Research Article | Open Access
On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis
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.
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  focused on the application to the theory of linear viscoelasticity.
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 , using the Mellin transform, is the Fox function of the given parameters, is obtained as a solution to the particular problem In  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 :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 . 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 .
Corollary 7. If and , there exists such that
2.2. Some Bounds and Convergence
The assertions in this subsection were proved in .
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  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 ).
In  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
Now the purpose is to prove that the kernel verifies the hypothesis of Lemma 12.
The Mainardi function is a positive and decreasing function in . (see ).
(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 , 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 ).
Let be defined in (60) and let be any constant depending on , or . Thendue to Lemma  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 ) 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)