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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 109582, 5 pages
Analytic Solution of a Class of Fractional Differential Equations
1School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China
2Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China
Received 8 July 2013; Revised 23 October 2013; Accepted 13 November 2013
Academic Editor: Shawn X. Wang
Copyright © 2013 Yue Hu and Zuodong Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
Recently, Garra  studied the analytic solution of a class of fractional differential equations with variable coefficients by using operatorial methods.
The proof of his main results is strongly based on operatorial properties of Caputo fractional derivative and the following theorem by Miller and Ross .
Theorem M.-R. (Theorem 1.1 in ). Let be any function of the form or , where , and has a radius of convergence and . Thenholds for all , provided (1) and are arbitrary, or(2) and are arbitrary and for , where (ceiling of the number).
Garra's main results are as follows.
Theorem Garra (Theorem 1.2 in ). Consider the following boundary value problem (BVP):in the half plane , with analytic boundary condition such that the conditions of Theorem 1.1 (Theorem M.-R.) are satisfied. The operatorial solution of BVP (3) is given by where is the Laguerre derivative and denotes Caputo fractional derivative. On the basis of the previous result, Garra proved (Example 1 in ) that if , , then the analytic solution of the BVP (3) is given by where .
Motivated by the results, in the present note, we extend Garra's results to the general case.
Definition 1 (see ). For every positive integer , the operator (containing ordinary derivatives) is called the -order Laguerre derivatives and the -exponential function is defined by
In , 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 , we have . Thus, (7) leads to the classical property of the exponential function .
Similarly, the spectral properties can be obtained  by using the general Laguerre derivatives (here is a real or complex constant) or more generally the operator and the corresponding eigenfunctions or
Throughout this paper, we use the Caputo fractional derivatives as in .
Definition 3 (see ). The Caputo derivative of fractional order of function is defined as where .
Lemma 4 (see ). Let for some . Then where .
Lemma 5. Let . Let , , and . Then
Proof. This follows immediately from Lemma 4.
Remark 6. In general, for , is not true. For example, but
Lemma 7. Suppose , and let have a radius of convergence and . If for some , then
3. Main Results
We first study the following BVP in the plane :
Theorem 8. Let with and assume that is a positive integer. Then the operatorial form solution of BVP (21) is
The following generalization of the Theorem 10 can be proved similarly.
Theorem 11. Let be a real or complex constant and . Consider the following BVP: in the half plane , with analytic boundary condition such that the conditions of Lemma 7 are satisfied. If , , then the operatorial solution of BVP (31) is given by
Proof. Using spectral properties of Laguerre derivative, together with Lemma 7, we have
This completes the proof.
Remark 12. Conditions are very harsh. However, if we remove them, Theorems 10 and 11 no longer remain valid. It should be noted that the main result in  (Theorem 1.2 in ) is incorrect in general case. A counterexample is as follows. Let and . One has
On the other hand, we have
Set in (34) and (35). By Definition 3 and Lemma 4, we deduce that all terms in (34) are positive and cannot contain negative exponent of variable . For example,
Similarly, all terms in (35) are also positive, except that some terms contain negative exponent of variable . For example,
Thus, we conclude that
4. 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 .
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, 6–8]). 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 and , 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 as the class of all functions which are infinitely differentiable everywhere and are such that and all its derivatives are of order for all , , then, for all functions of class , Weyl fractional derivatives possess the semigroup property . This has brought us great convenience for studying Weyl fractional differential equations. We will considered this topic in a forthcoming paper.
The authors are grateful to the anonymous reviewers for several comments and suggestions which contributed to the improvement of this paper. This work is supported by the National Natural Science Foundation of China (no. 11171092) and the Natural Science Foundation of Jiangsu Higher Education Institutions of China (no. 08KJB110005).
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