Analytic Solution of a Class of Fractional Differential Equations
Abstract
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.
1. Introduction
Recently, Garra [1] 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 [2].
- (1)
β < λ + 1 and α are arbitrary, or
- (2)
β ≥ λ + 1 and α are arbitrary and ak = 0 for k = 0,1, …, m − 1, where m = ⌈β⌉ (ceiling of the number).
Garra′s main results are as follows.
Motivated by the results, in the present note, we extend Garra′s results to the general case.
2. Preliminaries
Definition 1 (see [3].)For every positive integer n, the operator (containing n + 1 ordinary derivatives) is called the n-order Laguerre derivatives and the nL-exponential function is defined by
In [4], the following result is proved.
Lemma 2. Let a be an arbitrary real or complex number. The function en(ax) is an eigenfunction of the operator ; that is,
For n = 0, we have D0L : = D. Thus, (7) leads to the classical property of the exponential function Deax = aeax.
Throughout this paper, we use the Caputo fractional derivatives as in [1].
Definition 3 (see [5].)The Caputo derivative of fractional order α of function f(t) is defined as
Lemma 4 (see [5].)Let f(t) = tλ for some λ ≥ 0. Then
Lemma 5. Let λ > 0. Let α > 0, β > 0, and α + β ≤ λ. Then
Proof. This follows immediately from Lemma 4.
Remark 6. In general, for α, β > 0, DαDβ = Dα+β is not true. For example,
Lemma 7. Suppose α > 0, and let have a radius of convergence R > 0 and 0 < X < R. If [αk] = ⌈α(k + 1)⌉ − 1 for some k ∈ ℕ, then
3. Main Results
Theorem 8. Let g(t) = tλ with λ > 0 and assume that λ/α is a positive integer. Then the operatorial form solution of BVP (21) is
Proof. Let λ/α = m. 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 g(t) = t and α = 0.6. By Lemma 5, we have
On the other hand, we have
Note that by Remark 6, we have
Hence,
Theorem 10. Let g(t) satisfy the assumptions of Lemma 7. If [αk] = ⌈α(k + 1)⌉ − 1, (k = 1,2, …), then the operatorial form solution of BVP (21) is given by
The following generalization of the Theorem 10 can be proved similarly.
Theorem 11. Let m be a real or complex constant and n ∈ ℕ. Consider the following BVP:
Proof. Using spectral properties of Laguerre derivative, together with Lemma 7, we have
This completes the proof.
Remark 12. Conditions [αk] = ⌈α(k + 1)⌉ − 1 (k = 1,2, …) 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 [1] (Theorem 1.2 in [1]) is incorrect in general case. A counterexample is as follows. Let g(t) = et and α = 0.6. One has
On the other hand, we have
Set x = 1 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 t. For example,
Similarly, all terms in (35) are also positive, except that some terms contain negative exponent of variable t. 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 [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, 6–8]). 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 α > 0 and β > 0, α derivative of the β derivative of a function is, in general, not equal to the α + β 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 f which are infinitely differentiable everywhere and are such that f and all its derivatives are of order t−N for all N, N = 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.
Acknowledgments
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).
