Analytic Solution of a Class of Fractional Differential Equations

We consider the analytic solution of a class of fractional differential equations with variable coefficients by operatorial methods. We obtain three theorems which extend the Garra’s results to the general case.


Introduction
Recently, Garra [1] studied the analytic solution of a class of fractional differential equations with variable coefficients by using operatorial methods.
Garra's main results are as follows.
Theorem Garra (Theorem 1.2 in [1]). Consider the following boundary value problem (BVP): in the half plane > 0, with analytic boundary condition ( ) such that the conditions of Theorem 1.
where is the Laguerre derivative and denotes Caputo fractional derivative. On the basis of the previous result, Garra proved (Example 1 in [1]) that if ( ) = , ∈ N, then the analytic solution of the BVP (3) is given by where Γ( ) = ∫ ∞ 0 − −1 . Motivated by the results, in the present note, we extend Garra's results to the general case.

Preliminaries
Definition 1 (see [3]). For every positive integer , the operator := , . . . , (containing + 1 ordinary derivatives) is called the -order Laguerre derivatives and the -exponential function is defined by 2 Abstract and Applied Analysis In [4], the following result is proved.

Lemma 2.
Let be an arbitrary real or complex number. The function ( ) is an eigenfunction of the operator ; that is, For = 0, we have 0 := . Thus, (7) leads to the classical property of the exponential function = . Similarly, the spectral properties can be obtained [3] by using the general Laguerre derivatives (here is a real or complex constant) or more generally the operator ( ∈ N) and the corresponding eigenfunctions Throughout this paper, we use the Caputo fractional derivatives as in [1].

Main Results
We first study the following BVP in the plane > 0: Abstract and Applied Analysis 3 Proof. Let / = . By Lemmas 4 and 5, we have In the fifth previous equality, we use the fact that Remark 9. We point out that the result of Example 1 in [1] is incorrect. A counterexample is as follows. Let ( ) = and = 0.6. By Lemma 5, we have On the other hand, we have (1!) 2 ) = 0.6 ( ) + 0.6 ( 0.6 ) .
Note that by Remark 6, we have 0.6+0.6 = 0, Hence, This completes the proof.
The following generalization of the Theorem 10 can be proved similarly.

Theorem 11. Let be a real or complex constant and ∈ N.
Consider the following BVP: Proof. Using spectral properties of Laguerre derivative, together with Lemma 7, we have This completes the proof.
and so forth.
Similarly, all terms in (35) are also positive, except that some terms contain negative exponent of variable . For example, Thus, we conclude that

Conclusion and Discussion
In this paper, we point out that Garra's results are incorrect and give some necessary counterexamples. In addition, we established three theorems (Theorems 8, 10, and 11) which correct and extend the corresponding results of [1]. Different from integer-order derivative, there are many kinds of definitions for fractional derivatives, including Riemann-Liouville, Caputo, Grunwald-Letnikov, Weyl, Jumarie, Hadamard, Davison and Essex, Riesz, Erdelyi-Kober, and Coimbra (see [1,[6][7][8]). These definitions are generally not equivalent to each other. Every derivative has its own serviceable range. In other words, all these fractional derivatives definitions have their own advantages and disadvantages. For example, the Caputo derivative is very useful when dealing with real-world problem, since it allows traditional initial and boundary conditions to be included in the formulation of the problem and the Laplace transform of Caputo fractional derivative is a natural generalization of the corresponding well-known Laplace transform of integer-order derivative. So, the Caputo fractional-order system is often used in modelling and analysis. However, the functions that are not differentiable do not have fractional derivative, which reduces the field of application of Caputo derivative (see [1,8,9]).
When solving fractional-order systems, the law of exponents (semigroup property) is the most important. Unlike integer-order derivative, for > 0 and > 0, derivative of the derivative of a function is, in general, not equal to the + derivative of such function. About the semigroup property of the fractional derivatives, under suitable assumptions of fractional order, there have existed some studies, but only a few studies provide valuable judgment methods (see [1,9]).
Fortunately, if we define Ω as the class of all functions which are infinitely differentiable everywhere and are such that and all its derivatives are of order − for all , = 1, 2, . . ., then, for all functions of class Ω, Weyl fractional derivatives possess the semigroup property [1]. This has brought us great convenience for studying Weyl fractional differential equations. We will considered this topic in a forthcoming paper.