A new transform method II: the global relation and boundary-value problems...

Proc. R. Soc. A (2010) 466, 2283–2307doi:10.1098/rspa.2009.0513

Published online 10 March 2010

A new transform method II: the global relationand boundary-value problems in

polar coordinatesBY E. A. SPENCE* AND A. S. FOKAS

Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Cambridge CB3 0WA, UK

A new method for solving boundary-value problems (BVPs) for linear and certainnonlinear PDEs was introduced by one of the authors in the late 1990s. For linearPDEs, this method constructs novel integral representations (IRs) that are formulatedin the Fourier (transform) space. In a previous paper, a simplified way of obtaining theserepresentations was presented. In the current paper, first, the second ingredient of thenew method, namely the derivation of the so-called ‘global relation’ (GR)—an equationinvolving transforms of the boundary values—is presented. Then, using the GR as well asthe IR derived in the previous paper, certain BVPs in polar coordinates are solved. TheseBVPs elucidate the fact that this method has substantial advantages over the classicaltransform method.

Keywords: boundary-value problems; transforms; integral representations

1. Introduction

(a) Background

In this paper, we consider the second-order linear elliptic PDE

Du(x) + lu(x) = −f (x), x ∈ Ω, (1.1)

where l is a real constant, f (x) a given function and Ω is some two-dimensionaldomain with a piecewise smooth boundary. For l = 0, this is Poisson’s equation,for l > 0, this is the Helmholtz equation and for l < 0, this is the modifiedHelmholtz equation.

For a boundary-value problem (BVP) to be well-posed, certain boundaryconditions must be prescribed; the ones of most physical importance are asfollows:

— Dirichlet: u(x) = known, x ∈ vΩ,— Neumann: vu/vn(x) = known, x ∈ vΩ, and— Robin: vu/vn(x) + au(x) = known, a = constant, x ∈ vΩ,

where vu/vn = Vu · n, where n is the unit outward-pointing normal to Ω.*Author for correspondence ([email protected]).

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2284 E. A Spence and A. S. Fokas

Green’s theorem gives the following integral representation (IR) of the solutionof equation (1.1):

u(x) =∫

vΩ

(E(x,x)

vuvn

(x) − u(x)vEvn

(x,x))

dS(x) +∫Ω

f (x)E(x,x) dV (x), x ∈ Ω,

(1.2)

where E is the fundamental solution (sometimes called the free-space Greenfunction) satisfying

(Dx + l)E(x,x) = −d(x − x), x ∈ Ω. (1.3)

For the different signs of l, E is given by the following expressions:

— Laplace/Poisson (l = 0): E = −(1/2p) log |x − x|,— Helmholtz (l > 0): E = (i/4)H (1)

0 (√

l|x − x|) (with assumed time depen-dence e−iut , this corresponds to outgoing waves), and

— modified Helmholtz (l < 0): E = (1/2p)K0(√−l|x − x|),

where H (1)0 is a Hankel function and K0 a modified Bessel function.

Starting with a given BVP in a separable domain, i.e. a domain of the formΩ = {a1 ≤ x1 ≤ b1} × {a2 ≤ x2 ≤ b2}, where xj are the coordinates under which thedifferential operator is separable, the method of separation of variables consistsof the following steps:

(1) Separate the PDE into two ODEs.(2) Concentrate on one of these ODEs and derive the associated completeness

relation (i.e. transform pair) depending on the boundary conditions. Thisis achieved by finding the one-dimensional Green function, g(x , x; n),with eigenvalue n and integrating g over a large circle in the complexn-plane,

d(x − x) = − limR→∞

12pi

∮|n|=R

g(x , x; n) dn. (1.4)

(3) Apply this transform to the PDE and use integration by parts to derivethe ODE associated with this transform.

(4) Solve this ODE using an appropriate one-dimensional Green function, orvariation of parameters.

The solution of the BVP is given as a superposition of eigenfunctions of the ODEconsidered in step 2 (either an integral or a sum depending on whether the ODEhas a continuous or discrete spectrum).

Proc. R. Soc. A (2010)



A new transform method II 2285

For each BVP, there exists two different representations of the solutiondepending on which ODE was considered in step 2. To show that these tworepresentations are equivalent requires two steps.

(1) Go into the complex plane, either by deforming contours (if the solution isgiven as an integral), or by converting the series solution into an integralusing the identity

∞∑n=−∞

f (n) =∫C

f (k)1 − e−2pik

dk, (1.5)

where C is a contour which encloses the real k-axis (in the positive sense)but has no singularities of f (k). This latter procedure is known as theWatson transformation (Watson 1918).

(2) Deform contours to enclose the singularities of f (k) and evaluate theintegral as residues/branch-cut integrals.

Completeness. The rigorous proof that the transform derived in step 2 iscomplete can usually be achieved in two ways

(a) a direct proof of equation (1.4) for a given Green function via integrationin the complex plane (Titchmarsh 1962) and

(b) if the inverse of the differential operator is a compact self-adjointoperator on a Hilbert space, then the spectral theorem implies thatthe eigenfunctions are complete (e.g. Stakgold 1967, §§3.3, 4.2; Evans1998, §6.5.1).

The main limitations of this method for solving BVPs are the following:

— It fails for BVPs with non-separable boundary conditions (for example,those that include a derivative at an angle to the boundary).

— The appropriate transform depends on the boundary conditions and so theprocess must be repeated for different boundary conditions.

— The solution is not uniformly convergent on the whole boundary of thedomain (since it is given as a superposition of eigenfunctions of one of theODEs).

— A priori, it is not clear which of the two representations is better forpractical purposes. For computations, an IR is generally superior to aninfinite series. In the cases where only an infinite series is available, severaltechniques have been used in order to improve the convergence of thisseries (e.g. Duffy 2001, §5.8).

— For the Helmholtz equation, the PDE, and hence at least one of theseparated ODEs, involves a radiation condition and so is not self-adjoint.Thus, completeness of the transform associated with the ODE with theradiation condition is not guaranteed. The direct method (a) mentionedearlier can still be used in some cases (Cohen 1964).





The implementation of these steps is classical and can be found in many books,for example, Stakgold (1967, ch. 4), Morse & Feshbach (1953, §5.1), Friedman(1956, ch. 4 and ch. 5), Keener (1995, ch. 7 and ch. 8) and Ockendon et al.(2003, §§4.4, 5.7, 5.8).

An excellent overview, including historical remarks, is given in Keller (1979).

(b) The method of Fokas (1997 )

The method of Fokas (1997) has two basic ingredients,

— the integral representation (IR) and— the global relation (GR).

(i) The integral representation

This is the analogue of Green’s IR in the transform space. Indeed, the solutionu is given as an integral involving transforms of the boundary values, whereasGreen’s IR expresses u as an integral directly involving the boundary values. TheIR can be obtained by first constructing particular IRs depending on the domainof the fundamental solution E (‘domain-dependent fundamental solutions’) andthen substituting these representations into Green’s integral formula of thesolution and interchanging the orders of integration (Spence & Fokas 2010).

(ii) The global relation

This is Green’s divergence form of the equation integrated over the domain,where one employs the solution of the adjoint equation instead of the fundamentalsolution. Indeed,

0 =∫

vΩ

(v(x)

vuvn

(x) − u(x)vv

vn(x)

)dS(x) +

∫Ω

f (x)v(x) dV (x), (1.6)

where v is any solution of the adjoint of equation (1.1)

Dv(x) + lv(x) = 0, x ∈ Ω (1.7)

(equation (1.7) is equation (1.3) without the delta function on the right-handside). Separation of variables gives a one-parameter family of solutions of theadjoint equation depending on the parameter k ∈ C (the separation constant).For example, for the Poisson equation, separation of variables in Cartesian co-ordinates yields four solutions of the adjoint equation, v = e±ikx±ky , k ∈ C. Allequations (1.6) obtained from these different choices of v shall be referred toas the GR. This relation was called the GR as it contains global, as opposed tolocal, information about the boundary values.

Just like Green’s IR, the new IR contains contributions from both knownand unknown boundary values. However, it turns out that the GR preciselyinvolves the transforms of the boundary values appearing in the IR. The main





idea of the method of Fokas (1997) is that, for certain BVPs, one canuse the information given in the GR about the transforms of the unknownboundary values to eliminate these unknowns from the IR. This can be achievedin three steps

(1) Use the GR to express the transforms of the unknown boundary valuesappearing in the IR in terms of the smallest possible subset of the otherfunctions appearing in the GR (if there exist different possibilities, use theone that yields the smallest number of unknowns in each equation).

(2) Identify the domains in the complex k-plane where the integrands arebounded and also identify the location of poles. At this stage, someunknowns can be eliminated directly using analyticity and employingCauchy’s theorem.

(3) Deform contours and use the GR again so that the contribution fromthe unknown boundary values vanish by analyticity (employing Cauchy’stheorem).

For certain domains, it is possible to solve the given BVP by using only asubset of the above three steps. For example, the Helmholtz equation in awedge, which is solved in §4a, only requires step 1, whereas the Poisson equationin a circular wedge, which is solved in §4b, requires steps 1 and 2. On theother hand, the Helmholtz equation in the exterior of the circle, which actuallyplayed a prominent role in the development of classical transforms (includingthe discovery of the Watson transformation, Keller 1979), requires all three steps(Spence 2010).

This method yields the solution as an integral in the complex k-plane involvingtransforms of the known boundary values. This novel solution formula has twosignificant features

— the integrals can be deformed to involve exponentially decayingintegrands and

— the expression is uniformly convergent at the boundary of the domain.

These features give rise to both analytical and numerical advantages incomparison with classical methods. In particular, for linear evolution PDEs, theeffective numerical evaluation of the solution is given in Flyer & Fokas (2008).

(c) Plan of the paper

In the previous paper (Spence & Fokas 2010), the appropriate domain-dependent fundamental solutions were constructed both for polygons and fordomains in polar coordinates. In §2, these representations are used to constructnovel IRs for two particular domains in polar coordinates. In §3, the GRs forthese domains are presented. In §4, the usefulness of the results of §§2 and 3is illustrated by solving certain BVPs for the Helmholtz equation in a wedgeand for the Poisson equation in a circular wedge. These results are discussedfurther in §5.





2. Integral representations for polar coordinates

(a) Notation

Let (r , q) be the physical variable, and (r, f) be the ‘dummy variable’ ofintegration.

We will consider only the Poisson equation, l = 0, and the Helmholtz equation,l = b2.

The following two propositions were proved in Spence & Fokas (2010).

Proposition 2.1 (integral representations of Es for the Helmholtz equation).For the Helmholtz equation, the outgoing non-periodic fundamental solution Escan be expressed in terms of radial eigenfunctions (the radial representation) inthe form

Es(r, f; r , q) = lim3→0

i4

(∫ i∞

0dk e3k2

H (1)k (br)Jk(br) eik|q−f|

+∫−i∞

0dk e3k2

H (1)k (br)Jk(br) e−ik|q−f|

). (2.1)

Alternatively, it can be expressed in terms of angular eigenfunctions (the angularrepresentation) in the form

Es(r, f; r , q) = i4

(∫∞

0dk H (1)

k (br>)Jk(br<) eik(q−f)

+∫∞

0dk H (1)

k (br>)Jk(br<)e−ik(q−f))

, (2.2)

where r> = max(r , r), r< = min(r , r) and −∞ < q, f < ∞.

Proposition 2.2 (integral representations of Es for the Poisson equation). Forthe Poisson equation, the non-periodic fundamental solution Es can be expressedformally either in terms of radial eigenfunctions (the radial representation) in theform

Es(r, f; r , q) = lim3→0

14p

(∫ i∞

3

dkk

(r

r

)keik|q−f| +

∫−i∞

3

dkk

(r

r

)ke−ik|q−f|

), (2.3)

or in terms of angular eigenfunctions (the angular representation) in the form

Es(r, f; r , q) = lim3→0

14p

(∫∞

i3

dkk

(r>

r<

)−k

eik(q−f) +∫∞

−i3

dkk

(r>

r<

)−k

e−ik(q−f)

), (2.4)

where r> = max(r , r), r< = min(r , r) and −∞ < q, f < ∞.





a

a

a

(a)

(b)

Figure 1. The domains (a) D1 and (b) D2.

(b) Novel integral representations for certain domains in polar coordinates

In order to illustrate the new IRs, we consider two examples in polarcoordinates: the Helmholtz equation in the wedge,

D1 = {a < r < ∞, 0 < q < a}, (2.5)

see figure 1a, and the Poisson equation in the circular wedge,

D2 = {a < r < ∞, −a < q < a}, (2.6)

see figure 1b.The associated novel IRs can be obtained by substituting the representations

of the fundamental solutions of propositions 2.2 and 2.1 into Green’s IRs in polarcoordinates, i.e. in the equation

u(r , q) =∫

vD

(u

vEs

vf− Es

vuvf

)dr

r+ r

(Es

vuvr

− uvEs

vr

)df

+∫∫

Df (r, f)Es(r, f; r , q) r dr df. (2.7)

In addition, u satisfies the following boundary conditions at infinity:

l = 0 (Poisson), u → 0 as r → ∞ (2.8)

and

l = b2 (Helmholtz),√

r(

vuvr

− ibu)

→ 0 as r → ∞. (2.9)

Furthermore, f satisfies appropriate conditions such that the integral∫D f (r, f) r dr df is well defined.

Proposition 2.3 (the Helmholtz equation in D1). Let u be a solution of equation(1.1 ) with l = b2 in the domain D1 defined by equation (2.5 ), which satisfies the





radiation condition at infinity (2.9 ). Then, u is given by

u(r , q) = lim3→0

i4

(∫ i∞

0dk e3k2

Jk(br)eikq[−ikD0(k) − N0(k)]

+∫−i∞

0dk e3k2

Jk(br)e−ikq[ikD0(k) − N0(k)]

−∫ i∞

0dk e3k2

Jk(br)e−ikqeika[ikDa(k) − Na(k)]

−∫−i∞

0dk e3k2

Jk(br)eikqe−ika [−ikDa(k) − Na(k)])

+∫∫

Ω

dr df r f (r, f)Es(r, f; r , q), (2.10)

where

Dc(k) =∫∞

0

dr

rH (1)

k (br) u(r, c) and Nc(k) =∫∞

0

dr

rH (1)

k (br) uq(r, c), (2.11)

c = 0 or a, with Es given by either equation (2.1 ) or equation (2.2 ).

Proof. On the boundary {f = a, ∞ > r > a}, use the radial representationof Es (2.1) with q < f. On the boundary {f = 0, 0 < r < ∞}, use the radialrepresentation of Es (2.1) with q > f. �

Proposition 2.4 (the Poisson equation in D2). Let u be a solution of equation(1.1 ) with l = 0 in the domain D2 defined by equation (2.6 ), which satisfies theboundary condition (2.8 ) at infinity. Then, u is given by

4pu(r , q) =∫ i∞

0

dkk

( ra

)−k(eikqeika[−ikD−a(k) − N−a(k)]

− e−ikqeika[ikDa(k) − Na(k)])

+∫−i∞

0

dkk

( ra

)−k(e−ikqe−ika[ikD−a(k) − N−a(k)]

− eikqe−ika[−ikDa(k) − Na(k)])+

∫∞

0

dkk

( ra

)−k(eikq[−aN (−ik) + kD(−ik)]

+ e−ikq[−aN (ik) + kD(ik)])+ 4p

∫∫Ω

dr df r f (r, f)Es(r, f; r , q), (2.12)





where the functions Dc(k), Nc(k), c = 0, a, D(±ik), N (±ik) are defined as follows:

Dc(k) =∫∞

a

dr

r

(r

a

)ku(r, c) and Nc(k) =

∫∞

a

dr

r

(r

a

)kuq(r, c), c = 0 or a

(2.13)and

D(±ik) =∫a

−a

df e±ikf u(a, f) and N (±ik) =∫a

−a

df e±ikf ur(a, f), (2.14)

with Es given by either equation (2.3 ) or equation (2.4 ).

Proof. On the boundary {f = a, ∞ > r > a}, use the radial representation ofEs (2.3) with q < f. On the boundary {r = a, a > f > −a}, use the angularrepresentation of Es (2.4) with r > r. On the boundary {f = 0, 0 < r < ∞}, use theradial representation of Es (2.3) with q > f. After letting 3 → 0, the singularityat k = 0 on the contours of integration is removable: the consistency condition∫

vDr

vuvr

dq − 1r

vuvq

dr +∫∫

Df (r , q)r dr dq = 0

yields the following relation for the boundary values:

−N0(0) + Na(0) − aN (0) = −∫∫

Ddr df r f (r, f).

This equation is the GR for this problem evaluated at k = 0, see equation (3.18).�

3. The global relation

(a) Polygons

Let Ω(i) be the interior of a convex polygon in R2 (figure 2). Let vΩ denote

the boundary of the polygon, oriented anticlockwise, where the vertices of thepolygon z1, z2, . . . , zn are labelled anticlockwise. Let Sj be the side (zj , zj+1) andlet aj = arg(zj+1 − zj) be the angle of Sj . For the Helmholtz equation l = 4b2,b ∈ R

+, and for the modified Helmholtz equation l = −4b2, b ∈ R+.

Let u be a solution of equation (1.1) for Ω = Ω(i). In Cartesian coordinates,the GR (1.6) is∫

vDu(vh dx − vx dh) − v(uh dx − ux dh) = −

∫D

f v dx dh, (3.1)

where v is any solution of the adjoint equation (1.7). Equivalently, equation (3.1)written in complex coordinates becomes∫

vDu

(vz ′ dz ′ − vz ′ dz ′) − v

(uz ′ dz ′ − uz ′ dz ′) = i

∫D

f v dx dh. (3.2)





zj + 1zj – 1

zj

aj

Figure 2. The convex polygon Ω(i).

Lemma 3.1 (adjoint solutions). The following are particular solutions of theadjoint equation (1.7 ):

l = 0 and v = either e−ikz or eikz , (3.3a)

l = −4b2 and v = e−ib(kz ′−(z ′/k)) (3.3b)

and l = 4b2 and v = e−ib(kz ′+(z ′/k)). (3.3c)

Proof. Separation of variables yields that the exponential function v = em1x+m2h

satisfies equation (1.7) iffm2

1 + m22 + l = 0. (3.4)

For l = 0, a natural way to parametrize this one-parameter family of solutions ism1 = ±ik, m2 = ±k, which leads to the solutions

e−ikx+kh and eikx+kh,

which are given in equation (3.3a). Two more solutions are obtained by lettingk → −k.

For l = −4b2, a natural parametrization of equation (3.4) is m1 = ±2b sin fand m2 = ±2b cos f; which, letting k = eif, yields

m1 = ∓ib(

k − 1k

)and m2 = ±b

(k + 1

k

).

This parametrization leads to four particular solutions, namely equation (3.3b), aswell as to those solutions obtained from equation (3.3b) using the transformationsk → −k and k → 1/k.

In a similar way, for l = 4b2 a natural parametrization of equation (3.4) is m1 =±2ib sin f and m2 = ±2ib cos f, which leads to the particular solution (3.3c), aswell as to those solutions obtained from equation (3.3c) using the transformationsk → −k and k → 1/k. �

Substituting these adjoint solutions into equation (3.2) immediately yields thefollowing GRs.

Proposition 3.2 (the global relation for Ω(i) is bounded). Let u be a solutionof equation (1.1 ) in the domain Ω(i). Then, the following relations, called GRs,are valid.





Modified Helmholtz (l = −4b2, b ∈ R+)

n∑j=1

uj(k) + if (k) = 0, k ∈ C, (3.5)

where {uj(k)}n1 are defined by

uj(k) =∫ zj+1

zje−ib(kz ′−(z ′/k))

[(uz ′ + ibku) dz ′ −

(uz ′ + b

iku)

dz ′],

j = 1, . . . , n, zn+1 = z1,

and f (k) is defined by

f (k) =∫∫

Ω(i)e−ib(kz ′−(z ′/k))f (x, h) dx dh. (3.6)

Helmholtz (l = 4b2, b ∈ R+)

n∑j=1

uj(k) + if (k) = 0, k ∈ C, (3.7)

where {uj(k)}n1 are defined by

uj(k) =∫ zj+1

zje−ib(kz ′+(z ′/k))

[(uz ′ + ibku) dz ′ −

(uz ′ − b

iku)

dz ′],

j = 1, . . . , n, zn+1 = z1,

=∫ zj+1

zje−ib(kz ′+(z ′/k))

[ivuvn

(s) + ib(

kdz ′

ds− 1

kdz ′

ds

)u(s)

]ds,

z ′ = z ′(s) ∈ Sj ,

and f (k) is defined by

f (k) =∫∫

Ω(i)e−ib(kz ′+(z ′/k))f (x, h) dx dh. (3.8)

Poisson (l = 0)n∑

j=1

uj(k) + if (k) = 0 andn∑

j=1

uj(k) − if (k) = 0, k ∈ C, (3.9)

where {uj(k)}n1 , {uj(k)}n1 , f (k) are defined by

uj(k) =∫ zj+1

zje−ikz ′ [(uz ′ + iku) dz ′ − uz ′dz ′], j = 1, . . . , n, zn+1 = z1,

=∫ zj+1

zje−ikz ′

[ivuvn

(s) + ikdz ′

dsu(s)

]ds, z ′ = z ′(s) ∈ Sj (3.10)





and

uj(k) =∫ zj+1

zjeikz [(uz − iku) dz − uz dz ], j = 1, . . . , n, zn+1 = z1,

=∫ zj+1

zjeikz ′

[−i

vuvn

(s) − ikdz ′

dsu(s)

]ds, z ′ = z ′(s) ∈ Sj , (3.11)

and f (k), f (k) are defined by

f (k) =∫∫

Ω(i)e−ikz ′

f (x, h) dx dh and f (k) =∫∫

Ω(i)eikz ′

f (x, h) dx dh. (3.12)

Remark 3.3 (the existence of two GRs for the Poisson equation and one for themodified Helmholtz and Helmholtz equations). In the particular adjoint solution(3.3b) for the modified Helmholtz equation, let k → k/b to obtain

v = e−ikz ′+ib2 z ′/k . (3.13)

For b = 0, this reduces to the expression for v of the first GR for the Poissonequation. Letting k → b2/k in equation (3.13), and then letting b = 0 yields eikz ,which is the expression for v of the second GR for the Poisson equation. Thus,the two adjoint solutions for the Poisson equation can be obtained from the singleadjoint solution for the modified Helmholtz equation (3.13).

Remark 3.4 (unbounded domains). If Ω(i) is unbounded, then k must berestricted so that the integral on the boundary at infinity is zero, see Spence(2010).

(b) Polar coordinates

Let u be a solution of equation (1.1). If Ω is unbounded, assume that u satisfiesthe boundary conditions (2.8) and (2.9) at infinity. In polar coordinates, theGR (1.6) is∫

vDr

(v

vuvr

− uvv

vr

)dq + 1

r

(u

vv

vq− v

vuvq

)dr +

∫∫D

f (r , q)v(r , q; r , q)r dr dq = 0,

(3.14)where v is any solution of the adjoint equation (1.7) that, if Ω is unbounded,satisfies the same boundary conditions at infinity as u (this means that theintegral at infinity is zero).

Lemma 3.5 (adjoint solutions). There are four particular solutions of the adjointequation that can be obtained by separation of variables in polar coordinates; twoof these solutions are given by

b = 0 and v = e±ikqrk (3.15a)

andb = 0 and v = e±ikqBk(br), (3.15b)

where Bk(r) denotes a solution of the Bessel equation of order k; two more solutionscan be obtained by k → −k.





Remark 3.6 (the restrictions on k in the adjoint solutions). Dependingon whether the domain contains the origin or is unbounded, some of theparticular solutions for v are disallowed. Indeed, first consider the Poissonequation. At infinity, assume u(r , q) = O(r−3) as r → ∞ where 3 > 0; actually,u(r , q) = O(r−p/g) as r → ∞, where g is the angle of the wedge the domain makesat infinity (Jones 1986). Then,

∫CR

r(

vvuvr

− uvv

vr

)dq + 1

r

(u

vv

vq− v

vuvq

)dr → 0 as R → ∞ ⇐⇒ k < 3,

with v given by equation (3.15a). At the origin, assume u(r , q) = O(r 3) as r → 0where 3 > 0; actually, u(r , q) = O(rp/g) as r → 0, where g is the angle of the wedgethe domain makes at 0 (Jones 1986). Then, for the integral in the GR to exist,we require k > −3.

For the Helmholtz equation,∫

CR→ 0 as R → ∞ if u and v satisfy the

radiation condition (2.9). Therefore if the domain is unbounded, v must beeither e±ikqH (1)

k (br), or a similar expression with k → −k. At the origin, assumeu(r , q) = O(r 3) as r → 0 where 3 > 0; actually, u(r , q) = O(rp/g) as r → 0 whereg is the wedge angle (Jones 1986). Then, for the integral to exist, Bk(br) mustbe bounded as r → 0. Recall that Jk(br) is bounded as r → 0 for k ≥ 0 andH (1)

k (br) is bounded as r → 0 for k = 0.

Proposition 3.7 (the global relation for the Helmholtz equation in the domainD1 defined by equation (2.5)). Let u be the solution of equation (1.1 ) with l = b2

in the domain D1 defined by equation (2.5 ), see figure 1a, and let u satisfy equation(2.9 ). Then,

±ikD0(k) − N0(k) − e±ika [±ikDa(k) − Na(k)] = −F(k, ±ik), k = 0, (3.16)

where Dc(k), Nc(k), c = 0 or a, are defined by equation (2.11 ) and F(k, ±ik) aredefined by

F(k, ±ik) =∫∞

0dr

∫a

0df r f (r, f)H (1)

k (br) e±ikf. (3.17)

Proof. Substitute equation (3.15b) with Bk = H (1)k into equation (3.14)

(according to remark 3.6, v must satisfy the radiation condition). Parametrize thesides of D1 by {f = a, ∞ > r > a} and {f = 0, 0 < r < ∞}. The region of validityof k is specified by remark 3.6. �

Proposition 3.8 (the global relation for the Poisson equation in the domain D2defined by equation (2.6)). Let u be the solution of equation (1.1 ) with l = 0 inthe domain D2 defined by equation (2.6 ), see figure 1b, and let u satisfy equation(2.8 ). Then,

e∓ika [±ikD−a(k) − N−a(k)] − e±ika [±ikDa(k) − Na(k)] − [aN (±ik) − kD(±ik)]

= −F(k, ±ik), k <p

a, (3.18)





where D(±ik), N (±ik) are defined by equation (2.14 ), Dc(k), Nc(k), c = 0 or a,are defined by equation (2.13 ) and F(k, ±ik) are defined by

F(k, ±ik) =∫∞

0dr

∫a

−a

df r f (r, f)(r

a

)ke±ikf. (3.19)

Proof. Substitute equation (3.15a) into equation (3.14) and parametrize thesides of D2 by {f = a, ∞ > r > a}, {r = a, a > f > −a} and {f = −a, 0 < r < ∞}.The region of validity in k is specified by remark 3.6. �

4. The solution of certain boundary-value problems in polar coordinates

In this section, we solve the Dirichlet problem for the Helmholtz equation in thewedge D1 (equation (2.5)) and the Neumann problem for the Poisson equationin the circular wedge D2 (equation (2.6)). This is achieved by employing theIRs of propositions 2.3 and 2.4, as well as the GRs of propositions 3.7 and 3.8,respectively, and then following the steps outlined in the introduction.

(a) The Helmholtz equation in a wedge

Proposition 4.1 (the Dirichlet problem). Let u(r , q) satisfy equation (1.1) withl = b2 in the domain D1 defined by equation (2.5), with the Dirichlet boundaryconditions

u(r , 0) = d0(r), u(r , a) = da(r), 0 < r < ∞, (4.1)where d0(r), da(r), f (r , q) are given. Then, u is given by

u(r , q) = 12

∫R

dk ksin ka

(sin kq(D(R)a (k, r) + D(L)

a (k, r))

+ sin k(a − q)(D(R)0 (k, r) + D(L)

0 (k, r)))

− i4

lim3→0

∫ i∞

0

dksin ka

e3k2Jk(br)(sin k(a − q)F(k, ik) + eika sin kqF(k, −ik))

− i4

lim3→0

∫−i∞

0

dksin ka

e3k2Jk(br)(sin k(a − q)F(k, −ik)

+ e−ika sin kqF(k, ik)) +∫∫

Ω

dr df r f (r, f)Es(r, f; r , q), (4.2)

where the functions D(R)c (k, r) and D(R)

c (k, r), c = 0 or a, are defined as follows:

D(R)c (k, r) = Jk(br)

∫∞

rH (1)

k (br)dc(r)dr

r,

c = 0 or a, 0 < r < ∞, k ≥ 0and

D(L)c (k, r) = H (1)

k (br)∫ r

0Jk(br)dc(r)

dr

r,

c = 0 or a, 0 < r < ∞, k ≥ 0,

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(4.3)

and R is a contour in the right half-plane with arg k = ±4 at infinity, 0 < 4 < p/2,and such that when �k = 0, k < p/a, see figure 3.





p /a

Figure 3. The contour R in the k-plane.

The functions F(k, ±ik) are defined by equation (3.17 ), and the function Esappearing in the last term of equation (4.2 ) is given by either equation (2.2 ) orequation (2.1 ).

Proof. Step 1. The IR (2.10) contains the two unknown functions N0, Na. TheGRs (3.16) are two equations containing these two unknown functions. Hence,solving equation (3.16) for N0, Na we find

Na(k) = 12i sin ka

(−2ikD0(k) + 2ik cos kaDa(k) + F(k, −ik) − F(k, ik))

and

N0(k) = 12i sin ka

(2ikDa(k) − 2ik cos kaD0(k) − e−ikaF(k, −ik) + eikaF(k, ik)).

Substituting these expressions into the IR yields the solution equation (4.2), butwith the first term replaced by

12

lim3→0

∫ i∞

−i∞dk k e3k2

Jk(br)(

Da(k) sin kq + D0(k) sin k(a − q)sin ka

). (4.4)

We now seek to deform the contour from iR to a contour on which the integralconverges absolutely, so we can let 3 = 0. The asymptotics

kJk(y)H (1)k (x) ∼ − i

p

(yx

)k, |k| → ∞, −p

2< arg k <

p

2(4.5)

and

Jk(y)H (1)k (x) ∼ −2i

sin kp

pe−ipk

(−2kex

)−k (−2key

)−k

, (4.6)

|k| → ∞,p

2< arg k <

3p

2,





imply that the product Jk(br)H (1)k (br) is bounded at infinity for k > 0 provided

that r > r . Our plan is to split the relevant integrals in D0 and Da and to use thefollowing identity in order to exchange the arguments of Hk and Jk :

∫ i∞

−i∞dk Jk(x)H (1)

k (y)Q(k) =∫ i∞

−i∞dk Jk(y)H (1)

k (x)Q(k), (4.7)

where Q(k) = −Q(−k). To prove this identity, expand Hk as a linear combinationof Jk and J−k , using the definition of Hk , and then let k → −k in the term involvingJ−k . Note that equation (4.7) also establishes reciprocity between r and r in theKontorovich–Lebedev transform, equation (4.8) below.

The definitions of Dc, equation (2.11), and of D(R)c , equation (4.3), imply the

identity

Jk(br)Dc(k) = D(L)c (k, r) + D(R)

c (k, r), c = 0 or a, 0 < r < ∞, k = 0,

where D(L)c , c = 0 or a, are defined by

D(L)c (k, r) = Jk(br)

∫ r

0H (1)

k (br)dc(r)dr

r, c = 0 or a, 0 < r < ∞, k = 0.

Using equation (4.7) in the terms involving D(L)c (k, r), deforming the contour from

iR to R, and setting 3 = 0, proves that equation (4.4) is equal to the first term ofequation (4.2). �

Remark 4.2 (the recovery of classical representations). The Dirichlet problemof the Helmholtz equation in a wedge can be solved using either a sine series inq or the Kontorovich–Lebedev transform in r . For simplicity, consider f = 0. Toobtain the sine-series solution starting with equation (4.2), evaluate the integralas residues at the poles np/a, n ∈ Z

+,

u(r , q) = i(p

a

)2 ∞∑n=1

n sinnpq

a

{D(R)

0

(np

a, r

)+ D(L)

0

(np

a, r

)

− (−1)n[D(R)

a

(np

a, r

)+ D(L)

a

(np

a, r

)]}.

The solution obtained by the Kontorovich–Lebedev transform is the same asequation (4.2), Jones (1980) or Jones (1986, §9.19, p. 587). Since there are noboundaries in r , the solution is given as an integral that is uniformly convergentat vD1, thus this is the best possible representation.

Remark 4.3 (verifying the boundary conditions). In order to verify theboundary conditions (4.1), evaluate equation (4.2) at q = 0 and q = a. In this





respect, we note that by employing equation (4.7) and by following similar stepsto those used earlier, the Kontorovich–Lebedev transform

g(k) =∫∞

0dy f (y)H (1)

k (y) (4.8a)

and

xf (x) = lim3→0

12

∫ i∞

−i∞dk e3k2

k Jk(x)g(k) (4.8b)

(see remark 3.2 of Spence & Fokas (2010) or Jones (1980)) can be written as

f (r) = 12

∫R

dk k(

Jk(br)∫∞

r

dr

rH (1)

k (br)f (r) + H (1)k (br)

∫ r

0

dr

rJk(br)f (r)

).

Furthermore, equation (2.1) implies

∫∫Ω

dr df r f Es(q = 0) = lim3→0

i4

(∫ i∞

0dk e3k2

Jk(br)F(k, ik)

+∫−i∞

0dk e3k2

Jk(br)F(k, −ik))

and∫∫

Ω

dr df r f Es(q = a) = lim3→0

i4

(∫ i∞

0dk e3k2

Jk(br)eikaF(k, −ik)

+∫−i∞

0dk e3k2

Jk(br)e−ikaF(k, ik))

.

(b) The Poisson equation in a circular wedge

Proposition 4.4 (the Neumann problem). Let u(r , q) satisfy equation (1.1 ) withl = 0 in the domain D2 defined by equation (2.6 ), with the Neumann boundaryconditions

uq(r , −a) = n−a(r), a < r < ∞, (4.9a)

ur(a, q) = n(q), −a < q < a (4.9b)

and uq(r , a) = na(r), a < r < ∞, (4.9c)

with the condition (2.8 ), and with the following consistency condition:

∫∞

a

dr

rna(r) −

∫∞

a

dr

rn−a(r) − a

∫a

−a

df n(f) +∫∞

adr

∫a

−a

df rf (r, f) = 0. (4.10)





Then, u is given by

4pu(r , q) =∫ i∞

0

dkk

( ra

)−k{e−ikqeiak

[A(k)D(k)

+ Na(k) − Na(−k)]

+eikqeiak[B(k)D(k)

− N−a(k) + N−a(−k)]}

+∫−i∞

0

dkk

( ra

)−k{eikqe−iak

[A(k)D(k)

− Na(−k) + Na(k)]

+e−ikqe−iak[B(k)D(k)

+ N−a(−k) − N−a(k)]}

+∫∞

0

dkk

( ra

)−k

×{2e−ikq

[eiakNa(−k) − e−iakN−a(−k) − aN (ik) + F(−k, ik)

]+ 2eikq

[e−iakNa(−k) − eiakN−a(−k) − aN (−ik) + F(−k, −ik)

]− e−ikqF(−k, ik)− eikqF(−k, −ik)

}

+ 4p

∫∫Ω

dr df r f (r, f)Es(r, f; r , q), (4.11)

where D, A, B, N±a, N are defined as follows:

N±a(k) =∫∞

a

dr

r

(r

a

)kn±a(r), N (±ik) =

∫a

−a

df e±ikf n(f),

D(k) = e2iak − e−2iak ,

A(k) = −(e2iak + e−2iak)[Na(k) + Na(−k)] + 2[N−a(k) + N−a(−k)]+ 2a[eiakN (ik) + e−iakN (−ik)] − eiak [F(k, ik) + F(−k, ik)]− e−iak [F(−k, −ik) + F(k, −ik)]

and B(k) = (e2iak + e−2iak)[N−a(k) + N−a(−k)] − 2[Na(k) + Na(−k)]+ 2a[eiakN (−ik) + e−iakN (ik)] − e−iak [F(k, ik) + F(−k, ik)]− eiak [F(−k, −ik) + F(k, −ik)].

The functions F(k, ±ik) are defined by equation (3.19 ), and Es is given by eitherequation (2.4 ) or equation (2.3 ).

Remark 4.5 (the consistency condition). The consistency condition (4.10) is

−N−a(0) + Na(0) − aN (0) + F(0, 0) = 0.





Proof. Step 1. The two GRs (3.18) can be rewritten as

ieikaDa(k) − ie−ikaD−a(k) − D(ik) = I (k) (4.12)

and

−ie−ikaDa(k) + ieikaD−a(k) − D(−ik) = J (k), k <p

2a, (4.13)

where

I (k) = 1k(eikaNa(k) − e−ikaN−a(k) − aN (ik) + F(k, ik))

and

J (k) = 1k(e−ikaNa(k) − eikaN−a(k) − aN (−ik) + F(k, −ik)).

Letting k → −k in equations (4.12) and (4.13) yields

ie−ikaDa(−k) − ieikaD−a(−k) − D(−ik) = I (−k) (4.14)

and

−ieikaDa(−k) + ie−ikaD−a(−k) − D(ik) = J (−k), k > − p

2a. (4.15)

The IR (2.12) contains the four unknown functions D±a(k) and D(±ik), whichcan be expressed in terms of the two functions D±a(−k) as follows:

iD−a(k) = −iD−a(−k) + 1D(k)

(e−ika(I (k) − J (−k)) − eika(I (−k) − J (k)))

(4.16)and

iDa(k) = −iDa(−k) + 1D(k)

(eika(I (k) − J (−k))

− e−ika(I (−k) − J (k))),p

2a< k <

p

2a. (4.17)

Indeed, equations (4.14) and (4.15) imply

D(−ik) = ie−ikaDa(−k) − ieikaD−a(−k) − I (−k) (4.18)

and

D(ik) = −ieikaDa(−k) + ie−ikaD−a(−k) − J (−k), k > − p

2a. (4.19)

Eliminating D(ik) from equations (4.12) and (4.15), and D(−ik) from equations(4.13) and (4.14) yields two equations for the combinations of the unknownfunctions D−a(k) − D−a(−k) and Da(k) − Da(−k). Solving these two equationsfor these two combinations yields equations (4.16) and (4.17).





Step 2. Substituting equations (4.16)–(4.19) into the IR, the terms involvingthe unknowns D±a(−k) are proportional to the following expression:

∫ i∞

0dk

( ra

)−k(eikqeikaD−a(−k) + eikae−ikqDa(−k))

+∫−i∞

0dk

( ra

)−k(−e−ikqe−ikaD−a(−k) − e−ikaeikqDa(−k))

+∫∞

0dk

( ra

)−k(eikqe−ikaDa(−k) − eikaeikqD−a(−k)

− e−ikqeikaD−a(−k) + e−ikae−ikqD−a(−k)).

This expression vanishes using Cauchy’s theorem. Indeed, eik(q+a) and eik(a−q)

decay for �k > 0, D±a(−k) are bounded at infinity for k ≥ 0 and are of O(1/k)as k → ∞ for k ≥ 0 (using integration by parts) and (r/a)k is bounded for k >0. Also, Jordan’s lemma (e.g. Ablowitz & Fokas 2003, p. 222) implies that theintegral at infinity vanishes.

The remaining terms yield

4pu(r , q) =∫ i∞

0

dkk

( ra

)−k{e−ikqeiak

[A(k)D(k)

+ Na(k)]

+ eikqeiak[B(k)D(k)

− N−a(k)]}

+∫−i∞

0

dkk

( ra

)−k{eikqe−iak

[A(k)D(k)

+ Na(k)]

+ e−ikqe−iak[B(k)D(k)

− N−a(k)]}

+∫∞

0

dkk

( ra

)−k

× {e−ikq[eiakNa(−k) − e−iakN−a(−k) − 2aN (ik) + F(−k, ik)]

+ eikq[e−iakNa(−k) − eiakN−a(−k) − 2aN (−ik) + F(−k, −ik)]}

+ 4p

∫∫Ω

dr df r f (r, f)Es(r, f; r , q). (4.20)

These integrals contain poles on the contours at k = 0. However, the integralstaken together are well defined. In order for each integral to be well defined, wewill rewrite equation (4.20) using the following identity:

−(∫ 0

∞+

∫−i∞

0

)dkk

( ra

)−k(eikqeikaN−a(−k) − e−ikqeikaNa(−k))

+(∫ 0

∞+

∫ i∞

0

)dkk

( ra

)−k(e−ikqe−ikaN−a(−k) − eikqe−ikaNa(−k)) = 0. (4.21)





Adding equation (4.21) to equation (4.20) yields equation (4.11). In order toestablish equation (4.21), using analyticity considerations similar to those usedabove, we find

−(∫ 3

∞+

∫−i3

3

+∫−i∞

−i3

)dkk

( ra

)−k (eikqeikaN−a(−k) − e−ikqeikaNa(−k)

)

+(∫ 3

∞+

∫ i3

3

∫ i∞

i3

)dkk

( ra

)−k (e−ikqe−ikaN−a(−k) − eikqe−ikaNa(−k)

) = 0.

(4.22)

Letting 3 → 0, the residue contributions from the pole at k = 0 cancel andequation (4.22) follows, where the integrals are understood in the principal valuesense, i.e. (∫ 0

i∞+

∫∞

0

)dkk

f (k) =∫∞

0

dkk

(f (k) − f (ik)

).

Now the definitions of A(k) and B(k) imply

A(0) = −4S and B(0) = −4S ,

whereS = Na(0) − N−a(0) − aN (0) + F(0, 0) = 0.

Thus, A(0) = B(0) = 0. In addition, the functions A(k) and B(k) are evenfunctions of k, thus A′(0) = B ′(0) = 0, and these relations imply that each bracketappearing in the integrals of equation (4.11) vanishes at k = 0. �

Remark 4.6 (comparison with the classical representations). The Neumannproblem of the Poisson equation in a circular wedge can be solved using either acosine series in q or the Mellin sine transform in r

g(k) =∫∞

0

dr

rsin

(k log

(r

a

))g(r)

and

g(r) = 2p

∫∞

0dk sin

(k log

( ra

))g(k), a < r < ∞,

Stakgold (1967, p. 316). The cosine-series solution is uniformly convergent at r =a, but not at q = ±a; the Mellin sine transform solution is uniformly convergent atq = ±a, but not at r = a. The solution (4.11) is the best possible representation:it is an integral that is uniformly convergent at q = ±a and at r = a (furthermore,the integral can either be evaluated as residues or deformed to give the cosine-series solution and Mellin sine solution, respectively).

5. Conclusions

(a) The method of Fokas (1997) versus the classical transform method

A natural question is ‘how does the method of Fokas (1997) compare with theclassical transform method?’. Table 1 compares the method of Fokas (1997) andthe classical transform method, and shows that the method of Fokas (1997)





Table 1. Comparison of the method of Fokas (1997) and classical transforms in two dimensions.

the method of Fokas (1997) transforms

mathematical completeness relation for each completeness relation for oneinput separated ODE in whole space separated ODE dependent on

(independent of the domain domain and boundaryand boundary conditions) conditions

Green’s theorem

implementationgiven a BVP

same steps independent ofboundary conditions

different transforms for differentboundary conditions

algebraic manipulation integration by partsunknowns vanish by Cauchy solve ODE using one-dimensional

Green’s function

solution uniformly convergent at vΩ not uniformly convergent at vΩ

given as an integral, deform either an infinite sum or an integralcontour so integrand decays depending on domain. If an integral,exponentially can deform contour so integrand

decays exponentially

boundary separable and some only separableconditions non-separable

domains separable and some only separablenon-separable

requires less mathematical input, is simpler to implement, yields more usefulsolution formulae and is more widely applicable than the classical transformmethod. In §4, this method was applied to two BVPs in separable domains withseparable boundary conditions. The method can also be used to solve BVPs innon-separable domains (Dassios & Fokas 2005; Fokas 2008; Spence 2010; Fokas &Kalimeris submitted; Kalimeris & Fokas in press), as well as in separable domainswith non-separable boundary conditions (Fokas 2008; Spence 2010).

(b) How the method of Fokas (1997 ) is related to other approaches

(i) Shanin

The GR for the Helmholtz equation in the interior of an equilateral trianglewas first discovered by Shanin (1997; later published as Shanin (2000)). In thesepapers, the eigenvalues and eigenfunctions of the Laplacian in an equilateraltriangle for impedance boundary conditions are found, and the Dirichlet problemof the Helmholtz equation is solved in terms of an infinite series (an IR of thesolution is also obtained by substituting an IR of the fundamental solution intoGreen’s IR).

(ii) The synthesis of ‘Fourier’ with ‘Green’ in the background of ‘Cauchy’

The method of Fokas (1997) combines the ideas of Green and the classicaltransform method by constructing the analogue of Green’s IR in the transformspace. Furthermore, for the elimination of the unknown boundary values, a crucial





role is played by analyticity and Cauchy’s theorem. Several investigations havemade some attempts in this direction (e.g. Croisille & Lebeau 1999; Gautesen2005; Bernard 2006), however, it appears that this ‘synthesis’ has not beenachieved before.

(iii) Integral representations in the complex plane

Representations of solutions to ODEs as integrals in the complex plane werepioneered by Laplace following earlier investigations by Euler. In hindsight, the‘moral’ of the Watson transformation is that the best representation of thesolution of a separable PDEs is an IR in the complex k-plane (which can bedeformed to either of the two representations obtained by transforms). Thisrepresentation is precisely the one obtained by the new method. In this sense,it is surprising that no-one tried to find this representation directly until theemergence of the method of Fokas (1997).

(iv) The Sommerfeld–Malyuzhinets technique

The Sommerfeld–Malyuzhinets (S–M) technique is a method for solving theHelmholtz equation in a wedge by assuming that the solution can be written asan integral in the complex plane over the so-called Sommerfeld contour (e.g.Osipov & Norris (1999); Babich et al. (2008). It appears that, for a wedge,the method of Fokas (1997) is related to the S–M method: the change ofvariables k = eif, z = reiq in the exponential exp ib(kz + z/k) appearing in theIR of the Helmholtz equation in a convex polygon (see Spence & Fokas 2010,proposition 2.7) transforms it to exp 2ibr cos(q − f), which appears in thesolutions given by the S–M method (recall that the frequency is 2b). Althougha detailed investigation of this connection will be presented elsewhere, we canalready make the following remarks: (i) The S–M technique is a method forsolving only the Helmholtz equation in a particular domain, whereas the methodof Fokas (1997) is applicable to a wide variety of PDEs and domains, (ii) thesolution formula of S–M is an integral involving a contour in the complex planethat is independent of the wedge angle, whereas the solution given by the methodof Fokas (1997) involves integrals over contours that depend on the domain, and(iii) the S–M technique requires the solution of a functional-difference equationas well as Cauchy’s theorem, whereas the method of Fokas (1997) requires onlyCauchy’s theorem as well as the algebraic manipulation of the GR.

(c) Extension of the method to three dimensions

The method of Fokas (1997) is applied to evolution PDEs in two space andone time dimensions in Fokas (2002) and Kalimeris & Fokas (in press). Regardingelliptic PDEs, we note the following: appropriate domain-dependent fundamentalsolutions for separable domains in three dimensions can be constructed using themethod of Spence & Fokas (2010) and are expressed as double integrals over twocomplex parameters. The construction of the GR is similar to the constructionpresented here, but now the GR contains two complex parameters. The solutionof BVPs in three dimensions also follows the steps outlined in §1b, but is moreinvolved due to the fact that the solution, like the domain-dependent fundamentalsolutions, is expressed as a double integral.





E.A.S. was supported by the EPSRC and the Cambridge Philosophical Society, and A.S.F. by aGuggenheim fellowship.

References

Ablowitz, M. & Fokas, A. 2003 Complex variables: introduction and applications, 2nd edn.Cambridge, UK: Cambridge University Press.

Babich, V., Lyalinov, M. & Grikurov, V. 2008 Diffraction theory: the Sommerfeld–Malyuzhinetstechnique. Oxford, UK: Alpha Science.

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