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].

Theorem M.-R. (Theorem 1.1 in [1]). Let 𝑓(𝑡) be any function of the form 𝑡^𝜆𝜂(𝑡) or 𝑡^𝜆(ln𝑡)𝜂(𝑡), where 𝜆>−1, and 𝜂(𝑡)=^∞_𝑛=0𝑎_𝑛𝑡^𝑛 (1)has a radius of convergence 𝑅>0 and 0<𝑋<𝑅. Then𝐷^𝛼_𝑡𝐷^𝛽_𝑡𝑓(𝑡)=𝐷_𝑡^𝛼+𝛽𝑓(𝑡) (2)holds for all 𝑡∈(0,𝑋), provided (1)𝛽<𝜆+1 and 𝛼 are arbitrary, or(2)𝛽≥𝜆+1 and 𝛼 are arbitrary and 𝑎_𝑘=0  for 𝑘=0,1,…,𝑚−1, where 𝑚=⌈𝛽⌉ (ceiling of the number).

Garra's main results are as follows.

Theorem Garra (Theorem 1.2 in [1]). Consider the following boundary value problem (BVP):𝐷_𝐿𝑥𝑓(𝑥,𝑡)=𝐷^𝛼_𝑡𝑓(),𝑓()(𝑡),𝑥,𝑡0,𝑡=𝑔 (3)in the half plane 𝑡>0, with analytic boundary condition 𝑔(𝑡) such that the conditions of Theorem 1.1 (Theorem M.-R.) are satisfied. The operatorial solution of BVP (3) is given by 𝑓(𝑥,𝑡)=^∞_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘(𝑘!)²𝑔(𝑡), (4)where 𝐷_𝐿𝑥 is the Laguerre derivative and 𝐷^𝛼_𝑡 denotes Caputo fractional derivative. On the basis of the previous result, Garra proved (Example 1 in [1]) that if 𝑔(𝑡)=𝑡^𝑚, 𝑚∈ℕ, then the analytic solution of the BVP (3) is given by 𝑓(𝑥,𝑡)=^∞_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘𝑡^𝑚(𝑘!)²=^[𝑚/𝛼]_𝑟=0Γ(𝑚+1)Γ()⋅𝑥𝑚+1−𝛼𝑟^𝑟𝑡^{𝑚−𝛼𝑟}(𝑟!)², (5)where  Γ(𝛼)=∫^∞₀𝑒^−𝑡𝑡^𝛼−1𝑑𝑡.

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 𝐷_𝑛𝐿𝑥∶=𝐷𝑥,…,𝐷𝑥𝐷𝑥𝐷 (containing 𝑛+1 ordinary derivatives) is called the 𝑛-order Laguerre derivatives and the 𝑛𝐿-exponential function is defined by 𝑒_𝑛(𝑥)∶=^∞_𝑘=0𝑥^𝑘(𝑘!)^𝑛+1. (6)
In [4], the following result is proved.

Lemma 2. Let 𝑎 be an arbitrary real or complex number. The function 𝑒_𝑛(𝑎𝑥) is an eigenfunction of the operator 𝐷_𝑛𝐿𝑥; that is, 𝐷_𝑛𝐿𝑥𝑒_𝑛=𝑎𝑒_𝑛(𝑎𝑥). (7)

For 𝑛=0, we have 𝐷_0𝐿∶=𝐷. Thus, (7) leads to the classical property of the exponential function 𝐷𝑒^𝑎𝑥=𝑎𝑒^𝑎𝑥.

Similarly, the spectral properties can be obtained [3] by using the general Laguerre derivatives 𝐷_1𝐿+𝑚𝐷=𝐷(𝑥𝐷+𝑚) (8) (here 𝑚 is a real or complex constant) or more generally the operator 𝐷(𝑥𝐷+𝑚)^𝑛=^𝑛_𝑘=0𝑛𝑘𝐷_𝑘𝐿𝑚^𝑛−𝑘 (9)(𝑛∈ℕ) and the corresponding eigenfunctions ^∞_𝑘=0𝑥^𝑘𝑘!(𝑘+𝑚)! (10) or ^∞_𝑘=0𝑥^𝑘𝑘!((𝑘+𝑚)!)^𝑛. (11)

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 𝐷^𝛼𝑓(𝑡)1∶=Γ()∫𝑚−𝛼^𝑡0𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝛼+1−𝑚}𝑑𝑑𝜏,𝑚−1<𝛼<𝑚_𝑚𝑓𝑑𝑡^𝑚,𝛼=𝑚, (12) where 𝑚∶=⌈𝛼⌉.

Lemma 4 (see [5]). Let 𝑓(𝑡)=𝑡^𝜆 for some 𝜆≥0. Then 𝐷^𝛼𝑓(𝑡)=Γ()0,𝑖𝑓𝜆∈0,1,…,𝑚−1,𝜆+1Γ()𝑡𝜆+1−𝛼^𝜆−𝛼,𝑖𝑓𝜆∈ℕ,𝜆≥𝑚𝑜𝑟𝜆∉ℕ,𝜆>𝑚−1, (13) where 𝑚=∶⌈𝛼⌉.

Lemma 5. Let 𝜆>0. Let 𝛼>0, 𝛽>0, and 𝛼+𝛽≤𝜆. Then 𝐷^𝛼+𝛽𝑡^𝜆=𝐷^𝛼𝐷^𝛽𝑡^𝜆(𝑡>0). (14)

Proof. This follows immediately from Lemma 4.

Remark 6. In general, for 𝛼,𝛽>0,  𝐷^𝛼𝐷^𝛽=𝐷^𝛼+𝛽 is not true. For example, 𝐷^0.6𝐷^0.6Γ(2)𝑡=Γ()𝑡0.8^−0.2, (15) but 𝐷^0.6+0.6𝑡=𝐷^1.2𝑡=0. (16)

Lemma 7. Suppose 𝛼>0, and let 𝑔(𝑡)=∑^∞_𝑘=0𝑎_𝑘𝑡^𝑘 have a radius of convergence 𝑅>0 and 0<𝑋<𝑅. If [𝛼𝑘]=⌈𝛼(𝑘+1)⌉−1 for some 𝑘∈ℕ, then 𝐷^(𝑘+1)𝛼𝑔(𝑡)=𝐷^𝛼𝐷^𝑘𝛼𝑔(𝑡). (17)

Proof. Let ⌈𝛼(𝑘+1)⌉=𝑚. Then 𝑚−1<𝛼(𝑘+1)≤0.
By lemmas 4 and 5, we have 𝐷^(𝑘+1)𝛼𝑔(𝑡)=𝐷^{(𝑘+1)𝛼∞}_𝑙=0𝑎_𝑙𝑡^𝑙=𝐷^{(𝑘+1)𝛼∞}_𝑙=𝑚𝑎_𝑙𝑡^𝑙. (18)
Since [𝛼𝑘]=⌈𝛼(𝑘+1)⌉−1=𝑚−1, together with Definition 3, we have 𝐷^𝛼𝐷^{𝛼𝑘𝑚−1}_𝑙=0𝑎_𝑙𝑡^𝑙=0, (19) and therefore (18) implies that 𝐷^(𝑘+1)𝛼𝑔(𝑡)=𝐷^𝛼𝐷^𝛼𝑘∞_𝑙=0𝑎_𝑙𝑡^𝑙=𝐷^𝛼𝐷^𝛼𝑘𝑔(𝑡). (20)
This completes the proof.

3. Main Results

We first study the following BVP in the plane 𝑡>0: 𝐷_𝑛𝐿𝑥𝑓(𝑥,𝑡)=𝐷^𝛼_𝑡𝑓(),𝑓()(𝑡).𝑥,𝑡0,𝑡=𝑔 (21)

Theorem 8. Let 𝑔(𝑡)=𝑡^𝜆 with 𝜆>0 and assume that 𝜆/𝛼 is a positive integer. Then the operatorial form solution of BVP (21) is 𝑓(𝑥,𝑡)=^𝜆/𝛼_𝑘=0𝑥^𝑘𝐷^𝛼𝑘𝑡^𝜆(𝑘!)^𝑛+1=∑^𝜆/𝛼_𝑘=0Γ(𝜆+1)Γ()⋅𝑥𝜆+1−𝛼𝑘(𝑘!)^𝑘𝑛+1𝑡^{𝜆−𝛼𝑘}. (22)

Proof. Let 𝜆/𝛼=𝑚. By Lemmas 4 and 5, we have 𝐷_𝑛𝐿𝑥𝑓(𝑥,𝑡)=^𝑚_𝑘=1𝑘^𝑛+1𝐷_𝑡^𝛼𝑘𝑡^𝜆(𝑘!)^𝑛+1=^𝑚_𝑘=1𝑥^𝑘−1𝐷_𝑡^𝛼𝑘𝑡^𝜆[(𝑘−1)!]^𝑛+1=^𝑚−1_𝑘=0𝐷_𝑡^𝛼(𝑘+1)𝑡^𝜆(𝑘!)^𝑛+1=𝐷^𝛼_𝑡^𝑚−1_𝑘=0𝑥^𝑘𝐷^𝛼𝑘𝑡^𝜆(𝑘!)^𝑛+1=𝐷^𝛼_𝑡^𝑚_𝑘=0𝑥^𝑘𝐷^𝛼𝑘𝑡^𝜆(𝑘!)^𝑛+1=𝐷^𝛼_𝑡𝑓(𝑥,𝑡). (23)
In the fifth previous equality, we use the fact that 𝐷^𝛼_𝑡𝑥^𝑚𝐷^𝛼𝑚𝑡^𝜆(𝑘!)^𝑛+1=𝐷^𝛼_𝑡𝑥^𝑚𝐷^𝜆𝑡^𝜆(𝑘!)^𝑛+1=0. (24)

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 𝐷_𝐿𝑥^[1/0.6]_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘(𝑘!)²𝑡=𝐷_𝑥𝑥𝐷_𝑥𝑡+𝑥𝐷_𝑡^0.6(1!)²𝑡=𝐷_𝑡^0.6(𝑡). (25)
On the other hand, we have 𝐷_𝑡^0.6[1/0.6]_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘(𝑘!)²𝑡=𝐷_𝑡^0.6𝑡+𝑥𝐷_𝑡^0.6(1!)²𝑡=𝐷_𝑡^0.6(𝑡)+𝑥𝐷_𝑡^0.6𝐷_𝑡^0.6𝑡. (26)
Note that by Remark 6, we have 𝐷_𝑡^0.6+0.6𝐷𝑡=0,_𝑡0.6𝐷_𝑡^0.61𝑡=Γ()𝑡0.8^−0.2≠0. (27)
Hence, 𝐷_𝐿𝑥^[1/0.6]_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘(𝑘!)²𝑡≠𝐷_𝑡^0.6[1/0.6]_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘(𝑘!)²𝑡. (28)

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 𝑓(𝑥,𝑡)=𝑒_𝑛𝑥𝐷^𝛼_𝑡𝑔(𝑡)=^∞_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘(𝑘!)^𝑛+1𝑔(𝑡). (29)

Proof. By Lemmas 2 and 7, we have 𝐷_𝑛𝐿𝑥𝑓(𝑥,𝑡)=𝐷_𝑛𝐿𝑥𝑒_𝑛𝑥𝐷^𝛼_𝑡𝑔(𝑡)=𝐷^𝛼_𝑡𝑒_𝑛𝑥𝐷^𝛼_𝑡𝑔(𝑡)=𝐷^𝛼_𝑡𝑓(𝑥,𝑡). (30)
This completes the proof.

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: 𝐷_𝑥𝑥𝐷_𝑥+𝑚^𝑛𝑓(𝑥,𝑡)=𝐷^𝛼_𝑡𝑓(),𝑓()(𝑡),𝑥,𝑡0,𝑡=𝑔 (31) in the half plane 𝑡>0, with analytic boundary condition 𝑔(𝑡) such that the conditions of Lemma 7 are satisfied. If [𝛼𝑘]=⌈𝛼(𝑘+1)⌉−1, (𝑘=1,2,…), then the operatorial solution of BVP (31) is given by 𝑓(𝑥,𝑡)=^∞_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘𝑘!((𝑘+𝑚)!)^𝑛𝑔(𝑡). (32)

Proof. Using spectral properties of Laguerre derivative, together with Lemma 7, we have 𝐷_𝑥𝑥𝐷_𝑥+𝑚^𝑛𝑓(𝑥,𝑡)=𝐷_𝑥𝑥𝐷_𝑥+𝑚^𝑛×^∞_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘𝑘!((𝑘+𝑚)!)^𝑛𝑔(𝑡)=^∞_𝑘=1𝑘(𝑘+𝑚)^𝑛𝑥^𝑘−1𝐷_𝑡^𝛼𝑘𝑘!((𝑘+𝑚)!)^𝑛𝑔(𝑡)=^∞_𝑘=0𝑥^𝑘𝐷^𝛼(𝑘+1)𝑘!((𝑘+𝑚)!)^𝑛+1𝑔(𝑡)=𝐷^𝛼∞_𝑘=0𝑥^𝑘𝐷^𝛼𝑘𝑘!((𝑘+𝑚)!)^𝑛𝑔(𝑡)=𝐷^𝛼𝑓(𝑥,𝑡). (33)
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)=𝑒^𝑡 and 𝛼=0.6. One has 𝐷_𝐿𝑥^∞_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘𝑒^𝑡(𝑘!)²=𝐷_𝑥𝑥𝐷_𝑥𝑥(1!)²𝐷_𝑡^0.6𝑒^𝑡+𝑥²(2!)²𝐷_𝑡^1.2𝑒^𝑡+𝑥³(3!)²𝐷_𝑡^1.8𝑒^𝑡+⋯=𝐷_𝑡^0.6𝑒^𝑡+𝑥(1!)²𝐷_𝑡^1.2𝑒^𝑡+𝑥²(2!)²𝐷_𝑡^1.8𝑒^𝑡+⋯. (34)
On the other hand, we have 𝐷^𝛼_𝑡^∞_𝑘=0𝑥^𝑘𝐷_𝑡^𝛼𝑘𝑒^𝑡(𝑘!)²=𝐷_𝑡^0.6𝑒^𝑡+𝑥(1!)²𝐷_𝑡^0.6𝑒^𝑡+𝑥²(2!)²𝐷_𝑡^1.2𝑒^𝑡+⋯. (35)
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, 𝐷_𝑡^0.6𝑒^𝑡=𝐷_𝑡^0.6𝑡1+𝑡+²2!+⋯=𝐷_𝑡^0.6𝑡𝑡+²=12!+⋯Γ()𝑡1.4^0.4+1Γ()𝑡2.4^1.4𝐷+⋯,_𝑡1.8𝑒^𝑡=𝐷_𝑡^1.8𝑡1+𝑡+²2!+⋯=𝐷_𝑡^1.8𝑡²+𝑡2!³=13!+⋯Γ()𝑡1.2^0.2+1Γ()𝑡2.2^1.2+⋯,andsoforth. (36)
Similarly, all terms in (35) are also positive, except that some terms contain negative exponent of variable 𝑡. For example, 𝐷_𝑡^0.6𝐷_𝑡^0.6𝑒^𝑡=𝐷_𝑡^0.6𝐷_𝑡^0.6𝑡1+𝑡+²2!+⋯=𝐷_𝑡^0.6𝐷_𝑡^0.6𝑡𝑡+²=12!+⋯Γ()𝑡0.8^−0.2𝐷+⋯,_𝑡0.6𝐷_𝑡^6.6𝑒^𝑡=𝐷_𝑡^0.6𝐷_𝑡^6.6𝑡1+𝑡+²2!+⋯=𝐷_𝑡^0.6𝐷_𝑡^6.6𝑡𝑡+²=12!+⋯Γ()𝑡0.8^−0.2+⋯,andsoforth. (37)
Thus, we conclude that 𝐷_𝐿𝑥^∞_𝑘=0𝑥^𝑘𝐷_𝑡^0.6𝑘𝑒^𝑡(𝑘!)²≠𝐷_𝑡^0.6∞_𝑘=0𝑥^𝑘𝐷_𝑡^0.6𝑘𝑒^𝑡(𝑘!)². (38)

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).