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 this paper, we present a simplified way of obtaining these representations for elliptic PDEs; namely, we introduce an algorithm for constructing particular, domain-dependent, IRs of the associated fundamental solutions, which are then substituted into Green's IRs. Furthermore, we extend this new method from BVPs in polygons to BVPs in polar coordinates. In the sequel to this paper, these results are used to solve particular BVPs, which elucidate the fact that this method has substantial advantages over the classical transform method.
In 1747, d'Alembert derived the wave equation, which was the first PDE in the history of mathematics. Soon after, d'Alembert and Euler discovered a general method for constructing large classes of solutions, namely the method of separation of variables. Daniel Bernoulli, in his attempt to solve the wave equation, introduced the infinite sine series, and Euler discovered the standard formula for the coefficients of a Fourier series. Fourier, in his attempts to understand heat diffusion, inaugurated in 1807 the era of linearization that dominated mathematical physics for the first half of the nineteenth century. In 1828, Green introduced the powerful approach of integral representations (IRs) that can be obtained via Green's functions (or more precise via the fundamental solutions). Separation of variables led to the spectral analysis of ordinary differential operators and to the solution of PDEs via a transform pair. The prototypical such pair is the Fourier transform, with variations including the sine, cosine, Laplace and Mellin transforms, as well as their discrete analogues.
In 1967, a new method, called the inverse scattering transform (IST) method, was introduced for solving the initial-value problem of certain nonlinear evolution PDEs called ‘integrable’. These equations include the celebrated Korteweg–de Vries and nonlinear Schrödinger equations. The distinguishing feature of these equations is the existence of a Lax pair; namely each of these equations can be written as the compatibility condition of two linear eigenvalue equations forming the associated Lax pair. Gelfand and one of the authors have emphasized (Fokas & Gelfand 1994) that the IST is based on a deeper type of separability. Indeed, the spectral analysis of the t-independent part of the Lax pair yields an appropriate nonlinear Fourier transform pair. It should be noted that, after constructing this nonlinear transform, the t-dependent part of the Lax pair yields the time evolution of the nonlinear Fourier data; in this sense, in spite of the fact that the IST is applicable to nonlinear PDEs, IST still follows the logic of separation of variables.
A unified approach for solving both linear and integrable nonlinear PDEs in two dimensions, motivated by the IST method, was introduced by one of the authors in 1997 (Fokas 1997). This method goes beyond separation of variables. Indeed, this method is based on the simultaneous spectral analysis of both parts of the Lax pair, and thus, it corresponds to the synthesis as opposed to separation of variables.
For integrable nonlinear PDEs, the simultaneous spectral analysis of the associated Lax pair is the only known method for constructing the novel IRs of the method of Fokas (1997). However, for linear PDEs, in addition to this method, there exist several other approaches. Among these approaches, it appears that for evolution PDEs, the simplest approach is to use the Fourier transform and contour deformations (Fokas 2002), whereas for elliptic PDEs, the simplest approach is to employ Green's IRs (Fokas & Zyskin 2002). This paper presents a simplification of the latter approach and also extends the method to PDEs in polar, as opposed to Cartesian, coordinates.
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(x)=known, x∈∂Ω,
— Neumann: (∂u/∂n)(x)=known, x∈∂Ω, and
— Robin: (∂u/∂n)(x)+αu(x)=known, α=constant, x∈∂Ω,
where (∂u/∂n)=∇u⋅n, where n is the unit outward-pointing normal to Ω. More complicated boundary conditions can involve derivatives at angles to the boundary. One can also prescribe mixed boundary conditions, such as Dirichlet on part of the domain, and Neumann on another part.
(b) Green's integral representation
Green's theorem gives the following 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 where δ denotes the Dirac delta function, and dS and dV denote the surface and volume elements. 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 K0 a modified Bessel function.
(c) The method of Fokas (1997 )
The method of Fokas (1997) has two basic ingredients,
— the integral representation 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. In this paper, the IR is 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.
(ii) The global relation
The GR is Green's divergence form of the equation integrated over the domain, where one employs a one-parameter family of solutions of the adjoint equation instead of the fundamental solution. Indeed, 1.4 where v is any solution of the adjoint of equation (1.1), 1.5 (equation (1.5) 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 coordinates yields four solutions of the adjoint equation, . All equations (1.4) 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 involves precisely 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 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 the classical methods. In particular, for linear evolution PDEs, the effective numerical evaluation of the solution is given in Flyer & Fokas (2008).
(d) Plan of the paper
The novel IRs for the interior of a convex polygon and for certain domains in polar coordinates are obtained in §§2 and 3, respectively. This is achieved by first constructing appropriate ‘domain-dependent fundamental solutions’ and then substituting these into Green's IR. These results are discussed further in §4.
In the sequel to this paper (Spence & Fokas 2010), the GR for these domains are constructed, and then the IRs and GR are used to solve certain BVPs for the Helmholtz equation in a wedge and for the Poisson equation in a circular wedge.
2. Integral representations for polygons
We identify with . Let z=x+iy be the physical variable, and z′=ξ+iη the ‘dummy variable’ of integration. We will consider u both as a function of (x,y), and as a function of interchangeably. Using the chain rule, equation (1.1) becomes For the Helmholtz equation, λ=4β2, , and for the modified Helmholtz equation, λ=−4β2, . The subscripts ‘T’ and ‘N’ will denote tangential and normal, respectively.
(b) Coordinate-system-dependent fundamental solutions
In Cartesian coordinates, equation (1.3) is 2.1 The differential operator on possesses the completeness relation 2.2 i.e. the Fourier transform (Stakgold 1967a, ch. 4). Similarly, the completeness relation for the η coordinate is 2.3 These two completeness relations give rise to the following IR of E: 2.4 where the contours of integration must be suitably deformed to avoid the poles of the integrand (the contours of equations (2.2) and (2.3) should be deformed before using them to obtain equation (2.4)).
Proposition 2.1 (the integral representation of E for the modified Helmholtz equation)
For the modified Helmholtz equation, the fundamental solution E is given by 2.5 where . (Note that if or θ+π, then the integral in equation (2.5) is not absolutely convergent.)
In a similar way, rotate the (k1,k2)-plane by letting 2.7 so that and equation (2.4) becomes 2.8
Now perform the kN integral by closing the contour in the upper half-plane (as XN≥0) to obtain 2.9
The right-hand side of equation (2.5) defines a function of z−z′ in a half-plane. To show that this expression defines a function of z−z′ in the whole plane (except zero), we must show that the right-hand sides of equation (2.5) for different values of θ are equal for z−z′ in the common domain of definition. This is achieved by rotating the contour, using Cauchy's theorem and the analyticity of the integrand in k. ■
The main differences between Helmholtz and modified Helmholtz equations are the following:
— there are two fundamental solutions and
— the contours of integrations for both the fundamental solutions contain circular arcs as well as rays in the complex plane.
These differences follow from the fact that, for λ>0, the IR of E equation (2.4) has poles on the contour; thus, it is not well defined. There are two different choices of contour that resolve this ambiguity, and these two choices yield two fundamental solutions.
Proposition 2.2 (the integral representation of Eout and Ein for the Helmholtz equation)
For the Helmholtz equation, the two fundamental solutions are given by 2.10a and 2.10b where and the contours Lout and Lin are shown in figure 2. (Note that if or θ+π, then the integrals in equations (2.10) are not absolutely convergent.)
For the Helmholtz equation, the fundamental solution is given by equation (2.4) with . As before, first perform the kN integral. For |kT|≥2β, the poles are on the imaginary axis (like for the modified Helmholtz equation) at . For |kT|≤2β, the poles are on the real axis, at , and the kN integral is not well defined unless the path of integration around the poles is specified. The two paths around the poles yielding non-zero contributions are given in figure 3. The two choices differ in their asymptotic behaviour at infinity, which correspond to outgoing or incoming waves, respectively (with the assumed time dependence e−iωt). The asymptotics can be determined by applying the method of steepest descent to equations (2.10). Perform the kN integration to obtain 2.11 with the top sign corresponding to and the bottom sign to . As with the modified Helmholtz equation, a change of variables can be used to eliminate the square roots. In this case, this change of variables is kT=β(l+1/l). This is motivated by the fact that, if l=eiϕ, then , and hence . Owing to the square roots, there are several choices for the range of integration in l, but all these choices lead to equivalent answers. In order to get the same integrand for both integrals in equation (2.11), we choose and l∈(0,−1) in the first integral, ϕ∈(−π,0) in the second for outgoing and ϕ∈(0,π) in the second for incoming. A few lines of computation, as well as relabelling l as k, yields equations (2.10). The proof that equations (2.10) defines a function of z−z′ in the whole plane follows by contour deformation in a way similar to that for the modified Helmholtz equation. ■
For the Poisson equation, this algorithm of constructing representations of the fundamental solution results in a representation that is formal; namely it involves divergent integrals. This is a consequence of the fact that the fundamental solution for the Poisson equation, , does not decay at infinity, and so its Fourier transform is not well defined in a classical sense. Nevertheless, when the representation of the fundamental solution is substituted into Green's IR, the resulting IR for u is well defined.
Proposition 2.3 (the integral representation of E for the Poisson equation)
For Poisson's equation, a formal representation of the fundamental solution is given by 2.12 where . (Note that if or θ+π, then the integrals in equation (2.12) are not absolutely convergent.)
Start with equation (2.4) with λ=0 and perform the same change of variables, equations (2.6) and (2.7), as for the modified Helmholtz equation to obtain equation (2.8) with β=0. Deform the kT contour above or below the singularity at 0; so it passes through iε (figure 4). Perform the kN integral by closing the contour in the upper half-plane to enclose the pole at kN=ikT for ℜkT>0 and kN=−ikT for ℜkT<0, to obtain 2.13 Now, let kT↦−kT in the second integral and use equation (2.6) to obtain the right-hand side of equation (2.12) without the limit (after relabelling kT as k).
The rotation of contours which shows that equations (2.5) and (2.10) are well-defined functions of z−z′ in the whole plane requires that the contours start from zero and hence fails for equation (2.13). Taking the limit allows this argument to proceed for equation (2.13); however, the integrals do not converge in this limit. ■
Remark 2.4 (derivation via one transform)
The IRs of E have been obtained by taking two transforms to obtain equation (2.4), and then computing one integral. Alternatively, they can be obtained by taking one transform, and then solving one ODE. For example, after rotating coordinates using equation (2.6), equation (2.1) with λ=−4β2 becomes Taking the Fourier transform in XT yields which can be solved using a one-dimensional Green function to give Then, the inverse Fourier transform yields equation (2.9).
(c) Domain-dependent fundamental solutions for a convex polygon
Let Ω(i) be the interior of a convex polygon in (figure 5). Let ∂Ω denote the boundary of the polygon, oriented anticlockwise, where the vertices of the polygon z1,z2,…,zn are labelled anticlockwise. Let Sj be the side (zj,zj+1) and let αj=arg(zj+1−zj) be the angle of Sj.
Let z′∈Sj, z∈Ω(i).
— for the modified Helmholtz equation, 2.14 where 2.15
— for the Helmholtz equation, 2.16 where 2.17
— for the Poisson equation, 2.18 where
(d) The integral representations for a convex polygon
The IRs are obtained by substituting the domain-dependent fundamental solutions into Green's IR.
If u is a solution of equation (1.1) for Ω=Ω(i), then Green's IR is 2.19a 2.19b
If the polygon is open, then u must obey the following boundary conditions at infinity:
— λ=0 (Poisson), as ,
— λ=−4β2 (modified Helmholtz), as , and
— λ=4β2 (Helmholtz), 2.20
(e) The modified Helmholtz equation
Proposition 2.6 (the integral representation of the solution of the modified Helmholtz equation)
Let λ=−4β2. Let u be a solution of equation (1.1) in the domain Ω(i). Suppose that u has the IR (2.19). Then, u also has the alternative representation 2.21 where lj, j=1,…,n, are rays in the complex k-plane, oriented from 0 to and defined by equation (2.15). The transforms of the boundary values of u on the side j, denoted by , are given by 2.22 where n is the outward-pointing normal to the polygon. The forcing term is given by 2.23 where E is the fundamental solution for the modified Helmholtz equation.
(f) The Helmholtz equation
For the Helmholtz equation, there are two fundamental solutions. For Ω unbounded, we can choose one of these solutions using the requirement that we require outgoing waves. However, when Ω=Ω(i), the domain is bounded and hence it is not possible to choose between the outgoing and incoming fundamental solutions. It turns out that substituting the IR of either of these fundamental solutions into Green's IR yields the same result for Ω=Ω(i).
Proposition 2.7 (the integral representation of the solution of the Helmholtz equation)
Let λ=4β2. Let u be a solution of equation (1.1) in the domain Ω(i). Suppose that u has the IR (2.19). Then, u also has the alternative representation 2.24 where Loutj, j=1,…,n, are rays in the complex k-plane oriented from 0 to and defined by equation (2.17 ), where the contour Lout is shown in figure 2. The transforms of the boundary values of u on the side j, denoted by , are given by 2.25 where n is the outward-pointing normal to the polygon. The forcing term is given by 2.26 where E is the fundamental solution for the Helmholtz equation.
Proof of proposition 2.7
This follows in exactly the same way as for the modified Helmholtz equation, using, of course, equation (2.16) instead of equation (2.14). If Ein is used in equation (2.19) instead of Eout, then we obtain equation (2.24) with Lout replaced by Lin. In order to show that equation (2.24) and the corresponding representation obtained by replacing Lout with Lin are equivalent, it is necessary to use the following GR for the Helmholtz equation: 2.27 This equation is derived in Spence & Fokas (2010, see proposition 3.2). For simplicity, consider f=0 (the case of non-zero f is considered in Spence 2010). The two representations differ by Indeed, adding the circle that is oriented anticlockwise to Lout transforms Lout to Lin; similar considerations apply to the contours Lout j, which are just Lout rotated by certain angles. Owing to the GR (2.27), the above integral vanishes. ■
(g) The Poisson equation
The representation of the fundamental solution for the Poisson equation (2.18) is formal since the integrals do not converge in the limit of . However, after the domain-dependent fundamental solution (2.18) is substituted into Green's IR, the limit does exist because the solution u satisfies the following consistency condition: 2.28 This equation can be obtained by integrating equation (1.1) over Ω and applying Green's theorem (recall that ∂Ω is oriented anticlockwise). These conditions imply that k=0 is a removable singularity.
Proposition 2.8 (the integral representation of the solution of Poisson's equation)
Let λ=0. Let u be a solution of equation (1.1) in the domain Ω(i). Suppose that u has the IRs (2.19). Then, u also has the alternative representation 2.29 The transforms of the boundary values of u on the side j, denoted by and , are defined by 2.30 and 2.31 The contours lj, j=1,…,n, are the same as in proposition 2.6, equation (2.15) and , are the complex conjugates of lj, j=1,…,n, that is, the rays in the complex k-plane oriented towards infinity are defined by 2.32 The forcing term is given by 2.33 where E is the fundamental solution for the Poisson equation.
Remark 2.9 (the removable singularity at k =0 and consistency conditions)
Proof of proposition 2.8
Remark 2.10 (non-convex polygons)
The coordinate-system-dependent fundamental solutions of §2b can also be used to obtain IRs for non-convex polygons (Charalambopoulos et al. 2010; Spence 2010). However, in this case, the contours of integration depend on the position of z; namely they depend on the position of z relative to each side of the polygon.
Remark 2.11 (the forcing term)
In order to obtain a spectral representation of the forcing term in Green's IR using equations (2.5), (2.10) and (2.12), we require that for z′∈Ω and z∈Ω, for some θ. This is impossible unless we split the domain Ω into two regions by a line through z, where we are free to choose the angle of the split. This means that the transforms of the forcing term depend on z, which is inconvenient, but apparently unavoidable (Spence 2010). A spectral representation of the forcing term is useful for the numerical evaluation of the solution, since it is then possible to use contour deformations to obtain integrands that decay exponentially as .
(h) Relation to classical results
The idea of obtaining IRs of E is certainly not new. However, the representations presented here apparently contain certain novel features explained below.
(i) Rotation to half-planes
Starting with equation (2.4), one of the following operations is usually performed:
— Rotate the k1,k2 coordinates so that k2 lies in the direction of (x−ξ,y−η); for the modified Helmholtz equation, this yields Computing the k2 integral, this becomes 2.34 Such IRs appear in Stakgold (1967b, pp. 56–57 and 279) and Abramowitz & Stegun (1965, p. 376, eqns (9.6.23) and (9.6.24)).
— Compute the k2 integral without first performing a rotation; for the modified Helmholtz equation, this yields which is equation (2.9) with θ=0 in equation (2.6). This appears in Ablowitz & Fokas (2003, p. 298); the analogue for the Helmholtz equation, which is equation (2.11) with θ=0 in equation (2.6), appears in Duffy (2001, p. 278, eqn (5.1.22)) (actually, eqn (5.1.22) has an error in the sign of the square root in the exponent, but eqn (5.1.21), from which it is obtained, is correct).
(ii) Change of variables to eliminate square roots
The transformations kT=β(l−1/l) for the modified Helmholtz equation and kT=β(l+1/l) for the Helmholtz equation, transform and into β(l+1/l) and β(l−1/l), respectively (modulo, a sign depending on the range of l). These transformations eliminate the square roots at the cost of introducing a pole at . These transformations have been used earlier but only in polar coordinates, in particular using them in the two analogues of equation (2.34) for the Helmholtz equation yields the IRs for and involving the contours of integration Lout and Lin, respectively (Watson 1966, §6.21, p. 179). It is surprising that apparently these transformations have not been used before in the solution of the Helmholtz equation in Cartesian coordinates. Indeed, Ockendon et al. (2003, §5.8.3, p. 191) states that ‘in a half-plane, transform methods for Helmholtz’ equation … are cursed by the presence of branch points in the transform plane’.
It is important to note that it is not possible to obtain the IRs of propositions 2.6–2.8 using the representations of E of the form equation (2.34) without first transforming equation (2.34) into the representations of propositions 2.1–2.3. This is because the integrand of equation (2.34) cannot be written as a function of (x,y) multiplied by a function of (ξ,η), and thus it is not possible to interchange the physical and spectral integrals when equation (2.34) is substituted into Green's IR.
3. Integral representations for polar coordinates
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=β2 (note that this is different to λ=4β2 used for the Helmholtz equation in §2).
(b) Coordinate-system-dependent fundamental solutions
In polar coordinates, equation (1.3) becomes 3.1
For reasons that will be explained later (see remark 3.4 and §3d), we consider the non-periodic fundamental solution Es defined by 3.2
Proposition 3.1 (the integral representation of Es for the Helmholtz equation)
For the Helmholtz equation, the outgoing non-periodic fundamental solution Es can be expressed in terms of radial eigenfunctions (the radial representation) in the form 3.3 Alternatively, it can be expressed in terms of angular eigenfunctions (the angular representation) in the form 3.4 where , and .
The differential operator on possesses the completeness relation 3.5 i.e. the Fourier transform (Stakgold 1967a, ch. 4). The completeness relation associated with the operator 3.6 on , with the additional condition that the eigenfunctions satisfy the outgoing radiation condition, is 3.7 where either r1=r,r2=ρ or vice versa (remark 3.2). The completeness relations (3.5) and (3.7) give rise to the following IR of E: 3.8 Choose r1=ρ and r2=r, so that Es satisfies the outgoing radiation condition in ρ. (In fact, r and ρ can be interchanged in the right-hand side of equation (3.3) using an identity involving integrals of Bessel functions (Spence 2010).) If θ>ϕ, close the k2 integral in the upper half-k2-plane, enclosing the pole at k2=k1 for ℑk1≥0 and at k2=−k1 for ℑk1≤0. Relabel k1 as k to obtain equation (3.3) for θ>ϕ. Similarly, if θ<ϕ, close the lower half-k2-plane to obtain equation (3.3) for θ<ϕ.
Evaluating the k1 integral requires knowledge of the asymptotics of the product of the Bessel functions as . For x,y fixed, the following formulae are valid: 3.9 and 3.10 Hence, the product is bounded at infinity only when ℜk>0 and x>y.
For r1>r2, deform the k1 contour to some contour in the right half-plane, enclosing the pole at k1=|k2|, with as for ℑk1>0 and as for ℑk1<0 (figure 6). Now, the k1 integral converges absolutely even with ε=0; so, by the dominated convergence theorem, ε can be set to zero in the integrand. After calculating the residue, this results in Relabel k2 as k and let k↦−k for k<0 to obtain equation (3.3). The proof that equations (3.3) and (3.4) define the same function follows by deforming the integrals on and to using equation (3.9) and the fact, mentioned earlier, that r and ρ can be interchanged in the right-hand side of equation (3.3). ■
Remark 3.2 (the Kontorovich–Lebedev transform)
Jones (1980) has proved rigorously that the following transform pair is valid: 3.11 and 3.12 where if the eigenfunctions satisfy the outgoing radiation condition, j=1 and if the eigenfunctions satisfy the incoming radiation condition, j=2. Jones has shown that equation (3.12) without the regularizing term (which is obtained by the spectral analysis of equation (3.6)) diverges even for the function g(y)=e−ay, ℜa>0. Essentially, the reason for this divergence is that the product of Hk and Jk is unbounded on . However, most of the solutions to BVPs obtained by using equations (3.11) and (3.12) without the term are correct because the contour is deformed (albeit illegally) and the resulting expression converges. The term justifies rigorously the contour deformation, after which ε can be set to zero.
For the Poisson equation, the algorithm results in formal representations of the non-periodic fundamental solution, Es, as in the case of Cartesian coordinates. Nevertheless, when these representations are substituted into Green's IR, the resulting representation for u is well defined.
Proposition 3.3 (the integral representation of Es for the Poisson equation)
For the Poisson equation, the non-periodic fundamental solution Es can be expressed formally either in terms of radial eigenfunctions (the radial representation ) in the form 3.13 or in terms of angular eigenfunctions (the angular representation) in the form 3.14 where , , .
The operator in θ is the same as for the Helmholtz equation, and so the appropriate completeness relation is equation (3.5). The differential operator on possesses the completeness relation 3.15 i.e. the Mellin transform (Stakgold 1967a, ch. 4, p. 308). After rewriting equation (3.2) in the form the above two completeness relations give rise to the following IR of E: 3.16 If θ>ϕ, deform the k1 integral off near the origin so that it passes through ε≠0. Then, close the k2 integral in the upper half-k2-plane, enclosing the pole at k2=k1 for ℑk1≥0 and at k2=−k1 for ℑk1≤0. Relabel k1 as k to obtain equation (3.13), without the limit, for θ>ϕ. Similarly, if θ<ϕ, close in the lower half-k2-plane to obtain equation (3.13), without the limit, for θ<ϕ.
If ρ>r, deform the k2 integral off near the origin so that it passes through iε, ε≠0. Then, close the k1 integral in the left half-k1-plane enclosing the pole at k1=−k2 for ℜk2≥0 and at k1=k2 for ℜk2≤0. Let k2↦−k2 for ℜk1≤0 and relabel k2 as k to obtain equation (3.14) for r>ρ. Similarly, if ρ<r, close in the right half-k1-plane to obtain equation (3.14) for ρ<r without the limit. The proof that equations (3.13) and (3.14) define the same function follows by contour deformation in a manner similar to the analogous proof for the Helmholtz equation, except that the differences become zero in the limit . ■
Remark 3.4 (periodicity)
For some domains in polar coordinates, periodicity in the angle coordinate is natural. The operator on (0,2π) with θ periodic possesses the completeness relation The analogue of equation (3.8) is now 3.17 Computing the k1 integral yields the angular expansion 3.18 Equation (3.18) is the analogue of equation (3.4). However, the summation in equation (3.17) cannot be computed and therefore there does not exist an analogue of equation (3.3). This is consistent with the following construction: if one attempts to obtain the radial representation for the Helmholtz equation by taking the Kontorovich–Lebedev transform in ρ of equation (3.1) (see remark 2.4), one finds For solutions periodic in ϕ, , and under this restriction, the Kontorovich–Lebedev transform cannot be inverted. Similar considerations apply for the Poisson equation.
The fact that there do not exist radial representations under periodicity implies that there do not exist IRs in the transform space for any domain other than the interior and exterior of the circle. The so-called ‘hybrid method’ of Dassios & Fokas (2008) for solving BVPs in the interior and exterior of the sphere uses the analogue of equation (3.18) in three dimensions to obtain an IR in the transform space (the construction of the GR and the solution of BVPs in three dimensions follow steps similar to the method in two dimensions described in this paper).
Remark 3.5 (the derivation of both coordinate-dependent fundamental solutions using only one completeness relation)
In the proof of proposition 3.1, it was shown that equations (3.3) and (3.4) can be obtained from each other by contour deformation. Combining this with remark 2.4 shows that both equations (3.3) and (3.4) can actually be obtained using either one of the completeness relations (3.5) or (3.7), although this approach is less algorithmic than the approach in the proofs of proposition 3.1.
(c) Novel integral representations for certain domains in polar coordinates
In the sequel (Spence & Fokas 2010), the coordinate-dependent fundamental solutions of propositions 3.1 and 3.3 are used to construct novel IRs for two examples in polar coordinates: the Helmholtz equation in the wedge, 3.19 and the Poisson equation in the circular wedge, 3.20
After constructing the associated GR, certain BVPs are solved using the steps outlined in the introduction.
(d) Relation to classical results
The novel idea of §3b is to consider the non-periodic fundamental solution, Es, instead of E (recall that there does not exist a radial representation under periodicity). The non-periodic fundamental solution was first introduced by Sommerfeld in 1896 when he extended the method of images to solve the problem of diffraction by a half-line, that is the Helmholtz equation in the domain Sommerfeld required 4π periodicity under which the problem is solvable by images. Similarly, letting (i.e. θ is not periodic) converts a wedge of arbitrary angle (less than 2π) into an infinite strip, which can then be solved using an infinite number of images. Sommerfeld expressed both his 4π periodic and non-periodic fundamental solutions (called ‘Riemann-surface’ or ‘branched’ solutions) as integrals of exponentials over the so-called Sommerfeld contours (Sommerfeld 1964, p. 249). The expansion of the non-periodic fundamental solution in angular eigenfunctions (3.4) is presented in Stakgold (1967b, p. 270). However, perhaps due to the unfamiliarity with the Kontorovich–Lebedev transform, the radial representation (3.3) does not appear to be known.
The periodic angular representation (3.18) and its analogue in three dimensions is well known (e.g. Morse & Feshbach 1953, vol. 1, p. 827); these authors are aware of the non-availability of a periodic radial representation (vol. 1, p. 829).
IRs of the solution of the homogeneous version of equation (1.1) in a convex polygon were given in Fokas (2001) using the simultaneous spectral analysis of the associated Lax pair (or equivalently the spectral analysis of an equivalent differential form). In addition, a representation for the derivative, uz, of the solution of the Laplace equation was also obtained. In Fokas & Zyskin (2002), these representations were re-derived by substituting equation (2.4) into Green's IRs and performing one k integration; the analogous method for deriving the IRs for evolution PDEs was first presented in Bressloff (1997).
BVPs for the inhomogeneous equation were solved in Fokas & Pinotsis (2006) by first subtracting off a particular solution and then solving the homogeneous equation (in general, this leads to a more complicated expression for the solution).
The main achievements of this paper are the following:
— IRs are presented for the inhomogeneous equation (1.1),
— the IRs contain only the Dirichlet and Neumann boundary values,
— an IR for the solution u of the Poisson equation has been obtained (instead of an IR for the derivative uz), and
— the method has been extended from polygonal domains to domains in polar coordinates.
In addition, the connection with Green is now better understood and the domain-dependent fundamental solutions provide a simpler method than Fokas & Zyskin (2002) or Fokas (2001) for obtaining the novel IRs.
In the conclusions of the sequel (Spence & Fokas 2010), a direct comparison is made between this method and the classical transform method, and the method is also placed in context with respect to other approaches, such as the Sommerfeld–Malyuzhinets technique.
E.A.S. was supported by the EPSRC and the Cambridge Philosophical Society, and A.S.F. by a Guggenheim fellowship.
- Received October 1, 2009.
- Accepted February 4, 2010.
- © 2010 The Royal Society