## Abstract

A new approach to solving problems of Wiener–Hopf type is expounded by showing its implementation in two concrete and typical examples from fluid mechanics. The new method adapts mathematical ideas underlying the so-called unified transform method due to A. S. Fokas and collaborators in recent years. The method has the key advantage of avoiding what is usually the most challenging part of the usual Wiener–Hopf approach: the factorization of kernel functions into sectionally analytical functions. Two example boundary value problems, involving both harmonic and biharmonic fields, are solved in detail. The approach leads to fast and accurate schemes for evaluation of the solutions.

## 1. Introduction

The Wiener–Hopf technique is a method first put forward [1] for the solution *f*(*x*) of the integral equation on the half-plane
*k*(*x*−*y*) is a given difference kernel and *g*(*x*) is a specified function for *x*>0; the essence of the method is to extend the given equation into *x*<0 to a similar one for which the ‘forcing’ on the right side for *x*<0 is unknown. Having defined an equation over the full line, it is natural to take a Fourier transform with respect to *x*, with *α* as the spectral parameter, and arrive at a typical Wiener–Hopf functional equation of the form
*H*_{−}(*α*) and *F*_{+}(*α*) are Fourier transforms taken only over the two half-lines *x*<0 and *x*>0, respectively, whereas *K*(*α*) is the usual Fourier transform of the given kernel. The Wiener–Hopf method relies on being able to factorize *K*(*α*) into a product of upper and lower analytic functions in the spectral plane.

The Wiener–Hopf method has found application in diffraction problems for acoustic, elastic and electromagnetic waves, crystal growth, fracture mechanics, flow problems, diffusion models, geophysical applications and mathematical finance [2–4]. An authoritative review article by Lawrie & Abrahams [3] gives a recent survey of the importance of the method, its history, its various applications and extensions and it serves as an Introduction to a special journal issue celebrating the 75th year anniversary of the appearance of the original article of Wiener & Hopf [1]. In the 1950s, a monograph by Noble [2] summarizing the rich possibilities of the method appeared and it remains, to this day, ‘the “Bible” of the Wiener–Hopf practitioner’ [3].

In this expository paper, we give details of an essentially different approach to problems amenable to the standard Wiener–Hopf technique, one that avoids what is generally accepted as the most challenging part of the usual method: the factorization of the transform of the kernel function into sectionally analytic functions. While, during its long and successful history, the method has seen a variety of extensions and generalizations, in its most basic form, ‘the method is ideally suited to solve two-part two- or three-dimensional boundary-value problems involving a governing equation (such as Laplace's or Helmholtz') with mixed boundary conditions along one infinite coordinate line’ [3].

The four main steps of our approach to problems of Wiener–Hopf type are as follows:

(1)

*Domain splitting*: for a problem domain involving boundary conditions of mixed type, find a convenient ‘splitting’ of the problem domain (domain decomposition) into distinct boundary value subproblems and solve each using the unified transform method.(2)

*Boundary conditions*: couple the resulting subproblems by using the*same*spectral parameter for each and by imposing appropriate continuity conditions on any common edges.(3)

*Spectral analysis*: analyse the spectral relations arising from the boundary conditions, together with the global relations, to identify special points in the spectral plane whereby information on a reduced set of unknown spectral functions can be determined.(4)

*Solution scheme and function representation*: identify the precise nature of the singularities occurring at boundary points where the boundary conditions change type and represent unknown boundary data in terms of specially tailored variables that incorporate those edge singularities. Solve for a reduced set of spectral functions, with the rest following by back-substitution into the spectral relations.

We share the view of Carrier *et al.* [4] that it is most instructive to see this programme in action for specific cases rather than described in general terms. Our two showcase problems are typical mixed-type boundary value problems arising in fluid mechanics. The particular problems here have been chosen because both have been solved previously in the literature, and on more than one occasion. The model problems involve shear flow past a periodic array of semi-infinite flat plates: if the shear flow is longitudinal, then the boundary value problem is for a harmonic field; if the shear flow is transverse a biharmonic field is relevant. Luchini *et al.* [5] were the first to use Wiener–Hopf techniques to solve both problems in 1991 in the physical context of shear flow over riblets; 10 years later, Jeong [6], who was motivated by porous media flows, solved the same boundary value problem also using the Wiener–Hopf method but with some technical differences in his approach. The availability of these two, different, prior approaches to the same problems using Wiener–Hopf methods will benefit the reader in affording at least two points of comparison of our new approach with the more traditional ones.

## 2. Longitudinal flow problem

Consider a steady longitudinal shear flow past an array of semi-infinite walls where the velocity **u** in Cartesian coordinates (*x*,*y*,*z*) has the form
*y*=*h*/2+*nh*, where *h*>0. By the periodicity in the *y* direction, it is enough to consider the domain *w*(*x*,*y*) satisfies Laplace's equation
*U* is the shear rate and λ has an interpretation as the longitudinal slip length. While *U* is externally specifiable, the value of λ is determined by the solution. On the other hand, as *w*(*x*,*y*)→0.

It is natural to introduce the usual complex variable *z*=*x*+i*y* and a complex potential function *q*(*z*), where
*χ*(*x*,*y*) is the harmonic conjugate to *w*(*x*,*y*). The flow is symmetric with the axis of symmetry *y*=0, and it is easy to argue that this implies that the complex potential and its Schwarz conjugate satisfy the condition

### (a) Domain splitting: left and right semi-strips

The first step in the approach is to split the domain in a natural way into left and right semi-strips separated by the common edge *x*=0,*y*∈[−*h*/2,*h*/2]. Figure 2 shows a schematic.

*Left semi-strip*: in the left semi-strip, we define the complex potential to be given by
*q*_{L}(*z*) is analytic in the left semi-strip and vanishes as

*Right semi-strip*: in the right semi-strip, the complex potential is defined to be
*q*_{R}(*z*) is analytic in the right semi-strip and vanishes as

### (b) Boundary conditions

*Left semi-strip*: for *x*<0, no-slip conditions on the two boundaries are imposed

*Right semi-strip*: for *x*>0, the symmetry of the flow implies the condition
*w*/∂*y*=0 and, therefore, we can write

It is easy to show by a local analysis that, at a transition point between a boundary condition of Dirichlet type (left semi-strip) to one of Neumann type (right semi-strip), a harmonic function exhibits a branch point singularity of order 1/2 (a square root) [7]. This fact will be used later in the analysis.

### (c) Spectral analysis

The next step is to use the representation for analytic functions in a semi-strip introduced by Fokas & Kapaev [8] (see also [9]) to find *q*_{L}(*z*) and *q*_{R}(*z*). It is known that *q*_{L}(*z*) can be represented by
*ρ*_{11}(*k*), *ρ*_{21}(*k*) and *ρ*_{31}(*k*) are defined by
*q*_{R}(*z*).

The challenge is to determine the six unknown spectral functions shown in figure 2. It is known [8,9] that they satisfy the following *global relations*:

Further relations between the spectral functions are available from the boundary conditions. We multiply the boundary condition (2.10) by *e*^{−ikz} and integrate along the lower boundary (*x*<0):

We multiply (2.14) by *e*^{−ikz} and integrate along the lower boundary (*x*>0) to find
*q*_{R}(+)≡*q*_{R}(i*h*/2) and *q*_{R}(−)≡*q*_{R}(−i*h*/2). From this relation, we immediately deduce that we must have
*ρ*_{12}(*k*) will have a singularity at *k*=0. Therefore,

Additional relations between the spectral functions arise from the requirement of continuity of the global solution across the common edge at *x*=0. The complex potentials are continuous at *x*=0 and this implies the condition

Finally, the no-slip condition on the two walls for *x*<0 implies that the solution of the right semi-strip problem must satisfy this condition at points *z*=±i*h*/2, i.e.

### (d) Solution scheme and function representation

From equation (2.21) and the fact that *ρ*_{11}(*k*) is analytic in the upper half-plane, we deduce that *ρ*_{31}(*k*) must vanish at solutions of 1+*e*^{kh}=0 in the upper half *k*-plane, that is, at points in the set

Similarly, (2.26) and the fact that *ρ*_{12}(*k*) is a lower analytic function together imply that *ρ*_{32}(*k*) must satisfy

The conditions on the spectral function *ρ*_{32}(*k*) at discrete points in the *k*-plane are illustrated in figure 3 and these clearly have different forms in the upper and lower-half spectral plane; these conditions are enough to determine *ρ*_{32}(*k*). It is precisely this asymmetry between the conditions on *ρ*_{32}(*k*) in the upper and lower-half spectral *k*-plane that reflects the need, using the traditional Wiener–Hopf approach, to factorize the associated kernel functions arising there. Our approach obviates the need for any such factorization, and the inherent spectral asymmetry in the problem now manifests itself differently. Once *ρ*_{32}(*k*) has been determined, the other unknown spectral functions follow by back substitution into the various spectral relations just derived.

At this point, we pause to remark that it should, in principle, be possible to proceed henceforth in a purely analytical fashion. Equations (2.33) and (2.34) provide the values of the entire function *ρ*_{32}(*k*) at an infinite sequence of points in the spectral plane, so function theoretic methods for the representation of entire functions should be applicable for its construction. We do not pursue this option here, however, and it is left as a possible avenue for future investigation.

Instead, we now show how to find *ρ*_{32}(*k*) by a highly accurate numerical scheme that properly accounts for known singularities of the solution at the corners of the semi-strip. (Moreover, this approach extends readily to other examples.) At *z*=±i*h*/2, we know that the problem admits square-root singularities; this suggests use of a specially tailored basis which will implicitly take them into account. A convenient option is to define the new complex variable *ζ* via the relation
*ζ* on the semicircle *ζ*=*e*^{iθ},*θ*∈[*π*/2,3*π*/2] and to write
*q*_{R} on the common boundary for some set of coefficients {*a*_{n}} to be found. This approach is especially suited to this problem, because the inverse function *ζ*=*ζ*(*z*) has precisely the same square-root singularities at *z*=±i*h*/2 as required of the solution, i.e. with *h*=2,
*z*=±i*h*/2=±*i*. It follows that

The sum (2.37) is truncated to include only terms *n*=−*N*,…,*N* for suitable *N* and an overdetermined linear system for the 2*N*+1 unknown coefficients {*a*_{n}} and slip length λ is solved by a least-squares method. This linear system comprises (2.31) together with (2.33) and (2.34) evaluated at sufficiently many points in *Σ*_{1} and *Σ*_{2} closest to the real *k*-axis (typically, we used twice as many equations as the number of unknowns). On solving this system for *U*=1, *h*=2, we show in table 1 the rapid convergence to the theoretical value

It turns out to be possible to bypass Wiener–Hopf and transform methods altogether and solve this particular problem by means of a construction based on conformal slit mappings (cf. [14–17]). The solution is
*h*=2 (note that the square-root branch points at *z*=±*i* are clearly seen in this explicit form of the solution). Because we have not found it in the literature details of our conformal geometrical derivation of (2.41) are given in the appendix. To confirm that the results in table 1 for λ properly reflect convergence of the method we have also used (2.41) to check other features of the transform solution. Jeong [6] also derived (2.41) by summing an infinite series generated by his Wiener–Hopf method.

## 3. Transverse flow problem

To show the versatility of the approach, we now consider the transverse flow problem in which the flow takes place in the (*x*,*y*)-plane and is independent of the perpendicular direction. Carrier *et al.* [4] also use a Stokes flow problem to illustrate the Wiener–Hopf technique. Now, the two-dimensional incompressible Stokes flow velocity field has the form
*ψ* is the streamfunction describing shear flow past an array of semi-infinite plates occupying *h*>0. By the periodicity in the *y*-direction, it is enough to consider the domain *U* and slip length λ. As *y*=±*h*/2 changes type at *x*=0: for *x*<0, the boundary conditions are those of no-slip; for *x*>0, we must impose that *p*=*u*=0, where *p* is the fluid pressure [6].

The streamfunction *ψ*(*x*,*y*) associated with a two-dimensional Stokes flow satisfies the biharmonic equation [18],
*z*=*x*+i*y*, the general solution to (3.3) is available in the form
*f*(*z*) and *g*(*z*) are two analytic functions in the fluid region [18]. It can be shown [18] that, for a fluid with unit viscosity,
*ω* is the fluid vorticity.

In the two-dimensional case, it is known [14–17] that at any boundary point *z*_{c} where the boundary condition changes from a no-slip condition to a no-shear condition there are square-root singularities where *f*(*z*) and *g*′(*z*) have the local behaviour
*f*_{0},*f*_{1/2},…,*g*_{0},*g*_{1/2},…} are some coefficients. The reader is referred to [14–17] for further discussion of closely related problems of no-slip/no-shear boundary value problems in Stokes flows.

The flow is antisymmetric about *y*=0; this implies that the Goursat functions satisfy the following conditions

### (a) Domain splitting: left and right semi-strips

*Left semi-strip*: the Goursat functions are given by
*μ* and where *f*_{L}(*z*), *g*′_{L}(*z*) are analytic in the fluid region and vanish as *f*_{L}(*z*) can be represented by
*ρ*_{11}(*k*), *ρ*_{21}(*k*) and *ρ*_{31}(*k*) are defined by
*f*_{L}(*z*) and *g*′_{L}(*z*) vanish at

*Right semi-strip*: the Goursat functions are given by
*f*_{R}(*z*), *g*′_{R}(*z*) are analytic in the fluid region and vanishing as *U* is externally specified but we expect *μ* and λ to be determined by the solution. Representations analogous to (3.9) and (3.11), albeit with different spectral functions as indicated in figure 5, can be written for *f*_{R}(*z*) and *g*_{R}′(*z*).

### (b) Boundary conditions

*Left semi-strip*: for *x*<0, we have no-slip conditions on the two boundaries

*Right semi-strip*: for *x*>0, we have
*f*′_{s}(*z*)]=0, we have

### (c) Spectral analysis

We now adopt the transform approach to general two-dimensional biharmonic boundary value problems originally expounded by Crowdy & Fokas [19], who considered problems in elastostatics as the example application (see also more recent work by Dimakos & Fokas [20]). Crowdy & Davis [21] have shown how to apply the method to Stokes flow problems in fluid mechanics and it is the latter formulation that we follow here.

First, we make some useful preliminary observations. Note that
*f*_{L}(+)≡*f*_{L}(i*h*/2) and *f*_{L}(−)≡*f*_{L}(−i*h*/2). Expressions (3.29) and (3.30) appear in the transform of the boundary conditions on the no-slip boundaries.

We now state the global relations. The spectral functions for the left semi-strip satisfy
*e*^{−ikz} and integrate along the lower no-slip boundary:
*e*^{−ikz} and integrate over the upper no-slip boundary

Addition of (3.36) and (3.38) gives
*e*^{−ikz} and integrate along the lower boundary:
*ρ*_{12}(*k*) and *ρ*_{22}(*k*). On substitution of (3.44) into the global relation (3.33), we find

We must impose continuity of velocity, pressure and vorticity across the common edge. This is equivalent to insisting that *f*(*z*) and *g*′(*z*) are continuous at *x*=0. The spectral form of these conditions can be written as

In addition, we insist that functions *f*(*z*) for left and right semi-strip problems are compatible at the corner points *z*=±i*h*/2: this can be written as
*x*<0 implies that the solution of the right semi-strip problem must satisfy this condition at points *z*=±i*h*/2, i.e.
*f*(*z*) and *g*′(*z*) of the right semi-strip problem. On substitution of (3.7) and (3.14), we find that

### (d) Solution scheme and function representation

Substitution of (3.49) and (3.52) into (3.40) gives
*ρ*_{32}(*k*), *μ*, λ. But, *ρ*_{11}(*k*) is upper analytic which implies that
*Σ*_{3} denotes the set of zeros of *ρ*_{32}(*k*) and

We now take advantage of the fact that *ρ*_{12}(*k*) and *ρ*_{32}(*k*) and *Σ*_{2} is the same set defined in (2.35). Similarly, equation (3.48) gives

Figure 6 shows a schematic of the points in the spectral *k*-plane where information on *ρ*_{32}(*k*) and *Σ*_{3} in the upper half-plane while two conditions (3.58) and (3.59) hold at the points in *Σ*_{2}. Again, this up–down asymmetric distribution of information in the spectral plane underlies the need for a kernel decomposition in the usual Wiener–Hopf method.

To summarize, we have manipulated the relations between the spectral functions derived from the boundary conditions, the global relations and the requirements of continuity at *x*=0 to produce a set of conditions to be satisfied by the two unknown spectral functions *ρ*_{32}(*k*) and

The known square-root form of the singularities in the Goursat functions at *z*=±i*h*/2 means that we should again use the variable *ζ* defined by (2.36) together with the representations
*z* on the slit [−i*h*/2,i*h*/2]. It can be shown that

By truncating the expansions (3.61) for *f*_{R}(*z*) and *g*′_{R}(*z*) as before, we formed an overdetermined linear system for the unknown coefficients {*c*_{n}}, {*d*_{n}} and the parameters *μ* and λ. It is noted that *two* conditions associated with equation (3.54) are included in the linear system; the known square-root singularity of *f*(*z*) and *g*(*z*) at point *z*=i*h*/2 implies that the second and third terms in the left-hand side of (3.54) are unbounded, and therefore one condition (independent of λ) forces cancellation of these singularities while a balancing of the regular terms gives an equation containing λ. Other equations in the linear system are found by evaluating (3.57), (3.58) and (3.60) at as many points in the sets *Σ*_{2} and *Σ*_{3} as needed.

Jeong [6] derived the following formula for λ:
*K*_{+}(*ζ*) is defined as
*ζ*_{n} are the roots of *M* terms. Clearly, the convergence of the scheme is impracticably slow if high accuracy is required. On the other hand, table 3 shows that only a moderate number of coefficients are required to achieve many digits of accuracy using the new approach expounded herein. It should be pointed out that the set *Σ*_{3} coincides with the eigenvalue set associated with the so-called Papkovich–Fadle eigenfunctions in a semi-strip. While our formulation does not involve consideration of these functions we anticipated that finding the solution to our linear system might be problematic owing to well-known ill-conditioning associated with these eigenfunctions [22]. However, as seen in table 3, no such difficulties were encountered using a least-squares approach.

We end by remarking that our overall approach to the tranverse problem is identical to that of the longitudinal problem—the only difference is that the biharmonic nature of the problem means that we have *two* spectral functions *ρ*_{32}(*k*) and

## 4. Discussion

The new approach to problems amenable to traditional Wiener–Hopf techniques described herein avoids the ‘unpleasant task’ [4] of factorizing a kernel function into sectionally analytic functions in the transform variable. The method uses a different spectral parameter than the usual Fourier transform variable of previous solution schemes [5,6]: here, we perform a *simultaneous* spectral analysis of the problems in both *x* and *y* variables in contrast to traditional Fourier transform approaches which use a spectral parameter associated only with the *x*-variable. (Crowdy & Davis [21] have given a direct comparison of the difference between these two approaches in the context of low Reynolds number flows in channels.) This notion of simultaneous spectral analysis lies at the moral heart of the unified transform method of Fokas and co-workers [9,23]. The idea of splitting the domain into subpolygons works, because the methods of [8,19], can be applied to harmonic and biharmonic fields in *arbitrary* simply connected polygons. This flexibility makes it clear that many more problems, in more complicated solution domains, are amenable to solution by the same method.

Moreover, the unified transform method has been applied to a wide range of partial differential equations, including the Helmholtz and modified Helmholtz equations [9]. Therefore, in combination with the general strategy put forward in this article, problems of Wiener–Hopf type arising for those equations should also be amenable to an analysis akin to that given here. It is interesting to note that a recent study of non-steady state heat conduction with composite walls [24] has several features in common with our approach.

Our approach reduces the spectral problem to simple linear systems that can be solved to high accuracy with strategic choices of basis representations that cater to any corner singularities inherent in the problem. With appropriate choices of the basis functions even very small systems can give rapid convergence. Traditional Wiener–Hopf methods involving kernel factorization might sometimes lead to explicit expressions for the solutions, which might be viewed as advantageous, but they can suffer from practical disadvantages of slow convergence. The Wiener–Hopf method for the transverse problem [6] generates expressions for the solutions to the biharmonic boundary value problem in terms of such slowly convergent sums (table 2) that, arguably, they are of dubious value if fast and accurate solution evaluation is required.

Fokas [9] and Fokas & Spence [23] have discussed general connections between the unified transform method and Wiener–Hopf techniques: here, we have given some concrete effectivizations thereof. Our work shows that even within this simultaneous spectral analysis framework it is important to combine it with strategic choices of functional representations and use of numerical methods; our choice of representation of unknown boundary data in terms of the Joukowski-type conformal slit mapping (2.36) was tailored to the known form of the corner singularities in our examples. In a similar vein, Smitheman *et al.* [12] have introduced a so-called *spectral collocation method* in which a spectral analysis of the boundary conditions and use of the global relations gives rise to a set of equations to determine the unknown boundary data. Their numerical experiments suggest that the method inherits the order of convergence of the basis used to expand the unknown functions, namely exponential for a polynomial basis such as Chebyshev, and algebraic for a Fourier basis. Our calculations here corroborate this conclusion. On the other hand, Fornberg & Flyer [13] have presented an alternative numerical approach based on Legendre expansions of the unknown boundary data on polygonal boundary segments. When the corner points are free of singularities the method gives exponential accuracy, but that fails when corner singularities are present. In the latter case, the authors show that including leading-order singular terms of known type improves accuracy. A crucial distinction between the work of Smitheman *et al.* [12] and Fornberg & Flyer [13] is that the former workers derive equations for the unknown spectral data by inspecting the spectral relations to find special points in the spectral plane where information on a reduced set of spectral functions can be found (as we have done here); the latter authors, on the other hand, simply evaluate the global relations—which relate *all* unknown spectral functions—at a set of essentially arbitrarily points in the spectral plane (chosen only to provide good numerical conditioning).

It is hoped that the general scheme showcased for our two concrete examples will point the way to generalization to more complicated problems of Wiener–Hopf type. Abrahams [25] has successfully used Padé approximant methods to obtain explicit exact factorizations of both approximate scalar and matrix kernels; the latter approach has allowed a number of long-outstanding problems to be solved recently [26–28]. It is a future challenge to understand how the methods given herein might be adapted to such generalized situations.

This paper has been inspired by the authors' interests in solving mixed boundary value problems for Stokes flows motivated by technological challenges arising in microfluidics and other low Reynolds number flow situations [14–17]. In this context, several problems arise where it is not clear how one might even attempt to apply traditional Wiener–Hopf methods; the approach of this paper, on the other hand, can be directly extended to these cases. The results of this ongoing work will be presented elsewhere.

## Acknowledgements

Both authors acknowledge financial support from a Research Grant from the Leverhulme Trust which supports a studentship for E. Luca. D.G.C. is partially funded by an EPSRC Established Career Fellowship.

## Appendix A. Conformal geometric approach to longitudinal problem

Consider the composed conformal mapping given by

The conformal mapping (A 1) (figure 7) takes the parametric *ζ* plane and transplants it to the physical *z* plane. More specifically, the first mapping in (A 2) takes the unit *ζ* disc to the whole complex plane in the *η* plane: the unit *ζ* circle and [−1,1] of the *ζ* plane are mapped to the real line in the *η* plane. A second mapping transplants the unbounded *η* plane to the unbounded *χ* plane, where the slits on the real axis of the *η* plane are rearranged on the real axis of the *χ* plane. Finally, the mapping *P*(*χ*) takes the unbounded *χ* plane to the fluid region in *z* plane.

Now, define a second sequence of conformal mappings shown in figure 8 by
*ζ*=0 corresponds to

Substitution into (A 4) gives
*h*=2 and *U*=*a*, we find
*z*↦−*z*+*i*.

- Received April 13, 2014.
- Accepted July 2, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.