Abstract and Applied Analysis
Volume 2019 (2019), Article ID 1483764, 9 pages
https://doi.org/10.1155/2019/1483764

Research Article

On the Relationship between the Inhomogeneous Wave and Helmholtz Equations in a Fractional Setting

Mateo Dulce1,2 and Alexander Getmanenko¹

¹Department of Mathematics, Universidad de Los Andes, Colombia
²Quantil S.A.S., Colombia

Correspondence should be addressed to Alexander Getmanenko; a.getmanenko@uniandes.edu.co

Received 24 November 2018; Revised 16 January 2019; Accepted 27 January 2019; Published 6 February 2019

Academic Editor: Paul W. Eloe

Copyright © 2019 Mateo Dulce and Alexander Getmanenko. 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 study convergence of solutions of a space and time inhomogeneous fractional wave equation on the quarter-plane to the stationary regime described by solutions of the Helmholtz equation.

1. Introduction and Main Result

Consider an inhomogeneous wave equation on the half-line with a time-periodic forcing term

where is, for simplicity, a function with compact support. Closely related to it is the one-dimensional Helmholtz equation

On the one hand, for any solution of the Helmholtz equation (2) the function solves the wave equation (1); on the other, there are pairs of solutions and of (1) and (2), respectively, such that as .

Results of this type go back at least to [1] (for the wave equation in ) and are known under the name of radiation principles. For an extensive treatment see the classical work [2]. The case of (1) will be recalled in Section 3.

In this paper we study a very simple question that clarifies, to us, the role of the fractional analogue, (5) below, of the Helmholtz equation (2): to what extent does a similar result hold in fractional settings? For concreteness, we chose an inhomogeneous fractional wave equation on a half-line

Here denotes the Riemann-Liouville fractional derivative, denotes the Caputo fractional derivative defined in Section 2, and is a fractional-harmonic function (see below), and . In various settings, (3) is a subject of active research; see [3] and related literature. The term fractional diffusion-wave equation is also used in the literature.

By a fractional-harmonic function we mean in this context a solution of the fractional differential equation (FDE)

where is a fixed constant. Taking a periodic , as in [4], does not seem to lead to a nice result.

By [5, ], as we can take any linear combination of Mittag-Leffler functions and defined in (14) below. This is quite natural since in the limiting case , , and . We will work out only the case ; an arbitrary linear combination leads to similar results.

Let now be some function satisfying the fractional Helmholtz equation

The classically known solutions of (5) are given in Theorem 4 below. Note that more recently related multidimensional equations have been solved, e.g., in [6] by the spectral method and in [7] in the form of an integral representation.

Then

and so the function

satisfies the fractional inhomogeneous wave equation (3).

Conversely, do solutions of (3) behave, for , as ? Our main result shows that the answer is “not quite.”

Before formulating the results, notice that, depending on the phase of the complex number , the function either is exponentially growing, or is bounded by a constant, or behaves as as ; see [8, Theorems , ]. Therefore any interesting result on the behavior of for ought to include information on asymptotic terms of order and smaller.

The main results of this paper are as follows.

Theorem 1. Let , , and , and let be a function with compact support on . Then there exist the following:
(a) A solution of the fractional wave equation (3) satisfying ,
(b) A solution of the fractional Helmholtz equation (5) satisfying ,
such that, as ,

where the symbol should be understood pointwise with respect to and the functions , , and are given in (66), (67), and (68).

Thus, if is such that , the right-hand side in (8) is of the same order as .

Theorem 1 is proven in Section 4. The proof follows, with complications specific to fractional calculus, the outline of the case of the wave equation, Section 3, and of the paper [1].

The case is much less natural. Indeed, a prototype for this case would be a heat equation with a space-harmonic source term and the limit as the space variable tends to infinity. We are not aware of any good result for such an equation which we would want to generalize to the fractional case. Nevertheless, if we impose a condition , we are able to prove the following statement; see Section 5.

Theorem 2. Let , , , and , and let be a function with compact support on . Then there exist the following:
(a) A solution of the fractional wave equation (3) satisfying ,

(b) a solution of the fractional Helmholtz equation (5) satisfying ,
such that, as ,

where the symbol should be understood pointwise with respect to and the functions , are given in (89) and (90).

The functions and constructed in the proofs of the above theorems are such that the corresponding fractional derivatives exist in the sense of definitions (11) and (13) below, in which the integral is the Lebesgue integral and the derivatives are taken in the sense of elementary calculus. In this paper we do not address the questions of uniqueness of solutions and appropriate functional spaces where such uniqueness would hold; therefore we phrase our results in the language “there exists a solution...” rather than “the solution....”

It would be curious to have an interpretation of the functions and .

2. Preliminaries

In this section we collect some definitions and results from fractional calculus in order to make the paper self-contained.

The Riemann-Liouville fractional derivative of order , , of a function is defined ([5, ] for and, in different notation, [9] for ) as

An additional subscript in will emphasize that the differentiation is with respect to .

Remark 3. Note that the usual formula for differentiation under the integral sign

is not applicable in (11) because the derivative of the integrand is not integrable at zero.

The Caputo fractional derivative of order , , of a function can be defined [5, ] by

An additional subscript in will emphasize that the differentiation is with respect to .

Some analytical conditions need to be imposed on the function to guarantee the existence of its fractional derivatives; see [5].

The following Mittag-Leffler functions treated in detail in [5, ] and [8, ] play the same role for FDEs as does the function in the theory of integer-order differential equations:

There is the following result on FDEs with the Riemann-Liouville derivative.

Theorem 4 (see [5, ], [9]). If , , , is an integrable function on for some , and is a constant, then the initial-value problem

has a unique solution

This theorem indicates the correct way to set up the initial conditions for equations with the Riemann-Liouville fractional derivative. By contrast, [5, ], one sets up the initial value problem for an FDE with Caputo derivative, say, in the usual way: , .

3. Case of the Classical Inhomogeneous Wave Equation

In this section we recall the case of the usual wave equation in order to motivate the fractional version of the argument and especially the formulas (30)-(31).

It is well-known (see, e.g., [10, Lecture ]) that the function

is a solution of the initial value problem

Define the Laplace transforms with respect to :

Then (17), after simple manipulations, yields

By the method of images we know that if is originally defined for and extended to the negative axis by , then from (17) solves the boundary-value problem on the half-line:

In this case (20) becomes

If the forcing term in (21) is periodic in time,

we get

On the other hand, the Helmholtz equation on the half-line

has a solution

Now let us understand the behavior of for large , pointwise with respect to , assuming . If , the Bromwich integral that expresses via (24) can be closed on the left, yielding that is simply a sum of residues:

Looking at (26) we recognize that

where is a solution of the Helmholtz equation above with and :

4. Case

This section contains the proof of Theorem 1. We begin by constructing a solution of the inhomogeneous fractional wave equation using the method of Laplace transform. Later on in Lemma 5 will be specialized to the Laplace transform of a Mittag-Leffler function.

Lemma 5. Let
(a) be a continuous function with compact support
(b) be an analytic function defined in estimated as as in that region
(c) be defined for in the sense of principal value
Then the function

where

for solves the FDE

Moreover, satisfies the boundary condition

and

Remark that since , the function (31) is a fractional generalization of (22). While the statement of the lemma is quite natural on the algebraic level, various analytic justifications need to be carried out.

Proof of the Lemma. Denote

Let us first show that the integral (30) converges.
According to [8, Th.], for , ,

and so the bracket in (35) has an asymptotic expansion

Using that for and that , we see that (38) is as , . Thus the integral converges.
Concerning the term, integrate (36) by parts to get

Analogously to (37), we see that the right-hand side of (39) is for , . As by assumptions of the lemma, the integral also converges.
Thus the definition of by (30) makes sense.
If , we can close the integration contour in (30) on the right and obtain . The decay of the integrand in (30) for is sufficiently fast to allow the differentiation under the integral sign; putting in the integral

and again closing the contour on the right, we obtain .
For the Bromwich integration contour in (30) can be replaced with the integration contour shown on Figure 1:

where now

By definition (11), since ,

The assumptions of Fubini’s theorem are clearly satisfied for interchanging the order of integration with respect to and (while keeping fixed); thus

Claim

Proof of the Claim. We have to work around the obstacle mentioned in Remark 3. We need to show that the first and the second -derivatives in the integrand of (44) are integrable with respect to .
Step A. The function

equals by Theorem 4

which is integrable along with respect to the measure , uniformly for in compact sets.
Step B. Using the beta-integral, for we find

Therefore, for each fixed value of and ,

(since the series converges uniformly with respect to the integration variables)

The integrand is regular enough to differentiate under the integral sign:

which is integrable along with respect to .
Step C. Similarly, using (48),

and (52) is also integrable along with respect to .
Step D. Adding the results of Steps B and C we conclude that also

is integrable along with respect to , uniformly with respect to on compact sets. Therefore two consecutive differentiations with respect to can be carried out under the integral sign in (44) and the claim is proven.
Resuming the Proof of Lemma 5. By the result of the claim and by Step in its proof, we have

By definition (13),

and similarly to [5, ]

Here we used the fact that convolutions interact with the Laplace integral in the usual way even if the contour of the Laplace integral is not rectilinear; see [11, Pré I.5].
Collecting the terms from (54) and (56), we obtain (32).
Finally, let us compute . With the same analytical details as above,

using Theorem 4.

Figure 1: The contour in the -plane.

Let us now subtract from the solution of the form (30) the solution of the type (7) and estimate the difference. Since we want in (3), we take in (32)

We will treat as belonging to the cut complex plane ; and let, for definiteness, be the principal branch, i.e., if . If then has only one root in the cut complex plane ; otherwise it has two roots and . If , one of the roots is located in the left half-plane .

As a solution of the Helmholtz equation (5) satisfying we take

The fact that is indeed a solution follows from Theorem 4; the form of was derived similarly to (31) by imitating (26) in the case of the wave equation. As we take the solution of the fractional wave equation constructed in Lemma 5. With these choices,

The integrand of (60) is analytic at , or , as the case may be. Therefore the only contribution to the integral comes from the discontinuity along the cut . To the jump of the integrand along that cut we apply the following.

Lemma 6 (generalized Watson’s lemma, [12, p.22]). Consider the integral

in the complex domain, where , and the path of integration is the straight line joining to . Suppose that the integral exists for some fixed and that, as along ,

where and . Then

as in for any in the interval .

An elementary calculation with power series shows that

where in the ordering of the terms we remembered that and and where

Now (8) follows from Lemma 6 and the formula

The proof of Theorem 1 is complete.

5. Case

In this section we prove Theorem 2, following the model of the previous section. The condition is imposed to make the function , for , decrease as , ; see [8, Theorems , ].

Lemma 7. Let be a continuous function with compact support.
Let be an analytic function defined in and bounded in that region; moreover is defined for in the sense of principal value. Then the function

where

for , solves the FDE

Moreover, satisfies the boundary conditions

Proof. Since as and , in the integral

the integrations can be carried out in arbitrary order; also, similarly to the proof of Lemma 5, one can perform fractional differentiations under the integral sign even without modifying the integration contour.

Next we assume that and work out the initial conditions of from (70) for ; namely, we compute

and

In the inner integral of (75) make the substitution and obtain the integral

where the integration contour is shown on Figure 2.

Figure 2: The contour in the -plane.

Since as , , we can close the contour and calculate the integral (77) using the residue at :

Similarly, the inner integral of (76) becomes

where we now introduce a cut in the -plane. Let ; then the integral representation [8, ] implies

where the contour is as on Figure 3 and avoids a large enough circle around the origin so as not to intersect the similarity image of the contour .

Figure 3: The contour .

Interchanging the order of integration in (80) (legal because ), we obtain

Closing the contour on the right (which is possible for ) we pick up the residue at , and (76) becomes

Let ; then in the numerator , and the integral equals (see [8, ])

Coming back to (76), we obtain

Putting together (78) and (84), we have proven the following.

Lemma 8. If is a smooth function with compact support, the solution from Lemma 7 with satisfies

where .

Let

By Theorem 4, satisfies the Helmholtz equation (5) and the boundary conditions . With the same choice of the contour as in the previous section,

and the only contribution to the integral on the right-hand side of (87) comes from the cut . We compute that

where

Using Lemma 6 and formula (69), we obtain (10). The proof of Theorem 2 is complete.

Data Availability

No data were used to support this study.

Disclosure

The research was performed as a part of Alexander Getmanenko’s Employment and Mateo Dulce’s Studies at the Universidad de Los Andes.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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