Caputo-Fabrizio and the Atangana-Baleanu fractional-order derivatives and their applications are also reviewed in the paper.

 

Introduction

The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. Fractional derivatives, fractional integrals and their properties are the subject of study in the field of fractional calculus.

A list of mathematicians, who have provided important contributions up to the middle of 20th century, includes Liouville, Riemann, Holmgren, Abel, Fourier, Weyl, Davis, Zygmund, Erdelyi, Kober, Hadamard, Hardy and Littlewood, Grunwald, Letnikov, Riesz. A lot of papers involving the functions of one and more variables have been introduced by many authors Agarwal P. & Purohit S.D. ([1]), Chaurasia and Srivastava ([10]), Chaurasia V. B. L.& Singh Y ([11]), Pandey N. & Khan R. [33], Yashwant Singh & Harmendra Kumar Mandia [42] etc. The well-known books by H.M. Srivastava, K.C. Gupta & S.P. Goyal ([40]), Oldham & Spanier ([25]), Samko, Miller & Ross ([20]), Kilbas & Saigo ([30]) and Nishimoto ([22]) on fractional calculus give a vivid account of its development and use in diverse fields of science and technology.

In the theory of integral equations and problems of mathematical physics, many research works namely Sunil Kumar Sharma & Ashok Singh Shekhawat [47], Hagos Tadesse, D. L. Suthar [46], Goyal, S.P. and Bhagtani, Manita [24], D. L. Suthar, Mitku Andualem, and Belete Debalkie [20], Samko [38], and several others proposed lot of generalizations of fractional integral operators from time to time.

In 1847, Riemann had arrived at expression for fractional integration as-

 ,             x > 0

This paper was published in 1876.

 

Most of the theory of fractional calculus is based upon the familiar differential operator defined as-

Where r is a positive integer.

Case (i): If,the (i) reduces to classical Riemann-Liouville fractional derivatives or integral of order w.

Case (ii): If ,the equation (i) may be defined with the definition of the familiar Wayl fractional operator of order w.

 

Misra (1981) has defined the fractional derivative operator in the following manner:

  ( xμ-1 )   =   xμ-1   = xμ-α-1 ,   α≠μ                 … (1.2)                                                                  

 Dk,α,x( xμ)  =   xμ+k  ,    α ≠ μ+1                                                                        

 

Erdelyi- Kober fractional integral operators:

I+    =                 … (1.3)

K–    =             … (1.4)

These operators are generalization of Riemann-Liouville and Weyl fractional integral operator.

Different Definitions:

  1. Electric transmission lines

 

Heaviside successfully developed his operational calculus without rigorous mathematical arguments by introducing the letter p for the differential operator d/dt and gave the solution of the diffusion equation.

 

for the temperature distribution u(x,t) in the symbolic form

 

in which p ≡ d/ dx was treated as constant, where a, A and B are also constant.

 

  1. Ultra-sonic wave propagation in human cancellous bone

        Fractional calculus is used to describe the viscous interactions between fluid and solid structure.

 

  • Modeling of speech signals using fractional calculus

The fractional order calculus is using in the modeling of speech signals. The operators can be applied for solve the differential problem.

  1. Modeling the Cardiac Tissue Electrode Interface Using Fractional Calculus

Conventional lumped element circuit models of electrodes can generalization through modification of the defining current-voltage relationships.

 

  1. Application of Fractional Calculus to the sound Waves Propagation in Rigid Porous Materials

      The observation that the asymptotic expressions of stiffness and damping in porous materials are proportional to fractional powers of frequency suggests the fact that  time  derivatives  of  fractional order might describe the behavior of soundwaves in this kind of materials, including relaxation and frequency dependence.

 

  1. Using Fractional Calculus for Lateral and Longitudinal Control of Autonomous Vehicles

         Fractional Order Controllers (FOC) applied to the path tracking problem in an autonomous electric vehicle. Several control schemes with these controllers have been simulated and compared.

 

  1. Application of fractional calculus in the theory of visco-elasticity

the fractional derivative method has been used in studies of the complex moduli and impedances for various models of visco-elastic substances.

 

  • Fractional differentiation for edge detection

          This paper demonstrates how introducing an edge detector based on non-integer (fractional) differentiation can improve the criterion of thin detection, or detection selectivity in the case of parabolic luminance transitions, and the criterion of immunity to noise, which can be interpreted in term of robustness to noise in general.

 

  1. Application of Fractional Calculus to Fluid Mechanics

     Application of fractional calculus to the solution of time -dependent, viscous-diffusion fluid mechanics problems are presented.

 

THE MULTIVARIABLE H-FUNCTION:

The H-function of multivariable introduced by [12], (Srivastava & Panda, 1984).  This function is defined and represented in the following manner­-

 

 

=                         …(1)

Where =

     …(2)

 

                                       …(3)

 

and an empty product is interpreted as unity. Suppose, as usual, that the parameters

                                                   …(4)

are complex numbers and the associated coefficients

                                              …(5)

are positive real numbers such that

                                                      …(6)

and

                                                             …(7)

 

where the integers , A, C, u(i), v(i), B(i) and D(i) are constrained by the inequalities ,  and  , and the inequalities in (1.9) hold true for suitably restricted value of the complex variables Z1,…,Zr

 

The sequence of parameters in (1.4) is such that none of poles of the integrand coincide, that is, the poles of the integrand in (1.4) are simple.

The contour Li in the complex Si–plane is of the Mellin–Barns type which runs from – to +, with identifications, if necessary, to ensure that all the poles of , j=1,…,u(i) are separated from those of , j=1,…,v(i) and , j=1,…,  .

Then it is known that the multiple Mellin–Barnes contour integral in (1.4) converges absolutely under the condition (1.10), when

 ,                                                                                …(8)

The points Zi=0 (i=1,…,r) and various exceptional parameter values being tacitly excluded.

For the multivariable H- function given in (1.4), we shall employ the following contracted notation:

H [Z1,…,Zr] or                                               …(9)

Furthermore, we have the known asymptotic expansions in the following form:

H[Z1,…,Zr] =                                      …(10)

Where,

                                                                            …(11)

 

,                                     …(12)

Again, throughout the present work, we employ in abbreviation (a) to denote the sequence of A parameters a1, …, aA; for each i=1,…,r,  abbreviates the sequence of B(i) parameters, , j=1,…,B(i) with similar interpretations for (c), , etc. i=1,…,r; it will be understood for example, that b(1)=, b(2)= and so on.

Also, for the sake of brevity, we use the following contracted notations:

 , , i=1,2,3,…r etc.                                       …(13)

Conclusion

In this paper, we reviewed various definitions of fractional derivatives and fractional integrals and applications are also reviewed. The application of fractional calculus was made by Abel in the solution of an integral equation. This problem deals with the determination of the shape of a frictionless plane curve through the origin in a vertical plane along which a particle of mass m can fall in a time that is independent of the starting position. Fractional calculus is used to describe the viscous interactions between fluid and solid structure. Comparing the fractional results for boundary shear -stress and fluid speed to the existing analytical results for the first and second Stokes problems.

 

Funding:

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

 

References

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