AN ELEMENTARY AND INTRODUCTORY APPROACH TO FRACTIONAL CALCULUS AND ITS APPLICATIONS

A b s t r a c t: The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It does 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 paper * is to present a brief elementary and introductory approach to the theory of fractional calculus and its applications especially in developing solutions of certain interesting families of ordinary and partial fractional differintegral equations. Relevant connections of some of the results presented in this lecture with those obtained in many other earlier works on this subject will also be indicated.


INTRODUCTION AND A BRIEF HISTORICAL SURVEY
The concept of it fractional calculus that is, calculus of integrals and derivatives of any arbitrary real or complex order) seems to have stemmed from a question raised in the year 1695 by Marquis de l'Hôpital (1661-1704) to Gottfried Wilhelm Leibniz (1646-1716), which sought the meaning of Leibniz's (currently popular) notation "...This is an apparent paradox from which, one day, useful consequences will be drawn...." Subsequent mention of fractional derivatives was made, in some context or the other, by ( 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  [11], [41], and [14]. The first work, devoted exclusively to the subject of fractional calculus, is the book by Oldham and Spanier [39].One of the most recent works on the subject of fractional calculus is the book by Podlubny [41].The latest (but certainly not the last) works on the subject of fractional calculus and its applications are the volume edited by Hilfer [11] and the monograph emphasizing upon the theory and applications of fractional differential equations by Kilbas et al. [14].Indeed, in the meantime, numerous other works (books, edited volumes, and conference proceedings) have also appeared (see, e. g., [13], [15], [26], [27], [28], [29], [31], [30], [43], [44], [47], [48] and [55]).And today there exist at least two international journals which are devoted almost entirely to the subject of fractional calculus: (i) Journal of Fractional Calculus and (ii) Fractional Calculus and Applied Analysis.
Here, in this expository lecture, we aim at presenting an elementary and introductory overview of the theory of fractional calculus and of some of its applications.

THE RIEMANN-LIOUVILLE AND WEYL OPERATORS OF FRACTIONAL CALCULUS
Now we begin by defining the linear integral operators mathcal 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, With a view to interpolating (n -1) between the positive integer values of n, one can set … respectively, it being tacitly assumed that the function f (t) is so constrained that the integrals in (2.6) and (2.7) exist.
In the remarkably vast literature on fractional calculus and its fairly widespread applications, there are potentially useful operators of fractional de- , which correspond to the abovedefined fractional integral operators μ and μ , respectively, and we have There also exist, in the considerably extensive literature on the theory and applications of fractional calculus, numerous further extensions and generalizations of the operators μ and μ , μ 0 ; x , and μ ∞ ; x , each of which we have chosen to introduce here for the sake of the non-specialists in this subject.

INITIAL-VALUE PROBLEMS BASED UPON FRACTIONAL CALCULUS
If we define, as usual, the Laplace transform operator by provided that the integral exists, for the Riemann-Liouville fractional derivative operator μ 0 ; t of order μ , we have Such initial values as those occurring in (3.2) are usually not interpretable physically in a given initial-value problem.This situation is overcome at least partially by making use of the so-called Caputo fractional derivative which arose in several important works, dated 1969 onwards, by M. Caputo (see, for details, [41, p. 78 et seq.];see also [14, p. 90

et seq.]).
In many recent works, especially in the theory of viscoelasticity and in hereditary solid mechanics, the following (Caputo's) definition is adopted for the fractional derivative of order α > 0 of a causal function (t) (i.e., (t) = 0 for t < 0): denotes the usual (ordinary) derivative of order n and Γ is the Gamma function occurring already in (2.4) and (2.5).One can apply the above notion in order to generalize some basic topics of classical mathematical physics, which are treated by simple, linear, ordinary or partial, differential equations, since [cf.Equation (3.2) and Definition (3.3)] which obviously is more suited for initial-value problems than (3.2).See, for details, Gorenflo et al. [9], Podlubny [41] and Kilbas et al. [14].
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 superslow or intermediate processes that, in mathematical physics, we may refer to as fractional phenomena.Our analysis of each of these phenomena, carried out by means of fractional calculus and Laplace transforms, leads to certain special functions in one variable of Mittag-Leffler and Fox-Wright types.These useful special functions are investigated systematically as relevant cases of the general class of functions which are popularly known as Fox's Hfunction after Charles Fox (1897-1977) who initiated a detailed study of these functions as symmetrical Fourier kernels [see, for details, Srivastava et al. [51] and [52].
We choose to summarize below some recent investigations by Gorenflo et al. [9] who did indeed make references to numerous earlier closely-related works on this subject.

I. The Fractional (Relaxation-Oscillation) Ordinary Differential Equation
Initial Conditions: with g(x) ≡ 0 for continuous dependence of the solution on the parameter β also in the transition from β = 2 1 -to β = 2 1 +.

FRACTIONAL DIFFERINTEGRAL OPERATORS BASED UPON THE CAUCHY-GOURSAT INTEGRAL FORMULA
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.In recent years, many authors have demonstrated the usefulness of fractional calculus operators (based upon the Cauchy-Goursat integral formula) in obtaining particular solutions of numerous families of homogeneous (as well as nonhomogeneous) linear ordinary and partial differential equations which are associated, for example, with many of the following celebrated equations as well as their close relatives:

II. The Kummer Equation
III.The Euler Equation: IV.The Coulomb Equation: V. The Laguerre-Sonin Equation: Numerous earlier contributions on fractional calculus along the aforementioned lines are reproduced, with proper credits, in the works of Nishimoto (cf.[29] and [31]).Moreover, a rather systematic analysis (including interconnections) of many of the results involving (homogeneous or nonhomogeneous) linear differential equations associated with (for example) the Gauss hypergeometric equation (4.1) can be found in the works of Nishimoto et al. ( [37] and [38]) and the recent contribution on this subject by Wang et al. [63] (see also some other recent applications considered by Lin et al. [19] and Prieto et al. [42]).

VI. The Chebyshev Equation:
In the cases of (ordinary as well as partial) differential equations of higher orders, which have stemmed naturally from the Gauss hypergeometric equation (4.1) and its many relatives and extensions, including some of the above-listed linear differential equations (4.2) to (4.7), there have been several seemingly independent attempts to present a remarkably large number of scattered results in a unified manner.We choose to furnish here the generalizations (and unifications) proposed in one of the latest works on this subject by Tu et al. [59] in which references to many earlier related works can be found.We find it to be convenient to begin by recalling the following definition of a fractional differintegral (that is, fractional derivative and fractional integral) of f (z) of order ν ∈ .Definition (cf.[29], [31], and [57]).If the function f (z) is analytic and has no branch point inside and on , where -is a contour along the cut joining the points z and , which starts from the point at , ∞ − encircles the point z once counter-clockwise, and returns to the point at , ∞ − + is a contour along the cut joining the points z and , which starts from the point at , ∞ encircles the point z once counterclockwise, and returns to the point at , then f ν (z) (ν > 0) is said to be the fractional derivative of f (z) of order ν and Throughout the remainder of this section, we shall simply write f ν for f ν (z) whenever the argument of the differintegrated function f is clearly understood by the surrounding context.Moreover, in case f is a many-valued function, we shall tacitly consider the principal value of f in this investigation.
Each of the following general results is capable of yielding particular solutions of many simpler families of linear ordinary fractional differintegral equations (cf.Tu et al. [59]) including (for example) the classical differential equations listed above [cf.Equations (4.1) to (4.7)].
Theorem 1.Let P (z; p) and Q (z; q) be polynomials in z of degrees p and q, respectively, defined by Suppose also that f ν (≠ 0) exists for a given function f.Then the nonhomogeneous linear ordinary fractional differintegral equation: has a particular solution of the form: where, for convenience, provided that the second member of (4.17) exists.
Theorem 2. Under the various relevant hypotheses of Theorem 1, the homogeneous linear ordinary fractional differintegral equation: has solutions of the form: where K is an arbitrary constant and H(z; p, q) is given by (4.18), it being provided that the second member of (4.20) exists.
Next, for a function u = u (z, t) of two independent variables z and t, we find it to be convenient to use the notation: to abbreviate the partial fractional differintegral of u (z, t) of order μ with respect to z and of order ν with respect to t (μ, ν ∈ ).And we now state the following general result (cf.Tu et al. [59]): Theorem 3. Let the polynomials P (z; p) and Q (z; q) be defined by (4.14), (4.15), respectively.Suppose also that the function H (z; p, q) is given by (4.18).
Then the partial fractional differintegral equation:   provided that the second member of (4.22) exists in each case.
We conclude this section by remarking further that either or both of the polynomials P(z; p) and Q(z; q), involved in Theorems 1 to 3, can be of degree 0 as well.Thus, in the definitions (4.14) and (4.15) (as also in Theorems 1 to 3),  may easily be replaced (if and where needed) by  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 evergrowing literature on fractional calculus.

APPLICATIONS INVOLVING A CLASS OF NON-FUCHSIAN DIFFERENTIAL EQUATIONS
In this section, we aim at applying Theorem 1 in order to find (explicit) particular solutions of the following general class of non-Fuchsian differential equations with six parameters: where f is a given function and the parameters α, β, γ, δ, ε, and l are unrestricted, in general.Indeed, if we make use of the transformation: constrain the various parameters involved in (5.1) and (5.2) so that then Theorem 1 would eventually imply that the nonhomogeneous linear ordinary differential equation ( 5.1) has a particular solution in the form: ( (5.4) and (by Theorem 2) the corresponding homogeneous linear ordinary differential equation: has solutions given by, where K=$ is an arbitrary constant, the parameters ρ and λ are given (as before) by (2.3), and For various special choices for the free parameters occurring in (5.1) and (5.5), one can apply the results of this section to many known non-Fuchsian differential equations.These include (for example) a special limit (confluent) case of the Gauss hypergeometric equation (4.1), referred to as the Whittaker equation (cf., e.g., [66, p. 337, Equation 16.1 (B)]; see also [5, Vol.I, p. 248, Equation 6.1 (4)]), the so-called Fukuhara equation (cf.[7]; see also [33]), the Tricomi equation (cf.[45, p. 7, Equation 1.2 (1)]; see also [5, Vol.I, p. 251, Equation 6.2 (13)]), the familiar Bessel equation (cf.[65]), and so on.For a systematic investigation of these and many other closely-related differential equations (including, for example, many of the familiar differential equations list at the beginning of Section 4 here), we refer the interested reader to the recent works of Nishimoto et al. ([32] to [38]), Salinas de Romero et al. ([45] and [46]), Galué [8], Lin et al. ([20] to [25]), Tu et al. ([59] to [62]), and Wang et al. ([63] and [64]).

THE CLASSICAL GAUSS AND JACOBI DIFFERENTIAL EQUATIONS REVISITED
The main purpose of this section (and Section 7 below) is to follow rather closely and analogously the investigations in (for example) [16], [23], [53], [63] and [64] of solutions of some general families of second-order linear ordinary differential equations, which are associated with the familiar Bessel differential equation of general order ν (cf.[5], Vol.II, Chapter 7]; see also [65] and [66, Chapter 17] which is named after Friedrich Wilheim Bessel (1784-1846).More precisely, just as in the earlier works [21] and [53] (see also [17] and [18]), which dealt systematically with Legendre's differential equation (cf.[5.Vol.I, p. 121, Equation 3.2 (1)]; see also [66, Chapter 15]): we aim here in this section at demonstrating how the underlying simple fractional-calculus approach to the solutions of the classical differential equations (6.1) and ( 6.2) would lead us analogously to several interesting consequences including (for example) an alternative investigation of solutions of the following two-parameter family of second-order ordinary differential equations (see also [63]): We begin by setting in Theorem 1.We can thus deduce the following application of Theorem 1 relevant to the linear ordinary differential equation (6.3)., then the following nonhomogeneous linear ordinary differential equation: z has a particular solution of the form: z provided that the second member of (6.6) exists.Furthermore, the following homogeneous linear ordinary differential equation: were K is an arbitrary constant, it being provided that the second member of (6.8) exists.
Remark 1.If we consider the case when |z| < 1, by making use of the familiar binomial expansion, we find from the assertion (2.5) of Theorem 4 that ( ) Thus, in view of the following well-exploited fractional differintegral formula: in terms of the Gauss hypergeometric function 2 F 1 (see [5, Vol.I, Chapter 2]).
Remark 2. If we consider the case when |z| > 1, by appropriately applying the familiar binomial expansion once again, we find from the assertion (6.8) of Theorem 4 that ( ) Thus, in view of the fractional differintegral formula (6.10), we find the following explicit solution of the differential equation (6.7) when |z| > 1: in terms of the Gauss hypergeometric function 2 F 1 (see [5, Vol.I, Chapter 2]).

A FAMILY OF UNIFIED ALTERNATIVE SOLUTIONS RESULTING FROM THEOREM 4
We now propose to develop alternative solutions of several classical differential equations of mathematical physics in a unified manner by suitably applying the assertions of Theorem 4, Remark 1, and Remark 2.

I. Gauss's Differential Equation
which possesses the following well-known power-series solution relative to the regular singular point z = 0 (see, for example, [12, p. 162]): and 1 , in (6.11), we obtain the following explicit solution of (7.1): where K 1 and K 2 are arbitrary constants, it being understood that each member of (7.4) exists.
Alternatively, if we set and 1 , in (6.13), then we obtain the following explicit solution of (7.1) [12, p. 162]: If, on the other hand, we choose to set and 1 , in (6.12), then we obtain the following further explicit solution of (7.1) [12, p. 162]: which does indeed follow also from (7.5) upon interchanging the röles of the parameters α and β .Thus, if we combine the solutions ) ( where * 1 K and * 2 K are arbitrary constants, it being understood that each member of (7.7) exists.
Lastly, since any solution of the Gauss differential equation (7.1) is linearly expressible in terms of two linearly independent solutions (see, e.g., [12, p. 168]), it is not difficult to deduce from the above observations that (see, for example, [5, Vol.I, p. 108, Equation 2.10 (2)]) where, for convenience, the coefficients A and B are given by The analytic continuation formula (7.8) is usually derived by the calculus of residues and the Mellin-Barnes contour integral representation for the Gauss hypergeometric function occurring on its left-hand side (see, for details, [5, Vol.I, p. 62, Section 2.1.4]).Moreover, it is easily seen from this analytic continuation formula (7.8) that asymptotically, for large |z|, we have where the coefficients A and B are given (as before) by (7.9).
II. Jacobi's Differential Equation: given explicitly by Jacobi's differential equation (7.11) assumes the following form: in (6.11) and (6.13), or (alternatively) by directly applying the hypergeometric solutions given by (7.2), (7.3), (7.5) and (7.6), we obtain the following explicit solutions of (7.13): Thus, if we make use of the relationships given by (7.14) in our observations (7.15) to (7.18), we are led fairly easily to the following explicit solutions of the general Jacobi differential equation (7.11): we can rewrite the solution ) ( given by (7.22)In conclusion, we observe that such general results as Theorems 1 to 3 and their various companions (proven by Tu et al. [59]) can be applied similarly in order to derive explicit solutions of many other interesting families of ordinary and partial differential equations.

FURTHER MISCELLANEOUS APPLICATIONS OF FRACTIONAL CALCULUS
For the purpose of those readers who are interested in pursuing investigations on the subject of fractional calculus, we give here references to some of the other applications of fractional calculus operators in the mathematical sciences, which are not mentioned in the preceding sections.
(i) Theory of Generating Functions of Orthogonal Polynomials and Special Functions (cf.[54]);

Theorem 4 .
If the given function f satisfies the constraint (4.13) and 0 ≠ −λ f the following well-known general solution of the Gauss differential equation (7.1) by means of fractional calculus: point at infinity, we find the following general solution of the Gauss differential equation (7.1) by means of fractional calculus: to the relatively more familiar differential equation satisfied by the classical Jacobi polynomials

[
66] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis : An Introduction to the General Theory of Infinite Processes and of Analytic Functions, With an Account of the Principal Transcendental Functions, Fourth edition, Cambridge University Press, Cambridge, London and New York, 1927.R e z i m e .23) expressible in terms of the Jacobi function of the second kind defined by (cf., e.g., [5, Vol.II, p. 170, Equation10.8 (18)])