Abstract
We consider the time-fractional derivative in the Caputo sense of order . Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in , two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.
1. Introduction
The one-dimensional heat equation has become the paradigm for the all-embracing study of parabolic partial differential equations, linear and nonlinear. A methodical development of a variety of aspects of this paradigm can be seen in [1–3].
This paper deals with two problems associated with the time-fractional diffusion equation, obtained from the standard heat equation by replacing the first-order time-derivative by a fractional derivative of order in the Caputo sense: where the fractional derivative in the Caputo sense of arbitrary order is given by where and is the Gamma function defined by .
The interest on (1) has been in constant increase during the last 30 years. So many authors have studied it [4–10] and, among the several applications that have been studied, Mainardi [11] focused on the application to the theory of linear viscoelasticity.
A comprehensive analysis of the Cauchy problem associated with this equation can be found in [12] and a physical meaning is discussed in [13].
The two initial-boundary-value problems considered areassociated with the Dirichlet and Newmann boundary conditions, respectively.
Some variants of problem (3) have already been solved. In [10], using the Mellin transform, is the Fox function of the given parameters, is obtained as a solution to the particular problem In [14] the two problems for the time-fractional diffusion equation are considered in two disjoint intervals for the spatial variable which cover the set . Here the following conditions are imposed: and for the particular case , which is the case of our interest, the solutions are presented, where is a particular constant of the problem and is the Wright function, which will be defined in the next section.
In both cases no complete mathematical proof that the obtained functions actually are solutions of the fractional-diffusion equation is presented. We propose here a different approach involving convolutions that allows us to achieve more general solutions to problem (3) and we also solve problem (4) for the Neumann boundary condition. Moreover, we provide in each case a rigorous proof that the proposed function is a solution of the considered problem. Finally we show how from given solutions to the fractional diffusion equation one can construct new ones that verify different boundary conditions.
The paper is presented as follows. Some useful properties about the behavior of Wright functions are given in Section 2. In Sections 3, 4, and 5 the two problems enunciated previously will be solved. At the end of Sections 3 and 5 the limit when of the respective solutions will be done, recovering the respective solutions of the classical boundary-value problems when and (1) becomes the heat equation.
2. Preliminaries: Some Results about the Special Functions Involved
Definition 1. For every , , and the Wright function is defined by
Definition 2. For every , the Mainardi function is defined by
Note 1. This series are absolutely convergent over compact sets and so its derivatives are easy to calculate: For the special case of the Mainardi function, we have
2.1. Asymptotic Behavior
The following asymptotic behavior for the Mainardi function was proved in [15]:where
Theorem 3. If , , and , , thenwhere The coefficients are defined by the asymptotic expansion valid for , , and all lying between and and tending to infinity.
This theorem was proved in [16]. The next results follow.
Corollary 4. The following limits hold
Corollary 5. The next limit holds
Corollary 6. If and , there exists such that where is a polynomial function of degree less than or equal to and .
Proof. Let us consider the function . One hasTaking Theorem 3 gives the equationOr equivalently,Taking , there exists such that Then
If , . If , . If , taking large enough so that if , it follows that Hence there exists two constants and depending on such that Finally,where is a polynomial function of degree less than or equal to .
Therefore
Corollary 7. If and , there exists such that
2.2. Some Bounds and Convergence
The assertions in this subsection were proved in [17].
Lemma 8. If , is a strictly decreasing positive function in .
Corollary 9. If ,
Corollary 10. If , is a positive and decreasing function in such that
Note 2. Note thatHence
Lemma 11. If and , (1),(2)
3. Solving the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis with Temperature-Boundary Condition
Let us consider problem (3). The principle of superposition is valid due to the linearity of the Caputo derivative. Then, solving problem (3) is equivalent to solving the two auxiliary problems: Problem (34) was solved in [18] and its solution is given bywhere the function is the Mainardi function defined in (10) and is a continuous bounded function in (which guarantees that is a solution; see the Cauchy problem in [19]).
In [17] it was proved thatwhere is the Wright function of parameters and defined in (9), is a solution to the problemThen, we can assure that is a solution to problem Taking into account Note 1 and Corollary 5, function can be expressed as Let Then function can be written as a convolution in the –variable:This new way of expressing leads us to propose the following function:as a solution to problem (35).
In order to prove this assertion, let us enunciate the following lemma.
Lemma 12. Let be a function that verifies the following conditions:Then where is the fractional integral of Riemann-Liouville of order defined by
Proof. Due to (46) and (48)NowSince (47) holds, (52) is equal to Substituting ,On the other hand, Hence
Now the purpose is to prove that the kernel verifies the hypothesis of Lemma 12.
(i) Hypothesis (45). Consider the integral We know that(see [20]).
The Mainardi function is a positive and decreasing function in . (see [17]).
is a bounded function in ; that is,Then (58) is convergent and (57) is in
(ii) Hypothesis (46). Consider the kernelApplying Corollary 6, there exists such that, for all , And this is an integrable function; in fact, making the substitution and considering the inequality it yields thatIt it easy to see that for any , there exists such that (65) is convergent. For example, if we can take .
Then, the first term of the sum (61) is bounded by an integrable function.
Let us consider the second term of sum (61). Making the substitution (63) and taking into account that the Mainardi function is a positive function, we haveNow, for any , there exists such that Then, using (59), it yields
(iii) Hypothesis (47). We have to prove that where . Or equivalently, Applying a similar reasoning like in the previous item, using Corollary 6, inequality (64), Corollary 9, and Tonelli’s theorem (see [21], page 55), the following assertions are true: Taking small according to Corollary 6,Now, note that and that (it is a consequence of Lemma from [17]).
Let be defined in (60) and let be any constant depending on , or . Thendue to Lemma [17] and that . Then, On the other hand, We proved in the previous item that Recalling that , it yields thatThen Proceeding like in item (ii), when checking Hypothesis (46), it can be proved that Finally, (79) and (80) yield From (75) and (81), Tonelli’s theorem holds and
(iv) Hypothesis (48). Let us prove that Note that, due to (59) and (60), Finally, we can apply Lemma 12 to kernel (44). Then,From Corollary 5, . Then Caputo derivative commutes (see equation () of [9]) and Now, recalling that is a solution of problem (40), it results in the following:Replacing (87) in (85), we haveTherefore verifies the fractional diffusion equation.
Proposition 13. The following limits hold: (1)(2)
Proof. Taking in (59), it yields .
Making the substitution (63) and applying Corollary 9, we have It is a consequence of applying substitution (63).
Let us check now the border conditions.
(i) Consider .
(ii) Consider .
Note thatApplying Proposition 13 to the second member of the sum, Let The next goal is to prove that .
Let be determined. ConsiderApplying Corollary 9, Now, is continuous in . Then, for given, there exists such that if . Using this fact and making the substitution (63),From (95) and (96), it results that , for every . Then, and we can assure that
Theorem 14. Let be a continuous bounded function in and let be a continuous function in . Thenis a solution to problem
Theorem 15. The limit when of the solution to problem is the classical solution to the analogous problem when and we recover the heat equation
Proof. Let be the solution of problem (102) (see [1]) and let be the solution of problem (101) given by Theorem 14, Applying Lebesgue Convergence Theorem and Lemma 11,
4. Solving the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis with Null Flux-Boundary Condition
In this section, we consider the problemLet us therefore consider the following auxiliary problem:where is an even extension of
This problem was solved in [22] and its solution is given by the following function: The next goal is to prove that Due to the continuity of the Mainardi function, Corollary 6, and , the next equalities are true:Note thatApplying Mean Value Theorem, it yields thatThen,On the other side,Applying Lebesgue Convergence Theorem to the first integral, For the second integral we apply Mean Value Theorem as before. Then,From (113) and (116),
Theorem 16. Let be a continuous bounded function in ; then is a solution to problem
5. The Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis with Flux-Boundary Condition
As in the previous sections two auxiliary problems are considered: From Theorem 16, is a solution to (120).
In view of the results obtained in Section 3, is proposed as a solution to problem (121).
Lemma 17. Let be a solution of the time-fractional diffusion equation such thatThen is a solution to the time fractional diffusion equation.
Proof. From (124), is well defined. The next equalities are valid due to (125), (126), and (127):
It can be proved that (123) is under the hypothesis of Lemma 17.
Regarding the border conditions:
(i) Observing thatLebesgue Convergence Theorem can be applied and
(ii) From (98),
Theorem 18. Let be a continuous bounded function in and let be a continuous function in . Thenis a solution to problem
Theorem 19. The limit when of the solution to problem is the classical solution to the analogous problem when and we recover the heat equation
Proof. It can be seen in [1] that is a solution to problem (135).
Let be a solution to problem (134) given by Theorem 18, Applying Lebesgue Convergence Theorem, Lemma 11, and Fubini’s Theorem,
6. Conclusions
On the basis of the asymptotic behavior of some Wright functions and the existence of bounds for the Mainardi and the Wright function in , two different initial-boundary-value problems (with Dirichlet and Neumann boundary conditions, resp.) for the time-fractional diffusion equation on the real positive semiaxis plane were solved. In each case, certain conditions must be verified for the data to obtain the solution and the convergence of this solution when was analyzed, recovering the classical solutions of the respective boundary-value problems corresponding to the heat equation on the real positive semiaxis.
It would be interesting to combine these obtained results for the two initial-boundary-value problems with numerical methods that approximate their solutions. Different authors have focused their work on fractional numerical methods, for example, [23], considering explicitly the time-fractional diffusion equation or also [24, 25] providing different approaches to these kind of problems. Taking into account that the explicit expressions of the solutions provided in this work are manageable and of low complexity, they might help to check if new numerical algorithms actually converge to the desired solution of the problem and at which rate.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This paper has been sponsored by the Project ING495 “Estudio de Diversos Problemas con Ecuaciones Diferenciales Fraccionarias” from Universidad Nacional de Rosario, Argentina.