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ON THE SOLUTION OF WIENER–HOPF PROBLEMS INVOLVING

NONCOMMUTATIVE MATRIX KERNEL DECOMPOSITIONS∗

I. DAVID ABRAHAMS†

SIAM J. APPL. MATH.

Vol. 57, No. 2, pp. 541–567, April 1997

c ? 1997 Society for Industrial and Applied Mathematics

013

Abstract. Many problems in physics and engineering with semi-infinite boundaries or interfaces

are exactly solvable by the Wiener–Hopf technique. It has been used successfully in a multitude of

different disciplines when the Wiener–Hopf functional equation contains a single scalar kernel. For

complex boundary value problems, however, the procedure often leads to coupled equations which

therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel

into a product of two functions, one analytic in an upper region of a complex (transform) plane and

the other analytic in an overlapping lower half-plane. This is straightforward for scalar kernels but

no method has yet been proposed for general matrices.

In this article a new procedure is introduced whereby Pad´ e approximants are employed to obtain

an approximate but explicit noncommutative factorization of a matrix kernel. As well as being

simple to apply, the use of approximants allows the accuracy of the factorization to be increased

almost indefinitely. The method is demonstrated by way of example. Scattering of acoustic waves

by a semi-infinite screen at the interface between two compressible media with different physical

properties is examined in detail. Numerical evaluation of the approximate factorizations together

with results on an upper bound of the absolute error reveal that convergence with increasing Pad´ e

number is extremely rapid. The procedure is computationally efficient and of simple analytic form

and offers high accuracy (error < 10−7% typically). The method is applicable to a wide range of

initial or boundary value problems.

Key words.

Pad´ e approximants

Wiener–Hopf technique, matrix Wiener–Hopf equations, scattering, acoustics,

AMS subject classifications. 30E10, 41A21, 45E10, 73D25, 76Q05, 78A45

PII. S0036139995287673

1. Introduction and difficulties in solving matrix Wiener–Hopf equa-

tions. The Wiener–Hopf technique has, since its invention in 1931 [34], proved an

immensely important method in mathematical physics. It offers one of the very few

approaches to obtaining an exact solution to a class of integral equations. These

equations are of either first or second kind, are defined over a half-line, and have

a difference kernel. An enormous variety of physically important problems can be

cast into this form of equation, and all have the characteristic feature of boundary

conditions defined on semi-infinite planes. Fields of application include diffraction of

acoustic [7], elastic [30], and electromagnetic waves; fracture mechanics; flow prob-

lems; diffusion models; and geophysical applications [16], to name but a few. The

solution method, although initially proposed to follow via construction of the inte-

gral equation, now usually proceeds directly from the boundary value problem to an

equation in the complex plane defined in an infinite strip [23]. This is accomplished

by Fourier transformation (or other appropriate transform) or Green’s theorem, and

the essential physical details are manifested in the singularity structure of the Fourier

transformed difference kernel often called the Wiener–Hopf kernel, K(α) (Noble [29]),

where α is the transform parameter. This kernel is usually arranged to be singular-

ity free in a finite-width strip of the complex α-plane which contains the real line as

∗Received by the editors June 14, 1995; accepted for publication (in revised form) February 26,

1996.

http://www.siam.org/journals/siap/57-2/28767.html

†Departmentof Mathematics, KeeleUniversity,

(i.d.abrahams@keele.ac.uk).

Keele,StaffordshireST55BG,UK

541

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I. DAVID ABRAHAMS

|α| → ∞. Further, this strip need not necessarily be straight (i.e., enclose the whole

real line) and is denoted henceforth as D. Solution of the Wiener–Hopf equation is

straightforward once K(α) is decomposed into a product of two functions K+(α),

K−(α), where K+(α) is regular and zero free in the region above and including the

strip D, denoted D+, and K−(α) is regular and zero free in the region below and

including D, denoted D−.

as |α| → ∞ in D±, respectively. Cauchy’s integral theorem provides a convenient

method for obtaining the explicit sum factorization of a function, e.g., of the form

g(α) = g+(α) + g−(α), where ± denotes the above analyticity properties, and so by

exponentiation we deduce (see Theorem C of Noble [29])

Further, K±(α) must have at worst algebraic growth

K±(α) = exp[g±(α)] = exp

?±1

2πi

?∞

−∞

logK(ζ)

ζ − α

dζ

?

,

(1)

where ζ is a contour in D and α lies above (below) ζ for K+(α) (K−(α)). Sometimes

a limiting procedure is necessary to ensure convergence of these integrals.

For problems in which K(α) is a scalar function, then, apart from some technical

difficulties associated with computing integrals of the above form (see Abrahams and

Lawrie [5]), the Wiener–Hopf procedure goes through without a hitch. However, in

more complex boundary value problems, it is not unusual to obtain a matrix functional

Wiener–Hopf equation with a square matrix kernel K(α). This presents two potential

obstacles to allow the solution scheme to go through. The first, which it is the aim

of this article to tackle, is the factorization procedure (1). For any matrix function,

K(α), the logarithm G(α) = log(K(α)) can be defined, for example by its power

series expansion log[I + (K(α) − I)] = K(α) − I −1

identity matrix, or otherwise. Hence the sum split of G(α) is accomplished by the

Cauchy-type integrals acting on each element of the matrix. However, the final step

to obtain the product factors, i.e.,

2(K(α) − I) + ···, where I is the

exp{G+(α) + G−(α)} = exp{G+(α)}exp{G−(α)},

is true if and only if G+(α) and G−(α) commute (Heins [20]). For general matri-

ces K(α) this will not be the case, and although no procedure is presently available

for tackling such arbitrary matrix Wiener–Hopf problems, Gohberg and Krein have

proved the existence of product factors in all cases [17]. Fortunately, many physically

interesting problems naturally give rise to kernels which, although intrinsically matrix

in form, have a commutative factorization or can be reworked (by premultiplication

and/or postmultiplication by suitable matrices) into such form. Khrapkov [26, 27], in

articles concerned with the stresses in elastostatic wedges with notches, was the first

author to express the commutative factorization in a form which indicates the subal-

gebra associated with this class of kernels. Many other authors have also examined

the commutative cases, and, in particular, reference should be made to the ingenious

approach to the simplest nontrivial commutative example by Rawlins [31], the direct

scheme of Daniele [14], and the Hilbert problem technique of Hurd [21]. The latter

approach, which transforms the factorization problem to a pair of uncoupled Hilbert

problems defined on a semi-infinite branch cut, is generalizable to an interesting range

of cases [32], one example of which is explored in [1]. It is generally accepted that

the generalized range of kernels susceptible to Hurd’s method and Khrapkov’s class

of commutative matrices suitably multiplied by entire matrices are in fact equivalent

[24].

(2)

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MATRIX WIENER–HOPF FACTORIZATION

543

The second difficulty associated with matrix kernels is that, although decomposi-

tion into matrix factors with overlapping domains of regularity may be achieved, the

growth of such factors could be exponential in the region of analyticity. This prohibits

the application of Liouville’s theorem at a later stage of analysis, and so a solution to

the matrix Wiener–Hopf equation cannot be obtained. Various techniques have been

proposed in the literature to overcome this difficulty, including a general scheme by

the author (Abrahams and Wickham [7]) and one specific to matrices of Khrapkov

form (Daniele [15]). A full bibliography illustrating the relevant physical applications

can be found in [6].

The aim of this article is to demonstrate a procedure for obtaining noncommu-

tative matrix factors which have algebraic growth.

first rather than the second difficulty outlined above. As already mentioned, many

useful and important problems have, or can be cast into, commutative factorization

form, but the most important, and by far the more diverse, range of physical cases

are intrinsically noncommutative. Matrix Wiener–Hopf equations arise in all areas

of physics, including those mentioned at the start of this introduction. Usually the

matrices yield noncommutative factorizations because they contain multiple branch

cut functions, infinite sequences of poles or zeros, or a combination of both of these

singularity structures. In certain special cases the factorization procedure can be

completed by subtraction of an infinite family of poles from both sides of the Wiener–

Hopf equation. The coefficients of each pole term are then found from the solution of

an infinite algebraic system of equations (see Idemen [22], Abrahams [2], Abrahams

and Wickham [8], and references quoted therein). The objective of this article is to

illustrate a new procedure for obtaining explicit but approximate noncommutative

matrix factors. The method should be applicable to many kernels of this previously

unsolved matrix class and is based on the replacement of certain components of the

matrix function by Pad´ e approximants. This allows the level of approximation to be

increased to very high accuracies while at the same time offering a surprisingly simple

factorization form. An alternative technique has recently been proposed by Wickham

[33] which, to the author’s knowledge, is the first to offer a computational route to fac-

torize noncommutative kernels. This involves the derivation of certain exact coupled

integral equations from which rigorous bounds on the convergence of such solutions

can be derived. However, the method described herein is a good deal simpler to use

in practice and appears much more direct than that suggested by Wickham. The

relationship between the two methods is discussed further in the concluding remarks.

The clearest and most concise method for illustrating the Pad´ e factorization pro-

cedure is to examine one specific example. We take a typical physical situation which

gives rise to a noncommutative kernel, namely a semi-infinite barrier at an interface

between two compressible media (Figure 1). This yields the simplest and therefore

the canonical matrix kernel and so will omit any extraneous or obfuscating technical

details. The plan of the paper is as follows. In section 2 the interfacial boundary value

problem is stated mathematically, and in section 3 it is reduced to a matrix Wiener–

Hopf equation. The explicit kernel factorization, employing Pad´ e approximants for

the approximate matrix kernel, is performed in section 4, and a discussion of the

derivation of an ansatz used in that section can be found in Appendix B. The formal

solution to the scattered field is given in section 5, and in section 6 the pressure in

the far-field is derived asymptotically for the upper half-plane. Section 6 also presents

numerical results and timings for different approximant numbers. A summary of the

important properties of Pad´ e approximants is written in Appendix A, and concluding

remarks are made in section 7.

Thus, we concentrate on the

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I. DAVID ABRAHAMS

FIG. 1. The physical configuration.

2. The boundary value problem. As discussed in the introduction, we will

study a two-dimensional problem involving a simple planar interfacial boundary be-

cause it offers the simplest form of a nontrivial matrix Wiener–Hopf kernel (with two

branch cut pairs). Defining (x,y) as a two-dimensional Cartesian coordinate system,

a semi-infinite screen lying along x < 0,y = 0 is placed between two different inviscid

compressible media. The upper medium has sound speed c1, say, and lower c2. Along

the extension of the screen (x > 0,y = 0) it is assumed that they have equal pressures

and normal velocities, and if the continua are fluids, then they do not mix. Thus, it

could be envisaged that on this line (x > 0) the two fluids are separated by a thin

membrane of negligible mass and surface tension. On the physical screen (x < 0), the

upper surface (y = 0+) is chosen to be perfectly soft (vanishing pressure p), whereas

the underside (y = 0−) is taken to be perfectly rigid (vanishing normal velocity v).

Figure 1 illustrates this geometry.

The governing equations for material motions can be expressed in terms of velocity

potentials as

?∂2

?∂2

where the pressure and velocity are related to these potentials via

p = −ρ1∂Φ1

∂x2+

∂2

∂y2−1

∂2

∂y2−1

c2

1

∂2

∂t2

∂2

∂t2

?

?

Φ1= 0,y ≥ 0,

(3)

∂x2+

c2

2

Φ2= 0,y ≤ 0,

(4)

∂t,

y ≥ 0;

p = −ρ2∂Φ2

∂t,

y ≤ 0,

(5)

u = (u,v) = ∇Φj,j = 1,2 for y > 0,y < 0.

(6)

Here ρ1,ρ2and c1,c2are the density and sound speeds in media 1, 2, respectively.

For brevity, only the case of scattering by plane waves, incident from below, will

be examined in this article, and the propagation speed c2is taken to be less than c1.

All other cases can be dealt with in an identical fashion. It is the purpose of this

paper to demonstrate the efficacy of the solution method, not to provide exhaustive

details of the scattered field. Writing the incident waves (satisfying (4)) with angular

frequency ω as

Φinc(x,y;t) = ?{eiω[(xcosθ0+y sinθ0)/c2−t]},

(7)

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MATRIX WIENER–HOPF FACTORIZATION

545

then, for steady-state motions, the total velocity potential is given by

Φ1(x,y;t) = ?{ψ1(x,y)e−iωt},

(8)

Φ2(x,y;t) = Φinc(x,y;t) + Φinc(x,−y;t) + ?{ψ2(x,y)e−iωt}.

Note that the reflected wave Φinc(x,−y;t) has been written explicitly for convenience,

and ψ1(x,y) and ψ2(x,y) are the scattered potentials which must be purely outgoing

as (x2+ y2)Substituting (8) and (9) into the governing equations and

boundary conditions gives

?∂2

?∂2

ψ1(x,0) = 0,∂ψ2

∂y(x,0) = 0,

(9)

1

2 → ∞.

∂x2+

∂2

∂y2+ω2

∂2

∂y2+ω2

c2

1

?

?

ψ1= 0,y ≥ 0,

(10)

∂x2+

c2

2

ψ2= 0,y ≤ 0,

(11)

x ≤ 0,

(12)

ρ1

ρ2ψ1(x,0) − ψ2(x,0) = 2eiωx

In order to exclude extraneous sound sources, it must also be insisted that

c2cosθ0,

∂ψ1

∂y(x,0) −∂ψ2

∂y(x,0) = 0,x ≥ 0.

(13)

ψ1,ψ2contain only outgoing waves as

?

x2+ y2→ ∞; (14)

ψ1,ψ2are bounded everywhere, including the origin (0,0).

(15)

The boundary value problem is fully specified by statements (10)–(15).

3. Reduction to a Wiener–Hopf equation. To solve the boundary value

problem defined in the previous section it is useful to introduce Fourier transforms.

As there is no intrinsic length scale in the model, it is advantageous first to scale x

and y on the length c1/ω, viz.

x = Xc1/ω,y = Y c1/ω,

(16)

and to write the sound speed ratio as

c1/c2= k.

(17)

As stated earlier, and with no real loss of generality, c1is taken to be greater than

c2, which means that k > 1. (If this is not the case, then x and y could be scaled on

c2/ω, k redefined as c2/c1in order to force it to be larger than unity, and the following

analysis repeated appropriately.) The Fourier transform of ψjcan be written as

?∞

where α can at present be taken as a real parameter, and so, assuming convergence

of transformed terms, (10) and (11) give

Ψj(α,Y ) =

−∞

eiαXψj(Xc1/ω,Y c1/ω) dX,j = 1,2,

(18)

∂2Ψ1

∂Y2− (α2− 1)Ψ1= 0,

∂2Ψ2

∂Y2− (α2− k2)Ψ2= 0,

Y ≥ 0,

(19)

Y ≤ 0.

(20)