Research Article | Open Access
On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities
A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.
In this work, we focus on singularly perturbed linear partial q-difference differential equations which couple two categories of operators acting both on the time variable, so-called q-difference operators of irregular type and Fuchsian differential operators. As a seminal reference concerning analytic and algebraic aspects of q-difference equations with irregular type, refer , and for a far reaching investigation of Fuchsian ordinary and partial differential equations, refer .
Our equations are presented in the following mannerfor vanishing initial data , where are integers, represents the dilation map acting on time t for some real number , and stand for polynomials in . The main block is polynomial in the arguments , holomorphic in the perturbation parameter ϵ on a disc centered at 0 and in the space variable z on a horizontal strip of the form , for some . The forcing term is analytic relatively to and defines an entire function w.r.t t in with (at most) q-exponential growth (see (60), for precise bounds).
This paper is a natural continuation of the study  by Lastra and Malek and will share the same spine structure. Indeed, in , we aimed attention at the next problemfor vanishing initial data , where stand for polynomials in their arguments as above and is like the forcing term but with (at most) exponential growth in t. Under convenient conditions put on the shape of (2), we are able to construct a set of genuine bounded holomorphic solutions expressed as a Laplace transform of order k along a halfline and Fourier inverse integral in space z:where the Borel/Fourier map is itself set forth as a Laplace transform of order :where has (at most) exponential growth along and exponential decay in phase m on . The resulting maps are therefore expressed as iterated Laplace transforms following a so-called multisummability procedure introduced by Balser, see . These functions define bounded holomorphic functions on domains for a well-selected bounded sector at 0 and where is a set of sectors which covers a full neighborhood of 0 and is called a good covering (cf. Definition 6). Additionally, the partial maps share on a common asymptotic expansion with bounded holomorphic coefficients on . This asymptotic expansion turns out to be (at most) of Gevrey order with , meaning that we can single out two constants such thatfor all , all .
We plan to obtain a similar statement for the problem under study (1). Namely, we will construct a set of genuine sectorial solutions to (1) and describe their asymptotic expansions as ϵ borders the origin. We first notice that our main problem (1) can be seen as a q-analog of (2), where the irregular differential operator is replaced by the discrete operator . This terminology originates from the basic observation that the expression approaches the derivative as q tends to 1. Here, as mentioned in the title, we qualify the q-analogy as partial since the Fuchsian operator is not discretized in the process. This suggests that, in the building procedure of the solutions (that will follow the same guideline as in ), the classical Laplace transform of order k shall be supplanted by a q-Laplace transform of order k as it was the case in the previous work  of the author where a similar problem was handled. However, due to presence of the Fuchsian operator , we will see that a single q-Laplace transform is not enough to construct true solutions and that a new mechanism of iterated q-Laplace and classical Laplace transforms is required. Furthermore, we witness that this enhanced multisummability procedure has a forthright effect on their asymptotic expression w.r.t ϵ. Namely, the expansions in the perturbation parameter are neither of classical Gevrey order as displayed in (3) nor of q-Gevrey order as in  (meaning that has to be replaced by in the control term of (3)). The asymptotic expansions we exhibit present a double scale structure which has a q-Gevrey leading part with order and a subdominant tail of Gevrey order , that we call Gevrey asymptotic expansion of mixed order (cf. Definition 8). Such a coupled asymptotic structure has already been observed in another setting by Lastra et al. in . Indeed, we considered linear q-difference differential Cauchy problems with the shapefor suitably chosen analytic Cauchy dataand properly selected complex number with , where and are integers and B stands for a polynomial. When , the Fuchsian operator is responsible of the classical Gevrey part of the asymptotic expansion of the true solution which is shown to be of mixed order (in the sense of Definition 8) outside some q-spiral for some w.r.t t near 0, uniformly in z in the vicinity of the origin. Here, the solutions are expressed through a single q-Laplace transform and the contribution in the asymptotics emerges from a discrete set of singularities that accumulates at 0 in the Borel plane.
It is worthwhile mentioning that the approach which consists in building solutions by means of iterated q-Laplace and Laplace transforms stems from a new work by Yamazawa. In , he examines linear q-difference differential equations of the formfor the given holomorphic forcing term near the origin and where is a polynomial in with holomorphic coefficients w.r.t t near 0. Under special conditions on the structure of (4), he is able to construct a genuine solution obtained as a small perturbation of iterated truncated q-Laplace and Laplace of order 1 transforms of the iterated Borel and q-Borel transforms of a formal solution of (4). Furthermore, he gets in particular that has as asymptotic expansion of mixed order w.r.t t, uniformly in x near 0.
Notice that in our paper, the solutions are built up as complete iterated q-Laplace and classical Laplace transforms that are shown to be exact solutions of our problem (1). This is why the process we follow can actually be understood as an enhanced version of the multisummation mechanism introduced by Balser, see .
In a larger framework, this work is a contribution to the promising and fruitful realm of research in q-difference and q-difference-differential equations in the complex domain. For recent important advances in this area, we mention in particular the works by Tahara and Yamazawa [8–10]. Notice that the fields of applications of q-difference equations have also encountered a rapid growth in the last years. Some forefront studies in this respect are given, for instance, by [11–13] and references therein.
Now, we describe a little more precisely our main results obtained in Theorems 1 and 3. Namely, under convenient restrictions on the shape of (1) detailed in the statement of Theorem 1, we can manufacture a family of bounded holomorphic solutions on domains for a suitable bounded sector at 0, a strip of width , and belonging to a good covering in , which can be displayed as a q-Laplace transform of order k along a halfline and Fourier integral:
The q-Borel/Fourier map is itself shaped as a classical Laplace transform of order along :where has (at most) q-exponential growth of some order along (see (180)) and exponential decay in phase . In Theorem 3, we explain the reason for which all the partial functions share a common asymptotic expansion on with bounded holomorphic coefficients on , which turns out to be of mixed order . This last result leans on a new version of the classical Ramis—Sibuya theorem fitting the above asymptotics, which is fully expounded in Theorem 2.
Our paper is arranged as follows.
In Section 2, we recall the definition of the classical Laplace transform and its q-analog. We also put forward some classical identities for the Fourier transform acting on functions spaces with exponential decay.
In Section 3, we set forth our main problem (33) and we discuss the formal steps leading to its resolution. Namely, a first part is devoted to the inquiry of solutions among q-Laplace transforms of order k and Fourier inverse integrals of Borel maps W with q-exponential growth on unbounded sectors and exponential decay in phase leading to the first main integrodifferential q-difference equation (69) that W is asked to fulfill. A second undertaking suggests to seek for W, as a classical Laplace transform of suitable order of a second Borel map with again appropriate behaviour. The expression is then contrived to solve a second principal integro q-difference equation (81).
In Section 4, bounds for linear convolution and q-difference operators acting on Banach spaces of functions with q-exponential growth are displayed. The second key equation (81) is then solved within these spaces at the hand of a fixed point argument.
In Section 7, the existence of a common asymptotic expansion of the Gevrey type with mixed order is established for the solutions set up in Section 6. The decisive technical tool for its construction is detailed in Theorem 2.
2. Laplace Transforms of Order , q-Laplace Transforms of Order k, and Fourier Inverse Maps
Let be an integer. We remind the reader the definition of the Laplace transform of order as introduced in .
Definition 1. We set as some unbounded sector with bisecting direction and aperture and as a disc centered at 0 with radius . Consider a holomorphic function that vanishes at 0 and withstands the bounds. There exist and such thatfor all . We define the Laplace transform of of order in the direction d as the integral transformalong a halfline , where γ depends on T and is chosen in such a way that , for some fixed real number . The function is well defined, holomorphic, and bounded on any sector:where and .
If one sets , the Taylor expansion of , which converges on the disc , the Laplace transform has the formal seriesas Gevrey asymptotic expansion of order . This means that for all , two constants can be selected with the bounds:for all , all .
In particular, if represents an entire function w.r.t with bound (11), its Laplace transform does not depend on the direction d in and represents a bounded holomorphic function on , whose Taylor expansion is represented by the convergent series on .
Let be an integer and be a real number. At the next stage, we display the definition of the q-Laplace transform of order k which was used in a former work of Malek .
Let us first recall some essential properties of the Jacobi Theta function of order k defined as the Laurent series:for all . This analytic function can be factorized as a product known as Jacobi’s triple product formula:for all , from which we deduce that its zeros are the set of real numbers . We recall the next lower bound estimates on a domain, bypassing the set of zeroes of , from  Lemma 3, which are crucial in the sequel.
Lemma 1. Let . There exists a constant depending on and independent of such thatfor all satisfying , for all .
Definition 2. Let be a real number and be an unbounded sector centered at 0 with bisecting direction . Let be a holomorphic function, continuous on the adherence , such that there exist constants and withfor all . Let with . We put . We define the q-Laplace transform of order k of f in direction γ aswhere is a halfline in the direction γ.
The following lemma is a slightly modified version of Lemma 4 from .
Lemma 2. Let chosen as in Lemma 1 above. The integral transform defines a bounded holomorphic function on the domain for any radius , whereNotice that the value does not depend on such that due to the Cauchy formula.
The next lemma describes conditions under which the q-Laplace transform defines a convergent series near the origin.
Lemma 3. Let be an entire function with Taylor expansion fulfilling bound (19) for all . Then, its q-Laplace transform of order k, , does not depend on the direction and represents a bounded holomorphic function on with the restriction whose Taylor expansion is given by the convergent series .
Proof. The proof is a direct consequence of the next formulas:whenever and , where the last equality follows (for instance) from identity (4.7) from , for all .
We restate the definition of some family of Banach spaces mentioned in .
Definition 3. Let . We set as the vector space of continuous functions such thatis finite. The space endowed with the norm becomes a Banach space.
Finally, we remind the reader the definition of the inverse Fourier transform acting on the latter Banach spaces and some of its handy formulas relative to derivation and convolution product as stated in .
Definition 4. Let with and . The inverse Fourier transform f is given byfor all . The function extends to an analytic bounded function on the stripsfor all given .(a)Define the function which belongs to the space . Then, the next identity occurs.(b)Take and set as the convolution product of f and . Then, ψ belongs to , and moreover, for all .
3. Layout of the Principal Initial Value Problem and Associated Auxiliary Problems
We set as an integer. Let be integers. We set
We consider a finite set I of that fulfills the next feature,whenever and we set nonnegative integers withfor all .
Let , and be polynomials such thatfor all , all .
We consider a family of linear singularly perturbed initial value problemsfor vanishing initial data . Here, stands for a real number and the operator is defined as the dilation by q acting on the variable t through .
The coefficients are built in the following manner. For each , we consider a function that belongs to the Banach space for some , which depends holomorphically on the parameter ϵ on some disc with radius and for which one can find a constant with
We constructas the inverse Fourier transform of the map for all . As a result, is bounded holomorphically w.r.t ϵ on and w.r.t z on any strip for in view of Definition 4.
The presentation of the forcing term requires some preliminary groundwork. We consider a sequence of functions , for , that belongs to the Banach space with the parameters given above and which relies analytically and is bounded w.r.t ϵ on the disc . We assume that the next bounds,hold for all and given constants . We define the formal seriesfor some real number . We introduce the next Banach space.
Definition 5. Let and be real numbers. Let be an open unbounded sector with bisecting direction centered at 0 in . We denote the vector space of complex valued continuous functions on the adherence , which are holomorphic w.r.t u on and such that the normis finite. One can check that the normed space represents a Banach space.
Remark 1. The spaces above are faint modifications of the Banach spaces already introduced in the works of Dreyfus and Lastra [16–18].
The next lemma is a proper adjustment of Lemma 5 out of  to the new Banach spaces from Definition 5.
Lemma 4. Let be fixed as in (36). We take a number such thatLet be chosen as above. Then, the function belongs to the Banach space for any unbounded sector , any disc . Moreover, one can find a constant (depending on ) with
Proof. ound (36) implies thatAccording to the elementary fact that the polynomial admits its maximum value at , we deduce by means of the change of variable thatfor all . Therefore, we deduce thatwhich converges, provided that (39) holds, whenever .
We defineas the Laplace transform of w.r.t u of order in direction . Notice that two constants (depending on ) can be found such thatfor all . As a result, owing to bound (40) and the last part of Definition 1, we deduce that does not depend on the direction d and can be written as a convergent series:w.r.t τ near the origin. Now, we fix some real number such that . Then, one can sort a constant (depending on ) such thatfor all . This inequality is a consequence of the Stirling formula, which states thatas x tends to and from the existence of a constant (depending on ) withfor all . Consequently, it turns out that represents an entire function w.r.t τ such thatfor all . Furthermore, owing to bound (42), we know thatfor all , all . Henceforth, we get the next global boundsprovided thatfor all , , and .
Next, we setas the q-Laplace transform of w.r.t u of order k in direction d and Fourier inverse integral w.r.t m. We putfor all . We first provide bounds for this sequence of functions. Namely, we can get a constant (relying on ) withfor all , whenever and z belongs to the horizontal strip for some (see Definition 4). Owing to Lemma 3, we deduce that the function converges near the origin w.r.t T, where it carries the next Taylor expansion:for all and . In particular, the function is independent of the direction d chosen.
We now show that represents an entire function w.r.t T and supply explicit upper bounds. Namely, in accordance with (47), we obtainAgain, estimate (42) yieldsfor all , all , where is defined by . By gathering the two last above inequalities, the next global estimates can be figured out:for all , all and , provided thatLastly, we define the forcing term f as a time rescaled version of ,that represents a bounded holomorphic function w.r.t and and an entire function w.r.t t with q-exponential growth of order .
Throughout this paper, we are looking for time rescaled solutions of (33) of the formAs a consequence, the expression , through the change of variable , is asked to solve the next singular problem:At the onset, we seek for a solution that can be expressed as an integral representation via a q-Laplace transform of order k and Fourier inverse integral:where the inner integration is performed along a halfline in direction . Overall this section, we assume that the partial functions have at most q-exponential growth of order k on some unbounded sector centered 0 with bisecting direction d and belong to the Banach space mentioned in Definition 3, whenever . Precise bounds will be given later in Section 5. Here, we assume that .
Our aim is now the presentation of a related problem fulfilled by the expression . We first need to state two identities which concern the action of q-difference and Fuchsian operators on q-Laplace tranforms.
Lemma 5. The actions of the q-difference operators for integers and the Fuchsian differential operator are given by
Proof. The first identity is a direct consequence of the commutation formula (76) displayed in Proposition 6 from . For the second, a derivation under the integral followed by an integration by parts implies the sequence of equalities:from which the forecast formula follows since the map is assumed to possess a growth of q-exponential order k and vanishes at .
The application of the above identities (66) and (67) in a row with (26) and (28) leads to the first integrodifferential q-difference equation fulfilled by the expression as long as solves (64):We turn now to the second stage of the procedure. Solutions of this latter equation are expected to be found in the class of Laplace transforms of order since by construction owns this structure after (44). Namely, we take for granted thatwhere stands for a halfline with direction , which belongs to where represents an unbounded sector centered at 0 with bisecting direction d. Within this step, we assume that the expression belongs to the Banach space introduced in Definition 5, for all , where the constants , and α are selected accordingly to the construction of the forcing term .
The next lemma has already been stated in our previous work .
Lemma 6. For all integers , positive integers and can be found such that
With the help of this last expansion, equation (69) can be recast in the form
This last prepared shape allows us to apply the next lemma that repharses formula (8.7) p. 3630 from , in order to express all differential operators appearing in (72) in terms of the most basic one .
Lemma 7. Let be integers. Then, there exist real numbers , , such thatBy convention, we take for granted that the above sum vanishes when .
Indeed, by construction of the finite set I, we can represent the next integers in a specific way:where for all and . As a consequence, we can further expand the next piece of (72) in its final convenient form: