### Article

Kyungpook Mathematical Journal 2020; 60(1): 73-116

**Published online** March 31, 2020 https://doi.org/10.5666/KMJ.2020.60.1.73

Copyright © Kyungpook Mathematical Journal.

### Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments

Hari Mohan Srivastava

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W3R4, Canada

and

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China

and

Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan e-mail : harimsri@math.uvic.ca

**Received**: February 1, 2019; **Revised**: October 7, 2019; **Accepted**: October 29, 2019

### Abstract

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

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. The main object of this survey-cum-expository article 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 “differintegral” equations. This general talk will be presented as simply as possible keeping the likelihood of

**Keywords**: fractional calculus, fractional-order integrals, fractional-order derivatives, diﬀ,erential equations, Integral equations, Cauchy-Goursat integral formula, diﬀ,erintegral equations, special functions, mathematical physics, Fuchsian and

### 1. Introduction, Notations and Preliminaries

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

Throughout this presentation, we denote by ℂ, ℝ, ℝ^{+},

Fractional calculus, the differentiation and integration of arbitrary (real or complex) order, arises naturally in various areas of science and engineering. For example, very recently, Wang and Zhang [104] investigated a class of nonlinear fractional-order differential impulsive systems with the Hadamard derivative (see also [103, 105, 112]).

The concept of

for the derivative of order _{0} := {0, 1, 2, ...} when

“...

Subsequent mention of fractional derivatives was made, in some context or the other, by (for example) Euler in 1730, Lagrange in 1772, Laplace in 1812, Lacroix in 1819, Fourier in 1822, Liouville in 1832, Riemann in 1847, Greer in 1859, Holmgren in 1865, Grünwald in 1867, Letnikov in 1868, Laurent in 1884, Nekrassov in 1888, Krug in 1890, and Weyl in 1917. In fact, in his

In addition, of course, to the theories of differential, integral, and integro-differential equations, and special functions of mathematical physics as well as their extensions and generalizations in one and more variables, some of the areas of present-day applications of fractional calculus include

Fluid Flow

Rheology

Dynamical Processes in Self-Similar and Porous Structures

Diffusive Transport Akin to Diffusion

Electrical Networks

Probability and Statistics

Control Theory

Viscoelasticity

Electrochemistry of Corrosion

Chemical Physics

Dynamical Systems

Mathematical Bio-Sciences

and so on (see, for details, [64, 27, 31]).

The very first work, devoted exclusively to the subject of fractional calculus, is the book by Oldham and Spanier [63]; it was published in the year 1974. Ever since then a significantly large number of books and monographs, edited volumes, and conference proceedings have appeared and continue to appear rather frequently. And, today, there exist

### 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

We begin by defining the linear integral operators ℐ and by

and

respectively. Then it is easily seen by iteration (and the principle of mathematical induction) that

and

where, just as elsewhere in this presentation,

and

The familiar (Euler’s) Gamma function Γ(

happens to be one of the most fundamental and the most useful special functions of mathematical analysis. It emerged essentially from an attempt by Euler to give a meaning to

Historically, the origin of the above-defined Gamma function Γ(

Thus, since

so that, obviously,

with a view to interpolating (

in terms of the Gamma function. Thus, in general, ^{μ}

and

respectively, it being

In the remarkably vast literature on fractional calculus and its fairly widespread applications, there are potentially useful operators of fractional derivatives ^{μ}

and

There also exist, in the considerably extensive literature on the theory and applications of fractional calculus, numerous ^{μ},

Now, for the Riemann-Liouville fractional derivative operator

Thus, upon setting

Observing that

since

the fractional derivative formula

In fact, it is the fractional derivative formula

which was derived in two pages (pp. 409–410) by S. F. Lacroix in his

### 3. Initial-Value Problems Based Upon Fractional Calculus

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

If we define, as usual, the Laplace transform operator ℒ by

provided that the integral exists, for the Riemann-Liouville fractional derivative operator

On the other hand, for the ^{(}^{n}^{)}(_{0}), it is well known that

or, equivalently,

where,

Upon comparing the Laplace transform formulas

In many recent works, especially in the theory of viscoelasticity and in hereditary solid mechanics, the following definition of Liouville (1832) and Caputo (1969) is adopted for the fractional derivative of order

where ^{(}^{n}^{)} (

which, just as the Laplace transform formulas

In the theory of ordinary differential equations, the following first- and second-order differential equations:

are usually referred to as the

are known as the

The basic processes of relaxation, diffusion, oscillations and wave propagation have been generalized by several authors by introducing fractional derivatives in the governing (ordinary or partial) differential equations. This leads to

We choose to summarize below some recent investigations by Gorenflo

### I. The Fractional (Relaxation-Oscillation) Ordinary Differential Equation

_{0} ≡ 0 for continuous dependence of the solution on the parameter

where _{α}

### II. The Fractional (Diffusion-Wave) Partial Differential Equation

where

where the Green function is given by

which can readily be expressed in terms of Wright’s (generalized Bessel) function

### 4. Fractional Kinetic Equations

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

During the past several years, fractional kinetic equations of different forms have been widely used in describing and solving several important problems of physics and astrophysics. Saxena

Here, in this presentation, we propose to investigate solution of a certain generalized fractional kinetic equation associated with the generalized Mittag-Leffler function (see [72]). It is also pointed out that the result presented here is general enough to be specialized to include many known solutions for fractional kinetic equations.

Fractional kinetic equations have gained popularity during the past decade or so due mainly to the discovery of their relation with the theory of CTRW (Continuous Time RandomWalks) in [29]. These equations are investigated in order to determine and interpret certain physical phenomena which govern such processes as diffusion in porous media, reaction and relaxation in complex systems, anomalous diffusion, and so on (see also [28, 36]).

Consider an arbitrary reaction characterized by a time-dependent quantity

In general, through feedback or other interaction mechanism, destruction and production depend on the quantity

This dependence is complicated, since the destruction or the production at a time

where _{t}

Haubold and Mathai [26] studied a special case of the

with the initial condition that _{i}_{0} is the number density of species _{i} >

Integration gives an alternative form of the

where

The fractional-calculus generalization of the

where

In terms of the generalized Bessel function _{l,b,c}

whose solution is given by

where _{ν,}_{2}_{k}_{+}_{l}_{+1} (·) is the above-mentioned generalized Mittag-Leffler function (see [53, 111]; see also [82]).

Srivastava and Tomovski [96] introduced the following generalization of the Mittag-Leffler function:

where, in terms of the Gamma function Γ(_{ν}

it being understood _{0} := 1 and assumed

Saxena and Nishimoto [77] studied a further generalization of the generalized Mittag-Leffler function

The special case of

The Mittag-Leffler function _{α}_{α,β}_{p}_{q}

Suppose that

whenever the limit exits (as a finite number). The convolution of two functions

which exists if the functions

### The Laplace Convolution Theorem

The so-called Sumudu transform is an integral transform which was defined and studied by Watugala [109] to facilitate the process of solving differential and integral equations in the time domain. The Sumudu transform has been used in various applications of system engineering and applied physics. For some fundamental properties of the Sumudu transform, one may refer to the works including (for example) [2, 9, 10, 86, 109]. It turns out that the Sumudu transform has very special properties which are useful in solving problems involving kinetic equations in science and engineering.

Let be the class of exponentially bounded functions

where _{1} and _{2} are some positive real constants. The Sumudu transform defined on the set is given by the following formula (see [109]; see also [17])

The Sumudu transform given in

The Sumudu transform

The Sumudu transform has been shown to be the theoretical dual of the Laplace transform. It is also connected to the

In connection with the definition

In our present investigation, we have chosen to make use of the Sumudu transform instead of the classical Laplace transform. In fact, for the various problems considered here, the Sumudu transform has not only been found to be more convenient to use, but the closed-form results derived here also appear to be remarkably simpler (see also [86]).

Throughout this presentation, it is

### 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

We first find the solution of the generalized fractional kinetic equation involving the generalized Mittag-Leffler function

### Lemma 5.1

_{γ,κ}_{1}, _{1}), · · ·, (_{m}, β_{m}

_{2}Ψ_{m}

**Proof**

Using the definition

For later convenience, a special case of _{1} and _{1} is given in Lemma 5.2 below.

### Lemma 5.2

_{1}), ℜ(_{1})}

### Theorem 5.3

^{+}_{j}, β_{j}, γ, κ

**Proof**

Applying the Laplace transform

where, just as in the definition

Using the geometric series:

we find for |

Now, by inverting the Laplace transform on each side of

we get

which, in view of the definition

### Theorem 5.4

^{+}_{j}, β_{j}, γ, κ

**Proof**

Proof of the result asserted by Theorem 5.4 runs parallel to that of Theorem 5.3. Here we use

### Remark 5.5

For , the results in Theorem 5.3 and Theorem 5.4 reduce to those for the generalized fractional kinetic equation involving the generalized Mittag-Leffler function studied by Saxena

By setting

### Corollary 5.6

^{+}

In its

### Corollary 5.7

^{+}

### Remark 5.8

The result asserted by Theorem 5.4 can also be suitably specialized to deduce solutions of certain generalized fractional kinetic equations analogous to those which are dealt with in Corollary 5.6 and Corollary 5.7.

### 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

In this section we propose to investigate the solution of the generalized fractional kinetic equation involving the generalized Mittag-Leffler function

### Lemma 6.1

**Proof**

By using

This last integral in

We thus find that

which, in view of

### Remark 6.2

We find it to be convenient to record here a special case of _{1} and _{1} as Lemma 6.3 below.

### Lemma 6.3

_{1}), ℜ(_{1}), ℜ(

### Theorem 6.4

^{+}_{j}, β_{j}, γ, κ

**Proof**

Taking the Sumudu transform on both sides of

where

Equivalently, we can write

Using the binomial series expansion of (1 + ^{ν}u^{ν}^{−1} in

Finally, we make use of the following formula:

After some simplification, we thus find that

which, in view of

### Theorem 6.5

^{+}_{j}, β_{j}, γ, κ

**Proof**

Our demonstration of Theorem 6.5 would run parallel to that of Theorem 6.4. Here, in this case, we use

Upon setting

### Corollary 6.6

^{+}

If we set

### Corollary 6.7

^{+}

We conclude this section by remarking that the results presented here are general enough to yield, as their special cases, solutions of a number of known or new fractional kinetic equations involving such other special functions as (for example) those considered by Haubold and Mathai [26] and Saxena

### 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

Operators of fractional differintegrals (that is, fractional derivatives and fractional integrals), which are based essentially upon the familiar Cauchy-Goursat integral formula:

were considered by (among others) Sonin in 1869, Letnikov in 1868 onwards, and Laurent in 1884. Here, as usual, the function _{0} is any point interior to the contour

### I. The Gauss Equation

### II. The Kummer Equation

### III. The Euler Equation

### IV. The Coulomb Equation

### V. The Laguerre-Sonin Equation

### VI. The Chebyshev Equation

### VII. The Weber-Hermite Equation

Numerous earlier contributions on fractional calculus along the aforementioned lines are reproduced, with proper credits, in the works of Nishimoto (

In the cases of (ordinary as well as partial) differential equations of

**Definition 7.1.([54, 55, 94])**

If the function

is a contour along the cut joining the points

and

where

and

then _{ν}_{ν}

Throughout the remainder of this section, we shall simply write _{ν}_{ν}

Each of the following general results is capable of yielding particular solutions of many simpler families of linear ordinary fractional differintegral equations (

**Theorem 7.2**

_{−}_{ν}

**Theorem 7.3**

Next, for a function

in order to abbreviate the partial fractional differintegral of

**Theorem 7.4**

_{1}_{2}

We conclude this section by remarking further that either or both of the polynomials _{0}. Furthermore, it is fairly straightforward to see how each of these general theorems can be suitably specialized to yield numerous simpler results scattered throughout the ever-growing literature on fractional calculus.

### 8. Applications Involving a Class of Non-Fuchsian Differential Equations

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

In this section, we aim at applying Theorem 7.2 in order to find (explicit) particular solutions of the following general class of non-Fuchsian differential equations with six parameters:

where

constrain the various parameters involved in

then Theorem 7.2 would eventually imply that the nonhomogeneous linear ordinary differential

and (by Theorem 7.3) the corresponding homogeneous linear ordinary differential equation:

has solutions given by

where

For various special choices for the

### 9. The Classical Gauss and Jacobi Differential Equations Revisited

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

The main purpose of this section (and Section 10 below) is to follow rather closely and analogously the investigations in (for example) [39, 46, 90, 106, 107] of solutions of some general families of second-order linear ordinary differential equations, which are associated with the familiar

which is named after Friedrich Wilheim Bessel (1784–1846). More precisely, just as in the earlier works [44, 90] (see also [40, 41]), which dealt

we aim here in this section at demonstrating how the underlying simple fractional-calculus approach to the solutions of the classical differential

We begin by setting

in Theorem 7.2. We can thus deduce the following application of Theorem 7.2 relevant to the linear ordinary differential

### Theorem 9.1

_{−}_{λ}

where K is an arbitrary constant, it being provided that the second member of

### Remark 9.2

If we consider the case when |

Thus, in view of the following well-exploited fractional differintegral formula:

we readily obtain

in terms of the Gauss hypergeometric function _{2}_{1} (see [18, Vol. I, Chapter 2]).

### Remark 9.3

If we consider the case when |

Thus, in view of the fractional differintegral formula

in terms of the Gauss hypergeometric function _{2}_{1} (see [18, Vol. I, Chapter 2]).

### 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

We now propose to develop

### I. Gauss’s Differential Equation [see also Equation (7.1) ]

which possesses the following well-known power-series solution relative to the regular singular point

Furthermore, upon setting

in

Thus, by combining the ^{(1)}(^{(2)}(

where _{1} and _{2} are arbitrary constants, it being understood that each member of

Alternatively, if we set

in

If, on the other hand, we choose to set

in

which does indeed follow also from ^{(3)}(^{(4)}(

where

Lastly, since any solution of the Gauss differential

where, for convenience, the coefficients

The analytic continuation formula

where the coefficients

### II. Jacobi’s Differential Equation

which, in its _{0}, would reduce to the relatively more familiar differential equation satisfied by the

Indeed, upon setting

Jacobi’s differential

Clearly, we have

By setting

in

and

Thus, if we make use of the relationships given by

and

**Remark 10.1**

The solution Θ^{(1)}(

**Remark 10.2**

In view of the familiar Euler transformation (see, for example, [18, Vol. I, p. 64, Equation 2.1.4 (23)]):

we can rewrite the solution Θ^{(4)}(

which obviously is expressible in terms of the

In concluding this section, we observe that such general results as Theorems 7.2, 7.3 and 7.4 and their various companions (proven by Tu

### 11. Further Miscellaneous Applications of Fractional Calculus

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

For the purpose of those in the audience who are interested in pursuing investigations on the subject of fractional calculus, we give here references to

Theory of Generating Functions of Orthogonal Polynomials and Special Functions (see, for details, [91]);

Geometric Function Theory (especially the Theory of Analytic, Univalent, and Multivalent Functions) (see, for details, [92, 93]);

Generalized Functions (see, for details, [50]);

Theory of Potentials (see, for details, [66]).

### 12. Other Recent Developments

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

In the past several decades, various real-world issues have been modeled in many areas by using some very powerful tools. One of these tools is fractional calculus. Several important definitions have been introduced for fractional-order derivatives, including: the Riemann-Liouville, the Grünwald-Letnikov, the Liouville-Caputo, the Caputo-Fabrizio and the Atangana-Baleanu fractional-order derivatives (see, for example, [7, 12, 14, 31, 64, 112]).

By using the fundamental relations of the Riemann-Liouville fractional integral, the Riemann-Liouville fractional derivative was constructed, which involves the convolution of a given function and a power-law kernel (see, for details, [31, 64]). The Liouville-Caputo (LC) fractional derivative involves the convolution of the local derivative of a given function with a power-law function [13]. Recently, Caputo and Fabrizio [12] and Atangana and Baleanu [7] proposed some interesting fractional-order derivatives based upon the exponential decay law which is a generalized power-law function (see [1, 3, 4, 5, 6, 8]). The Caputo-Fabrizio (CFC) fractional-order derivative as well as the Atangana-Baleanu (ABC) fractional-order derivative allow us to describe complex physical problems that follow, at the same time, the power law and the exponential decay law (see, for details, [1, 3, 4, 5, 6, 8]).

In a noteworthy earlier investigation, Srivastava and Saad [95] investigated the model of the gas dynamics equation (GDE) by extending it to some new models involving the time-fractional gas dynamics equation (TFGDE) with the Liouville-Caputo (LC), Caputo-Fabrizio (CFC) and Atangana-Baleanu (ABC) time-fractional derivatives. They employed the Homotopy Analysis Transform Method (HATM) in order to calculate the approximate solutions of TFGDE by using LC, CFC and ABC in the Liouville-Caputo sense and studied the convergence analysis of HATM by finding the interval of convergence through the

Given the

where

Srivastava and Saad [95] used the HATM (see, for example, [38, 67]) in order to solve the LC, CFC and ABC analogues of the TFGDE

successively, where the order

The corresponding LC, CFC and ABC time-fractional analogues of the TFGDE

respectively. Here

denote the time-fractional derivatives of order

and

where

where

is the Mittag-Leffler function and

In the bibliography of this presentation, we have chosen to include a remarkably large number of recently-published books, monographs and edited volumes (as well as journal articles) dealing with the extensively-investigated subject of fractional calculus and its widespread applications. Indeed, judging by the on-going contributions to the theory and applications of

### References

- Abstract
- 1. Introduction, Notations and Preliminaries
- 2. The Riemann-Liouville and Weyl Operators of Fractional Calculus
- 3. Initial-Value Problems Based Upon Fractional Calculus
- 4. Fractional Kinetic Equations
- 5. Solution of Generalized Fractional Kinetic Equations by Using the Laplace Transform
- 6. Solution of Generalized Fractional Kinetic Equations by Using the Sumudu Transform
- 7. Fractional Differintegral Operators Based Upon the Cauchy-Goursat Integral Formula
- 8. Applications Involving a Class of Non-Fuchsian Differential Equations
- 9. The Classical Gauss and Jacobi Differential Equations Revisited
- 10. A Family of Unified Alternative Solutions Resulting from Theorem 6.5
- 11. Further Miscellaneous Applications of Fractional Calculus
- 12. Other Recent Developments
- References

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