Research Article | Open Access
On the Relationship between the Inhomogeneous Wave and Helmholtz Equations in a Fractional Setting
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 termwhere is, for simplicity, a function with compact support. Closely related to it is the one-dimensional Helmholtz equationOn 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  (for the wave equation in ) and are known under the name of radiation principles. For an extensive treatment see the classical work . 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-lineHere 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  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 , 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 equationThe classically known solutions of (5) are given in Theorem 4 below. Note that more recently related multidimensional equations have been solved, e.g., in  by the spectral method and in  in the form of an integral representation.
Thenand so the functionsatisfies 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 .
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 .
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,  for ) asAn 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, ] byAn 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 .
There is the following result on FDEs with the Riemann-Liouville derivative.
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
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 :
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 functionwherefor solves the FDEMoreover, satisfies the boundary conditionand
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. DenoteLet us first show that the integral (30) converges.
According to [8, Th.], for , , and so the bracket in (35) has an asymptotic expansionUsing that for and that , we see that (38) is as , . Thus the integral converges.
Concerning the term, integrate (36) by parts to getAnalogously 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 integraland 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 nowBy 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); thusClaimProof 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 functionequals by Theorem 4which is integrable along with respect to the measure , uniformly for in compact sets.
Step B. Using the beta-integral, for we findTherefore, 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 alsois 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 haveBy 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.
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 takeThe 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 integralin 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 thatwhere in the ordering of the terms we remembered that and and where
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 functionwherefor , solves the FDEMoreover, 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 computeand
Since as , , we can close the contour and calculate the integral (77) using the residue at :
Similarly, the inner integral of (76) becomeswhere we now introduce a cut in the -plane. Let ; then the integral representation [8, ] implieswhere 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 .
Interchanging the order of integration in (80) (legal because ), we obtain
Lemma 8. If is a smooth function with compact support, the solution from Lemma 7 with satisfies where .