Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

David A. Smith
Department of Mathematics, University of Reading RG6 6AX
email: d.a.smith@reading.ac.uk
Abstract

We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas’ transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.

The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

1 Introduction

In this work, we consider

The initial-boundary value problem : Find which satisfies the linear, evolution, constant-coefficient partial differential equation

 ∂tq(x,t)+a(−i∂x)nq(x,t)=0 (1.1)

subject to the initial condition

 q(x,0)=q0(x) (1.2)

and the boundary conditions

 (1.3)

where the pentuple is such that

the order ,

the boundary coefficient matrix is in reduced row-echelon form,

if is odd then the direction coefficient , if is even then for some ,

the boundary data and the initial datum are compatible in the sense that

 A(q(n−1)0(0),q(n−1)0(1),q(n−2)0(0),q(n−2)0(1),…,q0(0),q0(1)){{{T}}}=h(0). (1.4)

Provided is well-posed, in the sense of admitting a unique, smooth solution, its solution may be found using Fokas’ unified transform method [5, 7]. The representation thus obtained is a contour integral of transforms of the initial and boundary data. Certain problems, for example those with periodic boundary conditions, may be solved using classical methods such as Fourier’s separation of variables [10], to yield a representation of the solution as a discrete Fourier series. By the well-posedness of , these are two different representations of the same solution.

For individual examples, Pelloni [13] and Chilton [2] discuss a method of recovering a series representation from the integral representation through a contour deformation and a residue calculation. Particular examples have been identified of well-posed problems for which this deformation fails but there is no systematic method of determining its applicability.

Pelloni [12] uses Fokas’ method to decide well-posedness of a class of problems with uncoupled, non-Robin boundary conditions giving an explicit condition, the number that must be specified at each end of the space interval, whose validity may be ascertained immediately. However there exist no criteria for well-posedness that are at once more general than Pelloni’s and simpler to check than the technical ‘admissible set’ characterisation of [7].

The principal result of this work is a new characterisation of well-posedness. The condition is the decay of particular integrands within certain sectors of the complex plane. Indeed, let . Then

Theorem 1.1.

The problem is well-posed if and only if is entire and the ratio

 ηj(ρ)ΔPDE(ρ)→0% as ρ→∞ from within D, away from the zeros of ΔPDE. (1.5)

for each .

We provide a small contribution to Fokas’ method, making it fully algorithmic. We express the solution in terms of the PDE characteristic determinant, , the determinant of the matrix

 Akj(ρ)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩c(Jj−1)/2(ρ)⎛⎜⎝ω(n−1−[Jj−1]/2)(k−1)∑r∈ˆJ+−∑r∈ˆJ+αˆJ+r(Jj−1)/2ω(n−1−r)(k−1)(iρ)(Jj−1)/2−r+e−iωk−1ρ∑r∈ˆJ−αˆJ−r(Jj−1)/2ω(n−1−r)(k−1)(iρ)(Jj−1)/2−r⎞⎟⎠Jj odd,cJj/2(ρ)⎛⎜⎝−ω(n−1−Jj/2)(k−1)e−iωk−1ρ∑r∈ˆJ+−∑r∈ˆJ+βˆJ+rJj/2ω(n−1−r)(k−1)(iρ)Jj/2−r+e−iωk−1ρ∑r∈ˆJ−βˆJ−rJj/2ω(n−1−r)(k−1)(iρ)Jj/2−r⎞⎟⎠Jj even. (1.6)

The matrix appears in the generalised spectral Dirichlet to Neumann map derived in Section 2. The application of the map to the formal result Theorem 2.1 yields the following implicit equation for , the solution of .

Theorem 1.2.

Let be well-posed with solution . Then may be expressed in terms of contour integrals of transforms of the boundary data, initial datum and solution at final time as follows:

 2πq(x,t)=∫Reiρx−aρnt^q0(ρ)dρ−∫∂D+eiρx−aρnt∑j∈J+ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ)dρ−∫∂D−eiρ(x−1)−aρnt∑j∈J−ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ)dρ, (1.7)

where the sectors and .

Equation (1.7) gives only an implicit representation of the solution as the functions are defined in terms of the Fourier transform of the solution evaluated at final time, which is not a datum of the problem. Nevertheless the importance of the PDE characteristic determinant is clear. The integrands are meromorphic functions so depends upon their behaviour as from within and upon their poles, which can only arise at zeros of . It is the behaviour at infinity that is used to characterise well-posedness in Theorem 1.1, the proof of which is given in Section 3.

In Section 4 we derive two representations of the solution of an initial-boundary value problem. Let be a sequence containing each nonzero zero of precisely once and define the index sets

 KR={k∈N:σk∈R}, K+={k∈N:Imσk⩾0}, K−={k∈N:Imσk<0}.

Then the following theorems give representations of the solution to the problem .

Theorem 1.3.

Let the problem be well-posed. Then the solution may be expressed using contour integrals of transforms of the initial and boundary data by

 (1.8)
Theorem 1.4.

Let and let the problems and be well-posed. Then the solution of may be expressed as a discrete series of transforms of the initial and boundary data by

 (1.9)

The final integral term in both equations (1.8) and (1.9) depends upon , a linear combination of -transforms of the boundary data which evaluates to if . Hence if is a homogeneous initial-boundary value problem then the final term makes no contribution to equations (1.8) and (1.9).

Special cases of Theorem 1.3 have appeared before but the representations differ from equation (1.8). The result is shown for several specific examples in [8, 13], including a second order problem with Robin boundary conditions. For simple boundary conditions, the result is mentioned in Remark 6 of [7] and Lemma 4.2 of [12] contains the essence of the proof. Unlike earlier forms, equation (1.8) represents using discrete series as far as possible; only the parts of the integral terms that cannot be represented as series remain. This may not have any advantage for computation but is done to highlight the contrast with equation (1.9).

In Theorem 1.4 the well-posedness of is used to show that the first two integral terms of equation (1.8) evaluate to zero. Under the map , maps to , the interior of its complement; we exploit this fact together with Theorem 1.1 to show the decay of

 ζj(ρ)ΔPDE(ρ) as % ρ→∞ from within ˜E.

This maximally generalises of the arguments of Pelloni and Chilton in the sense that the deformation of contours cannot yield a series representation of the solution to if is ill-posed.

Theorem 1.1 is useful because it reduces the complexity of the analysis necessary to prove that a particular initial-boundary value problem is well-posed but its use still requires some asymptotic analysis. It would be preferable to give a condition that may be validated by inspection of the boundary coefficient matrix and is sufficient for well-posedness. We discuss such criteria in Section 5.

Section 5 also contains a proof of the following result, complementing Theorem 1.4. This theorem highlights the essential difference between odd order problems, whose well-posedness depends upon the direction coefficient, and even order problems, whose well-posedness is determined by the boundary coefficient matrix only.

Theorem 1.5.

Let be even and . Using the notation of Theorem 1.4, the problem is well-posed if and only if is well-posed.

In Section 6 we investigate the PDE discrete spectrum, the set of zeros of the PDE characteristic determinant. We prove a technical lemma, describing the distribution of the which is used in the earlier sections. Under certain conditions we are able to exploit symmetry arguments to improve upon the general results Langer presents [11] for the particular exponential polynomials of interest.

2 Implicit solution of IBVP

In Section 2.1 we give the standard results of Fokas’ unified transform method in the notation of this work. In Section 2.2 we state and prove Lemma 2.6, the generalised spectral Dirichlet to Neumann map. In Section 2.3 we apply the map to the formal results of Section 2.1, concluding the proof of Theorem 1.2. The latter two sections contain formal definitions of many of the terms and much of the notation used throughout this work.

2.1 Fokas’ method

The first steps of Fokas’ transform method yield a formal representation for the solution of the initial-boundary value problem, given in the following

Theorem 2.1.

Let the initial-boundary value problem be well-posed. Then its solution may be expressed formally as the sum of three contour integrals,

 q(x,t)=12π(∫Reiρx−aρnt^q0(ρ)dρ−∫∂D+eiρx−aρntn−1∑j=0cj(ρ)˜fj(ρ)dρ−∫∂D−eiρ(x−1)−aρntn−1∑j=0cj(ρ)˜gj(ρ)dρ), (2.1)

where

 ˜fj(ρ) =∫T0eaρnsfj(s)ds, ˜gj(ρ) =∫T0eaρnsgj(s)ds, (2.2) fj(t) =∂jxq(0,t), gj(t) =∂jxq(1,t), ^q0(ρ) =∫10e−iρyq0(y)dy, cj(ρ) =−aρn(iρ)−(j+1).

The above theorem is well established and its proof, via Lax pair and Riemann-Hilbert formalism, appears in [4, 5, 7]. We state it here without proof to highlight the difference in notation to previous publications. We use to denote the spectral parameter, in place of in the earlier work. We use and exclusively to denote the boundary functions; even for simple boundary conditions in which some of the boundary functions are equal to boundary data we denote the boundary data separately by .

The transformed boundary functions are the unknowns in equation (2.1), of which at most may be explicitly specified by the boundary conditions (1.3). To determine the remaining or more we require a generalised Dirichlet to Neumann map in the form of Lemma 2.6. This is derived from the boundary conditions and the global relation.

Lemma 2.2 (Global relation).

Let be well-posed with solution . Let

 ^qT(ρ)=∫10e−iρyq(y,T)dy

be the usual spatial Fourier transform of the solution evaluated at final time. Then the transformed functions , , and satisfy

 n−1∑j=0cj(ρ)(~fj(ρ)−e−iρ~gj(ρ))=^q0(ρ)−eaρnT^qT(ρ),ρ∈C. (2.3)

The global relation is derived using an application of Green’s Theorem to the domain in the aforementioned publications. As the -transform,

 ˜X(ρ)=∫T0eaρntX(t)dt, (2.4)

is invariant under the map for any integer , the global relation provides a system of equations in the transformed functions to complement the boundary conditions.

2.2 Generalised spectral Dirichlet to Neumann map

We give a classification of boundary conditions and formally state the generalised spectral Dirichlet to Neumann map.

Notation 2.3.

Consider the problem , which need not be well-posed. Define . Define the boundary coefficients , to be the entries of such that

 (2.5)

We define the following index sets and functions.

such that is a pivot in for some , the set of columns of relating to the left of the space interval which contain a pivot.

such that is a pivot in for some , the set of columns of relating to the right of the space interval which contain a pivot.

, the set of columns of relating to the left of the space interval which do not contain a pivot.

, the set of columns of relating to the right of the space interval which do not contain a pivot.

such that such that , an index set for the boundary functions whose corresponding columns in do not contain a pivot. Also, the decreasing sequence of elements of .

such that such that , an index set for the boundary functions whose corresponding columns in contain a pivot. Also, the decreasing sequence of elements of .

The functions

the boundary functions whose corresponding columns in do not contain a pivot.

The functions

the boundary functions whose corresponding columns in contain a pivot.

, a sequence such that is a pivot in when .

, a sequence such that is a pivot in when .

Definition 2.4 (Classification of boundary conditions).

The boundary conditions of the problem are said to be

1. homogeneous if . Otherwise the boundary conditions are inhomogeneous.

2. uncoupled if

 if βkj is a pivot in A%thenαkr=0∀r.

Otherwise we say that the boundary conditions are coupled.

3. non-Robin if

that is each contains only one order of partial derivative. Otherwise we say that boundary condition is of Robin type. Note that whether boundary conditions are of Robin type or not is independent of whether they are coupled, unlike Duff’s definition [3].

4. simple if they are uncoupled and non-Robin.

The terms ‘generalised’ and ‘spectral’ are prefixed to the name ‘Dirichlet to Neumann map’ of the Lemma below to avoid confusion regarding its function.

Generalised: The boundary conditions we study are considerably more complex than those considered in [2, 4, 7, 8, 12, 13]. Indeed, as may specify any linear boundary conditions, the known boundary functions may not be ‘Dirichlet’ (zero order) and the unknown boundary functions need not be ‘Neumann’ (first order). Further, if has more than non-zero entries then the lemma must be capable of expressing more than unknown boundary functions in terms of fewer than known boundary data.

Spectral: Owing to the form of equation (2.1) we are interested not in the boundary functions themselves but in their -transforms, as defined in equations (2.2). It is possible, though unnecessarily complicated, to perform a generalized Dirichlet to Neumann map in real time and subsequently transform to spectral time but, as the global relation is in spectral time, to do so requires the use of an inverse spectral transform. Instead, we exploit the linearity of the -transform (2.4), applying it to the boundary conditions, and derive the map in spectral time.

The crucial component of the lemma is given in the following

Definition 2.5.

Let be an initial-boundary value problem having the properties but not necessarily well-posed. We define the PDE characteristic matrix by equation (1.6) and the PDE characteristic determinant to be the entire function

 ΔPDE(ρ)=detA(ρ). (2.6)
Lemma 2.6 (Generalised spectral Dirichlet to Neumann map).

Let be well-posed with solution . Then

1. The vector of transformed boundary functions satisfies the reduced global relation

 A(ρ)V(ρ)=U(ρ)−eaρnT⎛⎜ ⎜⎝^qT(ρ)⋮^qT(ωn−1ρ)⎞⎟ ⎟⎠, (2.7)

where

 U(ρ) (2.8) u(ρ,k) =^q0(ωk−1ρ)−∑l∈ˆJ+cl(ωk−1ρ)˜hˆJ+l(ρ)+e−iωk−1ρ∑l∈ˆJ−cl(ωk−1ρ)˜hˆJ−l(ρ) (2.9)

and is the function obtained by applying the -transform (2.4) to the boundary datum .

2. The PDE characteristic matrix is of full rank, is independent of and and differing values of only scale by a nonzero constant factor.

3. The vectors and of transformed boundary functions satisfy the reduced boundary conditions

 W(ρ)=(˜h1(ρ),˜h2(ρ),…,˜hn(ρ)){{{T}}}−ˆAV(ρ), (2.10)

where the reduced boundary coefficient matrix is given by

 ˆAkj={αk(Jj−1)/2Jj odd,βkJj/2Jj even. (2.11)
Proof.

Applying the -transform (2.4) to each line of the boundary conditions (1.3) yields a system of equations in the transformed boundary functions. As is in reduced row-echelon form it is possible to split the vector containing all of the transformed boundary functions into the two vectors and , justifying the reduced boundary conditions.

The reduced boundary conditions may also be written

 ˜fj(ρ) =˜hˆJ+j(ρ)−∑r∈˜J+αˆJ+jr˜fr(ρ)−∑r∈˜J−βˆJ+jr˜gr(ρ), for j ∈ˆJ+ and (2.12) ˜gj(ρ) =˜hˆJ−j(ρ)−∑r∈˜J+αˆJ−jr˜fr(ρ)−∑r∈˜J−βˆJ−jr˜gr(ρ), for j ∈ˆJ−. (2.13)

As the -transform is invariant under the map for any integer , the global relation Lemma 2.2 yields the system

 n−1∑j=0cj(ρ)ω(n−1−j)r˜fj(ρ)−n−1∑j=0e−iωrρcj(ρ)ω(n−1−j)r˜gj(ρ)=^q0(ωrρ)−eaρnT^qT(ωrρ),

for . Using the fact we split the sums on the left hand side to give

 ∑j∈ˆJ+cj(ρ)ω(n−1−j)r˜fj(ρ)+∑j∈˜J+cj(ρ)ω(n−1−j)r˜fj(ρ)−∑j∈ˆJ−e−iωrρcj(ρ)ω(n−1−j)r˜gj(ρ)−∑j∈˜J−e−iωrρcj(ρ)ω(n−1−j)r˜gj(ρ)=^q0(ωrρ)−eaρnT^qT(ωrρ),

for . Substituting equations (2.12) and (2.13) and interchanging the summations we obtain the reduced global relation.

The latter statement of (ii) is a trivial observation from the form of the PDE characteristic matrix. A full proof that is full rank is given in the proof of Lemma 2.17 of [14]. ∎

2.3 Applying the map

We solve the system of linear equations (2.7) for using Cramer’s rule hence, by equation (2.10), determining also.

Notation 2.7.

Denote by the determinant of the matrix obtained by replacing the  column of the PDE characteristic matrix with the vector and denote by the determinant of the matrix obtained by replacing the  column of the PDE characteristic matrix with the vector for and . Define

 (2.14)

for and . Define

 ζj(ρ)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩c(Jj−1)/2(ρ)ˆζj(ρ)cJj/2(ρ)ˆζj(ρ)c(J′j−n−1)/2(ρ)ˆζj(ρ)cJ′j−n/2(ρ)ˆζj(ρ)ηj(ρ)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩c(Jj−1)/2(ρ)ˆηj(ρ)Jj % odd,cJj/2(ρ)ˆηj(ρ)Jj even,c(J′j−n−1)/2(ρ)ˆηj(ρ)J′j−n%odd,cJ′j−n/2(ρ)ˆηj(ρ)J′j−n % even, (2.15)

for and define the index sets

 J+ ={j:Jj odd}∪{n+j:J′j odd}, J− ={j:Jj even}∪{n+j:J′j even}.

The generalised spectral Dirichlet to Neumann map Lemma 2.6 and Cramer’s rule yield expressions for the transformed boundary functions:

 ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩c(Jj−1)/2(ρ)˜f(Jj−1)/2(ρ)Jj odd,cJj/2(ρ)˜gJj/2(ρ)Jj even,c(J′j−n−1)/2(ρ)˜f(J′j−n−1)/2(ρ)J′j−n odd,cJ′j−n/2(ρ)˜gJ′j−n/2(ρ)J′j−n even, (2.16)

hence

 n−1∑j=0cj(ρ)˜fj(ρ) =∑j∈J+ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ), n−1∑j=0cj(ρ)˜gj(ρ) =∑j∈J−ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ).

Substituting these equations into Theorem 2.1 completes the proof of Theorem 1.2.

Remark 2.8.

There are several simplifications of the above definitions for specific types of boundary conditions.

If the boundary conditions are simple, as studied in [12], then . Hence, if the boundary conditions are simple and homogeneous then for each .

Non-Robin boundary conditions admit a significantly simplified form of the PDE characteristic matrix; see equation (2.2.5) of [14].

For homogeneous boundary conditions, is with replacing .

Remark 2.9.

It is possible to extend the results above to initial-boundary value problems for a more general linear, constant-coefficient evolution equation,

 ∂tq(x,t)+n∑j=0aj(−i∂x)jq(x,t)=0, (2.17)

with leading coefficient having the properties of . In this case the spectral transforms must be redefined with replacing and the form of the boundary coefficient matrix also changes. The appearing in equation (1.6) represent a rotation by , corresponding to a map between simply connected components of . The partial differential equation (2.17) has dispersion relation so is not simply a union of sectors but a union of sets that are asymptotically sectors; see Lemma 1.1 of [9]. Hence we replace with a biholomorphic map between the components of .

3 New characterisation of well-posedness

This section provides a proof of Theorem 1.1. The first subsection justifies that the decay condition is satisfied by all well-posed problems. The second subsection proves that the decay condition is sufficient for well-posedness.

We clarify the definitions of and from Section 1. By Lemma 6.1, there exists some such that the pairwise intersection of closed discs of radius centred at zeros of is empty. We define

 ˜D=D∖⋃k∈N¯¯¯¯B(σk,ε),˜E=E∖⋃k∈N¯¯¯¯B(σk,ε).

3.1 Well-posedness ⇒ decay

As the problem is well-posed, the solution evaluated at final time hence and are entire. Similarly, hence are entire and decay as from within . Hence, by equation (2.16),

 ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ)ck(ρ) (3.1)

is entire and decays as from within for each , where depends upon .

We define the new complex set

 D={ρ∈D such that −Re(aρnT)>2n|ρ|}.

As , the ratio (3.1) is analytic on and decays as from within . For , let be the  simply connected component of encountered when moving anticlockwise from the positive real axis and let . Then for each there exists such that the set

 Dp=(˜Dp∩D)∖¯¯¯¯B(0,R)

is simply connected, open and unbounded.

By definition, is an exponential polynomial whose terms are each

 W(ρ)e−i∑y∈Yωrρ

where is a monomial of degree at least and is an index set. Hence

 1ΔPDE(ρ)=o(en|ρ|ρ−1) as ρ→∞ or as ρ→0.

As and also grow no faster than , the ratios

 ζj(ρ)ΔPDE(ρ)ck(ρ),ηj(ρ)ΔPDE(ρ)ck(ρ)=o(e2n|ρ|ρ−1), as ρ→∞.

Hence the ratio

 eaρnTηj(ρ)ΔPDE(ρ)ck(ρ) (3.2)

decays as from within and away from the zeros of . However the ratio

 ζj(ρ)ΔPDE(ρ)ck(ρ) (3.3)

is the sum of ratios (3.1) and (3.2) hence it also decays as from within and away from the zeros of .

The terms in each of and are exponentials, each of which either decays or grows as from within one of the simply connected components of . Hence as from within a particular component the ratio (3.3) either decays or grows. But, as observed above, these ratios all decay as from within each . Hence the ratio (3.3) decays as from within .

Now it is a simple observation that the ratio

 ηj(ρ)ΔPDE(ρ)ck(ρ) (3.4)

must also decay as . Indeed ratio (3.4) is the same as ratio (3.3) but with replacing and, as observed above, also. Finally, the exponentials in and ensure that the ratio

 ηj(ρ)ΔPDE(ρ) (3.5)

also decays as from within . Indeed the transforms that multiply each term in ensure that the decay of ratio (3.4) must come from the decay of ratio (3.5), not from .

3.2 Decay ⇒ well-posedness

Many of the definitions of Section 2 require the problem to be well-posed. The statement of the following Lemma clarifies what is meant by when is not known to be well-posed a priori and the result is the principal tool in the proof of Theorem 1.1.

Lemma 3.1.

Consider the problem with associated PDE characteristic matrix whose determinant is . Let the polynomials be defined by . Let be defined by equation (2.8) and let be defined by equation (2.10). Let be defined by Notation 2.7, where is some function such that is entire and the decay condition (1.5) is satisfied. Let the functions be defined by equation (2.16). Let be the functions for which

 ˜fj(ρ)=∫T0eaρntfj(t)dt,˜gj(ρ)=∫T0eaρntgj(t)dt,ρ∈C. (3.6)

Then is an admissible set in the sense of Definition 1.3 of [7].

Proof.

By equation (2.16) and the definition of the index sets in Notation 2.7 we may write equations (1.13) and (1.14) of [7] as

 ˜F(ρ) =∑j∈J+ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ), (3.7) ˜G(ρ) =∑j∈J−ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ). (3.8)

By Cramer’s rule and the calculations in the proof of Lemma 2.6, equation (1.17) of [7] is satisfied.

As is entire, is entire so, by the standard results on the inverse Fourier transform, , defined by

 qT(x)=12π∫Reiρx^qT(ρ)dρ,

is a smooth function.

We know is entire by construction and is entire by assumption hence and are meromorphic on and analytic on . By the definition of and the decay assumption

 eaρnTηj(ρ)ΔPDE(ρ)→0 as ρ→∞ from within ˜D.

As and are entire so is . As is also entire and the definitions of and differ only by which of these functions appears, the ratio as from within also. This establishes that

 ζj(ρ)−eaρnTηj(ρ)ΔPDE(ρ)→0 as ρ→∞ from within ˜D.

Hence, by equations (3.7) and (3.8), as within .

An argument similar to that in Example 7.4.6 of [1] yields

 fj(t) =−ij2π∫∂Dρje−aρnt˜F(ρ)dρ, gj(t) =−ij2π∫∂Dρje−aρnt˜G(ρ)dρ.

Because as within , these definitions guarantee that and are smooth.

The compatibility of the and with is ensured by the compatibility condition . ∎

The desired result is now a restatement of Theorems 1.1 and 1.2 of [7]. For this reason we refer the reader to the proof presented in Section 4 of that publication. The only difference is that we make use of Lemma 3.1 in place of Proposition 4.1.

4 Representations of the solution

The proofs of Theorems 1.3 and 1.4 are similar calculations. In Section 4.1 we present the derivation of the series representation and, in Section 4.2, note the way this argument may be adapted to yield the integral representation. We derive the result in the case odd, ; the other cases are almost identical.

4.1 Series Representation

As is well-posed, Theorem 1.2 holds. We split the latter two integrals of equation (1.7) into parts whose integrands contain the data, that is , and parts whose integrands contain the solution evaluated at final time, that is .

 (4.1)

As is well-posed, Theorem 1.1 ensures the ratios

 η′j(ρ)Δ′PDE(ρ)→0 as ρ→∞ from within ˜D′,

for each . By definition and, by statement (ii) of Lemma 2.6, the zeros of are precisely the zeros of hence . Define