International Journal of Differential Equations
Volume 2015 (2015), Article ID 439419, 14 pages
http://dx.doi.org/10.1155/2015/439419

Research Article

On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis

D. Goos,1,2 G. Reyero,¹ S. Roscani,1,2 and E. Santillan Marcus1,3

¹Departamento de Matemática, FCEIA, Universidad Nacional de Rosario, Pellegrini 250, S2000BTP Rosario, Argentina
²CONICET, Departamento de Matemática, FCEIA, Universidad Nacional de Rosario, Pellegrini 250, S2000BTP Rosario, Argentina
³Departamento de Matemática, FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina

Received 10 July 2015; Accepted 31 August 2015

Academic Editor: Nasser-Eddine Tatar

Copyright © 2015 D. Goos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 are

associated 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 , , then

where

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 has

Taking

Theorem 3 gives the equation

Or 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 that

Hence

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 by

where 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 that

where is the Wright function of parameters and defined in (9), is a solution to the problem

Then, 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)

Now

Since (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 kernel

Applying 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 that

It 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 have

Now, 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 . Then

due to Lemma [17] and that . Then,

On the other hand,

We proved in the previous item that

Recalling that , it yields that

Then

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 have

Therefore 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 .

From Proposition 13 and (60),

(ii) Consider .

Note that

Applying Proposition 13 to the second member of the sum,

Let

The next goal is to prove that .

Let be determined. Consider

Applying 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 . Then

is 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 problem

Let 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 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,

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 that

Then 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 that

Lebesgue 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 . Then

is 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.

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