## Abstract

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

## 1. Introduction

### (a) Background

In this paper, we consider the second-order linear elliptic PDE
1.1
where *λ* is a real constant, *f*(** x**) a given function and

*Ω*is some two-dimensional domain with a piecewise smooth boundary. For

*λ*=0, this is Poisson’s equation, for

*λ*>0, this is the Helmholtz equation and for

*λ*<0, this is the modified Helmholtz equation.

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

— Dirichlet:

*u*()=known,*x*∈∂*x**Ω*,— Neumann: ∂

*u*/∂*n*()=known,*x*∈∂*x**Ω*, and— Robin: ∂

*u*/∂*n*()+*x**αu*()=known,*x**α*=constant,∈∂*x**Ω*,

where ∂*u*/∂*n*=∇*u*⋅** n**, where

**is the unit outward-pointing normal to**

*n**Ω*.

Green’s theorem gives the following integral representation (IR) of the solution of equation (1.1):
1.2
where *E* is the fundamental solution (sometimes called the free-space Green function) satisfying
1.3
For the different signs of *λ*, *E* is given by the following expressions:

— Laplace/Poisson (

*λ*=0): ,— Helmholtz (

*λ*>0): (with assumed time dependence*e*^{−iωt}, this corresponds to outgoing waves), and— modified Helmholtz (

*λ*<0): ,

where is a Hankel function and *K*_{0} a modified Bessel function.

Starting with a given BVP in a separable domain, i.e. a domain of the form *Ω*={*a*_{1}≤*ξ*_{1}≤*b*_{1}}×{*a*_{2}≤*ξ*_{2}≤*b*_{2}}, where *ξ*_{j} are the coordinates under which the differential operator is separable, the method of separation of variables consists of the following steps:

Separate the PDE into two ODEs.

Concentrate on one of these ODEs and derive the associated completeness relation (i.e. transform pair) depending on the boundary conditions. This is achieved by finding the one-dimensional Green function,

*g*(*x*,*ξ*;*ν*), with eigenvalue*ν*and integrating*g*over a large circle in the complex*ν*-plane, 1.4Apply this transform to the PDE and use integration by parts to derive the ODE associated with this transform.

Solve this ODE using an appropriate one-dimensional Green function, or variation of parameters.

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

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

Go into the complex plane, either by deforming contours (if the solution is given as an integral), or by converting the series solution into an integral using the identity 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 the Watson transformation (Watson 1918).Deform contours to enclose the singularities of

*f*(*k*) and evaluate the integral as residues/branch-cut integrals.

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

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

if the inverse of the differential operator is a compact self-adjoint operator on a Hilbert space, then the spectral theorem implies that the eigenfunctions are complete (e.g. Stakgold 1967, §§3.3, 4.2; Evans 1998, §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 the process must be repeated for different boundary conditions.

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

—

*A priori*, it is not clear which of the two representations is better for practical purposes. For computations, an IR is generally superior to an infinite series. In the cases where only an infinite series is available, several techniques have been used in order to improve the convergence of this series (e.g. Duffy 2001, §5.8).— For the Helmholtz equation, the PDE, and hence at least one of the separated ODEs, involves a radiation condition and so is not self-adjoint. Thus, completeness of the transform associated with the ODE with the radiation condition is not guaranteed. The direct method (a) mentioned earlier 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 solution *u* is given as an integral involving transforms of the boundary values, whereas Green’s IR expresses *u* as an integral directly involving the boundary values. The IR can be obtained by first constructing particular IRs depending on the domain of the fundamental solution *E* (‘domain-dependent fundamental solutions’) and then substituting these representations into Green’s integral formula of the solution 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 fundamental solution. Indeed,
1.6
where *v* is any solution of the adjoint of equation (1.1)
1.7
(equation (1.7) is equation (1.3) without the delta function on the right-hand side). Separation of variables gives a one-parameter family of solutions of the adjoint equation depending on the parameter (the separation constant). For example, for the Poisson equation, separation of variables in Cartesian co-ordinates yields four solutions of the adjoint equation, . All equations (1.6) obtained from these different choices of *v* shall be referred to as the GR. This relation was called the GR as it contains global, as opposed to local, information about the boundary values.

Just like Green’s IR, the new IR contains contributions from both known and unknown boundary values. However, it turns out that the GR precisely involves 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 can use the information given in the GR about the transforms of the unknown boundary values to eliminate these unknowns from the IR. This can be achieved in three steps

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

Identify the domains in the complex

*k*-plane where the integrands are bounded and also identify the location of poles. At this stage, some unknowns can be eliminated directly using analyticity and employing Cauchy’s theorem.Deform contours and use the GR again so that the contribution from the unknown boundary values vanish by analyticity (employing Cauchy’s theorem).

For certain domains, it is possible to solve the given BVP by using only a subset of the above three steps. For example, the Helmholtz equation in a wedge, which is solved in §4*a*, only requires step 1, whereas the Poisson equation in a circular wedge, which is solved in §4*b*, requires steps 1 and 2. On the other hand, the Helmholtz equation in the exterior of the circle, which actually played a prominent role in the development of classical transforms (including the 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 involving transforms of the known boundary values. This novel solution formula has two significant features

— the integrals can be deformed to involve exponentially decaying integrands and

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

These features give rise to both analytical and numerical advantages in comparison with classical methods. In particular, for linear evolution PDEs, the effective 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 for domains in polar coordinates. In §2, these representations are used to construct novel IRs for two particular domains in polar coordinates. In §3, the GRs for these domains are presented. In §4, the usefulness of the results of §§2 and 3 is illustrated by solving certain BVPs for the Helmholtz equation in a wedge and for the Poisson equation in a circular wedge. These results are discussed further in §5.

## 2. Integral representations for polar coordinates

### (a) Notation

Let (*r*,*θ*) be the physical variable, and (*ρ*,*ϕ*) be the ‘dummy variable’ of integration.

We will consider only the Poisson equation, *λ*=0, and the Helmholtz equation, *λ*=*β*^{2}.

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

### Proposition 2.1 (integral representations of *E*_{s}for the Helmholtz equation).

*For the Helmholtz equation, the outgoing non-periodic fundamental solution E*_{s} *can be expressed in terms of radial eigenfunctions (the radial representation) in the form*
2.1

*Alternatively, it can be expressed in terms of angular eigenfunctions (the angular representation) in the form*
2.2
*where* *,* *and* *.*

### Proposition 2.2 (integral representations of *E*_{s} for the Poisson equation).

*For the Poisson equation, the non-periodic fundamental solution E*_{s} *can be expressed formally either in terms of radial eigenfunctions (the radial representation) in the form*
2.3
*or in terms of angular eigenfunctions (the angular representation) in the form*
2.4
*where* *,* *and* *.*

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

In order to illustrate the new IRs, we consider two examples in polar coordinates: the Helmholtz equation in the wedge,
2.5
see figure 1*a*, and the Poisson equation in the circular wedge,
2.6
see figure 1*b*.

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 polar coordinates, i.e. in the equation
2.7
In addition, *u* satisfies the following boundary conditions at infinity:
2.8
and
2.9
Furthermore, *f* satisfies appropriate conditions such that the integral is well defined.

### Proposition 2.3 (the Helmholtz equation in *D*_{1}).

*Let u be a solution of equation (1.1 ) with λ=β*^{2} *in the domain D*_{1} *defined by equation (2.5 ), which satisfies the radiation condition at infinity (2.9 ). Then, u is given by*
2.10
*where*
2.11
*χ=0 or α, with E*_{s} *given by either equation (2.1 ) or equation (2.2 ).*

### Proof.

On the boundary , use the radial representation of *E*_{s} (2.1) with *θ*<*ϕ*. On the boundary , use the radial representation of *E*_{s} (2.1) with *θ*> *ϕ*. ■

### Proposition 2.4 (the Poisson equation in *D*_{2}).

*Let u be a solution of equation (1.1 ) with λ=0 in the domain D*_{2} *defined by equation (2.6 ), which satisfies the boundary condition (2.8 ) at infinity. Then, u is given by*
2.12
*where the functions D*_{χ}*(k),N*_{χ}*(k),χ=0,α, D(±ik),N(±ik) are defined as follows:*
2.13
*and*
2.14
*with E*_{s} *given by either equation (2.3 ) or equation (2.4 ).*

### Proof.

On the boundary , use the radial representation of *E*_{s} (2.3) with *θ*<*ϕ*. On the boundary {*ρ*=*a*,*α*>*ϕ*>−*α*}, use the angular representation of *E*_{s} (2.4) with *r*>*ρ*. On the boundary , use the radial representation of *E*_{s} (2.3) with *θ*>*ϕ*. After letting , the singularity at *k*=0 on the contours of integration is removable: the consistency condition
yields the following relation for the boundary values:
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 (figure 2). Let ∂*Ω* denote the boundary of the polygon, oriented anticlockwise, where the vertices of the polygon *z*_{1},*z*_{2},…,*z*_{n} are labelled anticlockwise. Let *S*_{j} be the side (*z*_{j},*z*_{j+1}) and let *α*_{j}=arg(*z*_{j+1}−*z*_{j}) be the angle of *S*_{j}. For the Helmholtz equation *λ*=4*β*^{2}, , and for the modified Helmholtz equation *λ*=−4*β*^{2}, .

Let *u* be a solution of equation (1.1) for *Ω*=*Ω*^{(i)}. In Cartesian coordinates, the GR (1.6) is
3.1
where *v* is any solution of the adjoint equation (1.7). Equivalently, equation (3.1) written in complex coordinates becomes
3.2

### Lemma 3.1 (adjoint solutions).

*The following are particular solutions of the adjoint equation (1.7 ):*
3.3a
3.3b*and*
3.3c

### Proof.

Separation of variables yields that the exponential function *v*=*e*^{m1ξ+m2η} satisfies equation (1.7) iff
3.4
For *λ*=0, a natural way to parametrize this one-parameter family of solutions is *m*_{1}=±*ik*, *m*_{2}=±*k*, which leads to the solutions
which are given in equation (3.3a). Two more solutions are obtained by letting .

For *λ*=−4*β*^{2}, a natural parametrization of equation (3.4) is and ; which, letting *k*=*e*^{iϕ}, yields
This parametrization leads to four particular solutions, namely equation (3.3b), as well as to those solutions obtained from equation (3.3b) using the transformations and .

In a similar way, for *λ*=4*β*^{2} a natural parametrization of equation (3.4) is and , which leads to the particular solution (3.3c), as well as to those solutions obtained from equation (3.3c) using the transformations and . ■

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

### Proposition 3.2 (the global relation for *Ω*^{(i)} is bounded).

*Let u be a solution of equation (1.1 ) in the domain Ω*^{(i)}*. Then, the following relations, called GRs, are valid.*

*Modified Helmholtz*
3.5
*where* *are defined by*
*and* *is defined by*
3.6

*Helmholtz*
3.7
*where* *are defined by*
*and* *is defined by*
3.8

*Poisson (λ=0)*
3.9
*where* *,* *,* *are defined by*
3.10
*and*
3.11
*and* *are defined by*
3.12

### Remark 3.3 (the existence of two GRs for the Poisson equation and one for the modified Helmholtz and Helmholtz equations).

In the particular adjoint solution (3.3b) for the modified Helmholtz equation, let to obtain
3.13
For *β*=0, this reduces to the expression for *v* of the first GR for the Poisson equation. Letting in equation (3.13), and then letting *β*=0 yields , 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 single adjoint solution for the modified Helmholtz equation (3.13).

### Remark 3.4 (unbounded domains).

If *Ω*^{(i)} is unbounded, then *k* must be restricted 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* satisfies the boundary conditions (2.8) and (2.9) at infinity. In polar coordinates, the GR (1.6) is
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 the integral at infinity is zero).

### Lemma 3.5 (adjoint solutions).

There are four particular solutions of the adjoint equation that can be obtained by separation of variables in polar coordinates; two of these solutions are given by
3.15a
*and*
3.15b
where *B*_{k}(*r*) denotes a solution of the Bessel equation of order *k*; two more solutions can be obtained by *k*↦−*k*.

### Remark 3.6 (the restrictions on *k* in the adjoint solutions).

Depending on whether the domain contains the origin or is unbounded, some of the particular solutions for *v* are disallowed. Indeed, first consider the Poisson equation. At infinity, assume as where *ε*>0; actually, as , where *γ* is the angle of the wedge the domain makes at infinity (Jones 1986). Then,
with *v* given by equation (3.15a). At the origin, assume as where *ε*>0; actually, as , where *γ* is the angle of the wedge the domain makes at 0 (Jones 1986). Then, for the integral in the GR to exist, we require ℜ*k*>−*ε*.

For the Helmholtz equation, as if *u* and *v* satisfy the radiation condition (2.9). Therefore if the domain is unbounded, *v* must be either , or a similar expression with *k*↦−*k*. At the origin, assume as where *ε*>0; actually, as where *γ* is the wedge angle (Jones 1986). Then, for the integral to exist, *B*_{k}(*βr*) must be bounded as . Recall that *J*_{k}(*βr*) is bounded as for ℜ*k*≥0 and is bounded as for ℜ*k*=0.

### Proposition 3.7 (the global relation for the Helmholtz equation in the domain *D*_{1}defined by equation (2.5)).

*Let u be the solution of equation (1.1 ) with λ=β*^{2} *in the domain D*_{1} *defined by equation (2.5 ), see*figure 1*a, and let u satisfy equation (2.9 ). Then,*
3.16
*where D*_{χ}*(k),N*_{χ}*(k), χ=0 or α, are defined by equation (2.11 ) and F(k,±ik) are defined by*
3.17

### Proof.

Substitute equation (3.15b) with into equation (3.14) (according to remark 3.6, *v* must satisfy the radiation condition). Parametrize the sides of *D*_{1} by and . The region of validity of *k* is specified by remark 3.6. ■

### Proposition 3.8 (the global relation for the Poisson equation in the domain *D*_{2} defined by equation (2.6)).

*Let u be the solution of equation (1.1 ) with λ=0 in the domain D*_{2} *defined by equation (2.6 ), see*figure 1*b, and let u satisfy equation (2.8 ). Then,*
3.18
*where D(±ik),N(±ik) are defined by equation (2.14 ), D*_{χ}*(k),N*_{χ}*(k), χ=0 or α, are defined by equation (2.13 ) and F(k,±ik) are defined by*
3.19

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

In this section, we solve the Dirichlet problem for the Helmholtz equation in the wedge *D*_{1} (equation (2.5)) and the Neumann problem for the Poisson equation in the circular wedge *D*_{2} (equation (2.6)). This is achieved by employing the IRs 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,θ) satisfy equation (1.1) with λ=β*^{2} *in the domain D*_{1} *defined by equation (2.5), with the Dirichlet boundary conditions*
4.1
*where d*_{0}*(r),d*_{α}*(r),f(r,θ) are given. Then, u is given by*
4.2
*where the functions* *and* *, χ=0 or α, are defined as follows:*
4.3
*and* *is a contour in the right half-plane with* *at infinity, 0<φ<π/2, and such that when ℑk=0, ℜk<π/α, see* figure 3*.*

*The functions F(k,±ik) are defined by equation (3.17 ), and the function E*_{s} *appearing in the last term of equation (4.2 ) is given by either equation (2.2 ) or equation (2.1 ).*

### Proof.

*Step 1.* The IR (2.10) contains the two unknown functions *N*_{0},*N*_{α}. The GRs (3.16) are two equations containing these two unknown functions. Hence, solving equation (3.16) for *N*_{0},*N*_{α} we find
and
Substituting these expressions into the IR yields the solution equation (4.2), but with the first term replaced by
4.4
We now seek to deform the contour from to a contour on which the integral converges absolutely, so we can let *ε*=0. The asymptotics
4.5
and
4.6
imply that the product is bounded at infinity for ℜ*k*>0 provided that *ρ*>*r*. Our plan is to split the relevant integrals in *D*_{0} and *D*_{α} and to use the following identity in order to exchange the arguments of *H*_{k} and *J*_{k}:
4.7
where *Q*(*k*)=−*Q*(−*k*). To prove this identity, expand *H*_{k} as a linear combination of *J*_{k} and *J*_{−k}, using the definition of *H*_{k}, and then let *k*↦−*k* in the term involving *J*_{−k}. Note that equation (4.7) also establishes reciprocity between *r* and *ρ* in the Kontorovich–Lebedev transform, equation (4.8) below.

The definitions of *D*_{χ}, equation (2.11), and of , equation (4.3), imply the identity
where , *χ*=0 or *α*, are defined by
Using equation (4.7) in the terms involving , deforming the contour from to , and setting *ε*=0, proves that equation (4.4) is equal to the first term of equation (4.2). ■

### Remark 4.2 (the recovery of classical representations).

The Dirichlet problem of the Helmholtz equation in a wedge can be solved using either a sine series in *θ* or the Kontorovich–Lebedev transform in *r*. For simplicity, consider *f*=0. To obtain the sine-series solution starting with equation (4.2), evaluate the integral as residues at the poles *nπ*/*α*, ,
The solution obtained by the Kontorovich–Lebedev transform is the same as equation (4.2), Jones (1980) or Jones (1986, §9.19, p. 587). Since there are no boundaries in *r*, the solution is given as an integral that is uniformly convergent at ∂*D*_{1}, thus this is the best possible representation.

### Remark 4.3 (verifying the boundary conditions).

In order to verify the boundary conditions (4.1), evaluate equation (4.2) at *θ*=0 and *θ*=*α*. In this respect, we note that by employing equation (4.7) and by following similar steps to those used earlier, the Kontorovich–Lebedev transform
4.8a
and
4.8b
(see remark 3.2 of Spence & Fokas (2010) or Jones (1980)) can be written as
Furthermore, equation (2.1) implies
and

### (b) The Poisson equation in a circular wedge

### Proposition 4.4 (the Neumann problem).

*Let u(r,θ) satisfy equation (1.1 ) with λ=0 in the domain D*_{2} *defined by equation (2.6 ), with the Neumann boundary conditions*
4.9a
4.9b*and*
4.9c
*with the condition (2.8 ), and with the following consistency condition:*
4.10
*Then, u is given by*
4.11
*where Δ,A,B,N*_{±α}*,N are defined as follows:*
*The functions F(k,±ik) are defined by equation (3.19 ), and E*_{s} *is given by either equation (2.4 ) or equation (2.3 ).*

### Remark 4.5 (the consistency condition).

The consistency condition (4.10) is

### Proof.

*Step 1.* The two GRs (3.18) can be rewritten as
4.12
and
4.13
where
and
Letting *k*↦−*k* in equations (4.12) and (4.13) yields
4.14
and
4.15
The IR (2.12) contains the four unknown functions *D*_{±α}(*k*) and *D*(±*ik*), which can be expressed in terms of the two functions *D*_{±α}(−*k*) as follows:
4.16
and
4.17
Indeed, equations (4.14) and (4.15) imply
4.18
and
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 unknown functions *D*_{−α}(*k*)−*D*_{−α}(−*k*) and *D*_{α}(*k*)−*D*_{α}(−*k*). Solving these two equations for these two combinations yields equations (4.16) and (4.17).

*Step 2.* Substituting equations (4.16)–(4.19) into the IR, the terms involving the unknowns *D*_{±α}(−*k*) are proportional to the following expression:
This expression vanishes using Cauchy’s theorem. Indeed, *e*^{ik(θ+α)} and *e*^{ik(α−θ)} decay for ℑ*k*>0, *D*_{±α}(−*k*) are bounded at infinity for ℜ*k*≥0 and are of as for ℜ*k*≥0 (using integration by parts) and (*r*/*a*)^{k} is bounded for ℜ*k*>0. Also, Jordan’s lemma (e.g. Ablowitz & Fokas 2003, p. 222) implies that the integral at infinity vanishes.

The remaining terms yield
4.20
These integrals contain poles on the contours at *k*=0. However, the integrals taken together are well defined. In order for each integral to be well defined, we will rewrite equation (4.20) using the following identity:
4.21
Adding equation (4.21) to equation (4.20) yields equation (4.11). In order to establish equation (4.21), using analyticity considerations similar to those used above, we find
4.22
Letting , the residue contributions from the pole at *k*=0 cancel and equation (4.22) follows, where the integrals are understood in the principal value sense, i.e.
Now the definitions of *A*(*k*) and *B*(*k*) imply
where
Thus, *A*(0)=*B*(0)=0. In addition, the functions *A*(*k*) and *B*(*k*) are even functions of *k*, thus *A*′(0)=*B*′(0)=0, and these relations imply that each bracket appearing in the integrals of equation (4.11) vanishes at *k*=0. ■

### Remark 4.6 (comparison with the classical representations).

The Neumann problem of the Poisson equation in a circular wedge can be solved using either a cosine series in *θ* or the Mellin sine transform in *r*
and
Stakgold (1967, p. 316). The cosine-series solution is uniformly convergent at *r*=*a*, but not at *θ*=±*α*; the Mellin sine transform solution is uniformly convergent at *θ*=±*α*, but not at *r*=*a*. The solution (4.11) is the best possible representation: it is an integral that is uniformly convergent at *θ*=±*α* 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 the classical transform method?’. Table 1 compares the method of Fokas (1997) and the classical transform method, and shows that the method of Fokas (1997) requires less mathematical input, is simpler to implement, yields more useful solution formulae and is more widely applicable than the classical transform method. In §4, this method was applied to two BVPs in separable domains with separable boundary conditions. The method can also be used to solve BVPs in non-separable domains (Dassios & Fokas 2005; Fokas 2008; Spence 2010; Fokas & Kalimeris submitted; Kalimeris & Fokas 2010), as well as in separable domains with 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 triangle was first discovered by Shanin (1997; later published as Shanin (2000)). In these papers, the eigenvalues and eigenfunctions of the Laplacian in an equilateral triangle for impedance boundary conditions are found, and the Dirichlet problem of the Helmholtz equation is solved in terms of an infinite series (an IR of the solution is also obtained by substituting an IR of the fundamental solution into Green’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 classical transform method by constructing the analogue of Green’s IR in the transform space. Furthermore, for the elimination of the unknown boundary values, a crucial role is played by analyticity and Cauchy’s theorem. Several investigations have made some attempts in this direction (e.g. Croisille & Lebeau 1999; Gautesen 2005; Bernard 2006), however, it appears that this ‘synthesis’ has not been achieved before.

#### (iii) Integral representations in the complex plane

Representations of solutions to ODEs as integrals in the complex plane were pioneered by Laplace following earlier investigations by Euler. In hindsight, the ‘moral’ of the Watson transformation is that the best representation of the solution of a separable PDEs is an IR in the complex *k*-plane (which can be deformed to either of the two representations obtained by transforms). This representation 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 the emergence of the method of Fokas (1997).

#### (iv) The Sommerfeld–Malyuzhinets technique

The Sommerfeld–Malyuzhinets (S–M) technique is a method for solving the Helmholtz equation in a wedge by assuming that the solution can be written as an 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 of variables *k*=*e*^{iϕ}, *z*=*re*^{iθ} in the exponential appearing in the IR of the Helmholtz equation in a convex polygon (see Spence & Fokas 2010, proposition 2.7) transforms it to , which appears in the solutions given by the S–M method (recall that the frequency is 2*β*). Although a detailed investigation of this connection will be presented elsewhere, we can already make the following remarks: (i) The S–M technique is a method for solving only the Helmholtz equation in a particular domain, whereas the method of Fokas (1997) is applicable to a wide variety of PDEs and domains, (ii) the solution formula of S–M is an integral involving a contour in the complex plane that is independent of the wedge angle, whereas the solution given by the method of Fokas (1997) involves integrals over contours that depend on the domain, and (iii) the S–M technique requires the solution of a functional-difference equation as well as Cauchy’s theorem, whereas the method of Fokas (1997) requires only Cauchy’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 and one time dimensions in Fokas (2002) and Kalimeris & Fokas (2010). Regarding elliptic PDEs, we note the following: appropriate domain-dependent fundamental solutions for separable domains in three dimensions can be constructed using the method of Spence & Fokas (2010) and are expressed as double integrals over two complex parameters. The construction of the GR is similar to the construction presented here, but now the GR contains two complex parameters. The solution of BVPs in three dimensions also follows the steps outlined in §1*b*, but is more involved due to the fact that the solution, like the domain-dependent fundamental solutions, is expressed as a double integral.

## Acknowledgements

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

## Footnotes

- Received October 1, 2009.
- Accepted February 4, 2010.

- © 2010 The Royal Society