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 α ∈ (0, 1). Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when α ↗ 1 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.


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][2][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 timederivative by a fractional derivative of order > 0 in the Caputo sense: 0 ( , ) = 2 ( , ) , 0 < < ∞, 0 < < , 0 < < 1, where the fractional derivative in the Caputo sense of arbitrary order > 0 is given by where ∈ N and Γ is the Gamma function defined by Γ( ) = ∫ The interest on (1) has been in constant increase during the last 30 years. So many authors have studied it [4][5][6][7][8][9][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].
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 0 < < ∞. 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 W 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 fractionaldiffusion 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 ↗ 1 of the respective solutions will be done, recovering the respective solutions of the classical boundary-value problems when = 1 and (1) becomes the heat equation.

Preliminaries: Some Results about the Special Functions Involved
Definition 1. For every ∈ C, > −1, and ∈ R the Wright function is defined by Definition 2. For every ∈ C, 0 < ] < 1 the Mainardi function is defined by .
Note 1. This series are absolutely convergent over compact sets and so its derivatives are easy to calculate: = W ( , , + ) . where The coefficients , = 0, 1, . . . are defined by the asymptotic expansion valid for arg , arg(− ), and arg(1 − − ) all lying between − and and tending to infinity.
This theorem was proved in [16]. The next results follow.
Proof. Let us consider the function W(− , − /2, 1 − ). One has Taking Theorem 3 gives the equation Or equivalently, Taking = 1, there exists > 0 such that Then International Journal of Differential Equations Hence there exists two constants 0 and 1 depending on such that Finally, where is a polynomial function of degree less than or equal to 1. Therefore Corollary 7. If 0 < < 1 and ∈ R + , there exists > 0 such that

Some Bounds and Convergence.
The assertions in this subsection were proved in [17].

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 by where the function M /2 (⋅) is the Mainardi function defined in (10) and is a continuous bounded function in R + 0 (which guarantees that 1 is a solution; see the Cauchy problem in [19]).
In [17] it was proved that where W(⋅, − /2, 1) is the Wright function of parameters − /2 and 1 defined in (9), is a solution to the problem 0 ( , ) = 2 2 2 ( , ) 0 < < ∞, 0 < < , 0 < < 1, International Journal of Differential Equations 5 Then, we can assure that Taking into account Note 1 and Corollary 5, function V can be expressed as Let Then function V can be written as a convolution in thevariable: This new way of expressing V 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 K( − ) ( ) be a function that verifies the following conditions: where 0 1− is the fractional integral of Riemann-Liouville of order 1 − defined by Proof. Due to (46) and (48) Since (47) holds, (52) is equal to On the other hand, International Journal of Differential Equations Now the purpose is to prove that the kernel verifies the hypothesis of Lemma 12.
(ii) Hypothesis (46). Consider the kernel Applying Corollary 6, there exists > 0 such that, for all ∈ ( − , ), And this is an integrable function; in fact, making the substitution and considering the inequality it yields that It it easy to see that for any ∈ (0, 1), there exists ∈ N such that (65) is convergent. For example, if = 1/4 we can take = 10. 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 have (iii) Hypothesis (47). We have to prove that where Ω = {( , ) ∈ R 2 : ∈ (0, ), 0 ≤ ≤ }. 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 Let be defined in (60) and let be any constant depending on , , , or . Then due to Lemma 4.2 [17] and that ∈ (0, 1). Then, On the other hand, We proved in the previous item that Recalling that ∈ (0, 1), it yields that Then Proceeding like in item (ii), when checking Hypothesis (46), it can be proved that 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, Replacing (87) in (85), we have Therefore 2 ( , ) verifies the fractional diffusion equation.

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 problem Let us therefore consider the following auxiliary problem: wherẽis an even extension of .
This problem was solved in [22] and its solution is given by the following function: International Journal of Differential Equations
Due to the continuity of the Mainardi function, Corollary 6, and ∈ C(R + ) ∩ ∞ (R + ), the next equalities are true: Note that Applying Mean Value Theorem, it yields that Then, On the other side, Applying Lebesgue Convergence Theorem to the first integral, For the second integral we apply Mean Value Theorem as before. Then,  (119)

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: is a solution to (120).
In view of the results obtained in Section 3,

Conclusions
On the basis of the asymptotic behavior of some Wright functions and the existence of bounds for the Mainardi and the Wright function (− , /2, 1) in R + , two different initialboundary-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 ↗ 1 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.