Partial, Composite Fractional Operators, and Their Properties and Applications

. The paper discusses the properties of the partial fractional integrals, the partial fractional derivatives, and the composite fractional integrals and derivatives. Some basic formulas are derived; and the relations between the Riemann-Liouville and Caputo ones are also presented. Finally, as the specific application, a two dimensional Abel integral equation is solved.


Introduction
Fractional calculus, as a natural extension of the classical calculus, has the history of more than 300 years [3,4,7,8,9]. In fact, since the beginning of the theory of differential and integral calculus, the mathematicians began their investigations of the calculation of noninteger order derivatives and integrals. Most of the researchers of this topic cite 30th September 1695 as the birthday of the so called 'fractional calculus'. In a letter, dated 30th September 1695, L'Hopital wrote to Leibniz asking him a particular notation that he had used in his publication for the n-th derivative of a function paradox, from which one day useful consequences will be drawn." Following L'Hopital's and Liebniz's first inquisition, fractional calculus was primarily a study reserved for the best minds in mathematics. Fourier, Euler, Laplace are among the many that dabbled with fractional calculus and the mathematical consequences [7]. Many proposed, using their own notation and methodology, the definitions that fit the concept of noninteger order integral or derivative. The most famous of these definitions that have been popularized in the world of fractional calculus (not yet the world as a whole) are the Riemann-Liouville and Grunwald-Letnikov derivatives [6].
Most of the mathematical theories applicable to the study of fractional calculus were developed prior to the turn of the 20th century. However in the past 100 years, the widely engineering and scientific applications have been found. In some cases, the mathematics has to be made some changes to meet the requirements of physical reality. Caputo reformulated the more 'classic' definition of the Riemann-Liouville fractional derivative in order to conveniently specify the initial conditions. And the so called Caputo derivative was presented [2,3,4]. More recently, the space-time fractional operator was discussed [1].
In this paper, we introduce the partial fractional integral, partial fractional derivative, their compositions, and the relations between their Riemann-Liouville and Caputo versions. The properties of these operators are detailedly discussed; and a specific application of solving a two dimensional Abel integral equation is proposed. For easy of presentation, we restrict the left end point of the interval of the definition of fractional operators at '0'; and the technique of Laplace transform is used [5].

Partial Fractional Integrals And Their Compositions
In our notation the Cauchy formula reads whereℕ is the set of positive integers. From this definition, it can be noted that � n (t)vanishes at t=0 with its derivatives of order 1,�, � , � � 1. Noting that (� � 1 ) ! = Γ(�), and introducing the arbitrary positive real number α, one defines the fractional integral of order α>0: Starting from the one dimensional case (2), we can naturally define the partial integral of the function �(�, y): It can be noted that the fractional integral can be considered as the Laplace convolution between Φ α (�) and �(�, y), i.e., and see e.g., [5], and recalling the convolution theorems for Laplace transforms, we get the following rules for the Laplace transforms of the fractional integrals, which are the straightforward generalization of the case with a n-fold repeated integral (� = �).

Composite Fractional Operators
In general, both the operators of fractional differentiation, � x α , � � β and � x α , � � β , do not satisfy either the semigroup property, or the (weaker) commutative property. To show how the Law of Exponents does not necessarily hold for the standard fractional derivative, we provide two simple example (with power functions) for which

Conclusions
Since the beginning of the theory of differential and integral calculus, the mathematicians began their investigations of the calculation of non-integer order derivatives and integrals. In this paper, we introduce the properties of the partial fractional integrals, the partial fractional derivatives, and the composite fractional integrals and derivatives, and a specific application of solving a two dimensional Abel integral equation is proposed.