1. Introduction
We analyse the modified Helmholtz equation in a regular hexagon using the unified transform, also known as the Fokas method. This method was introduced by one of the authors [
1], for analysing integrable nonlinear partial differential equations (PDEs) [
2]. Later, it was realized that it also yields novel results for linear evolution PDEs [
3]; results in this direction are obtained by several authors [
4,
5,
6,
7,
8,
9,
10]. Furthermore, it yields new integral representations for the solution of linear elliptic PDEs in polygonal domains [
11], which in the case of simple domains can be used to obtain the analytical solution of several problems which apparently cannot be solved by the standard methods [
12,
13]. Recently, researchers utilised the integral representations provided by the Fokas method for the local and global wellposedness analysis of Korteweg-de Vries and nonlinear Schrödinger type PDEs [
14,
15,
16,
17,
18], as well as for studying problems from control theory [
19].
The Fokas method is based on two basic ingredients:
- (1)
a global relation, which is an algebraic equation that involves certain transforms of all (known and unknown) boundary values.
- (2)
an integral representation of the solution, which involves transforms of all boundary values.
For linear PDEs, the Fokas method involves the following:
Given a PDE, define its formal adjoint and construct a one parameter family of solutions of this equation.
By employing the given PDE and its adjoint, obtain a one parameter family of equations in conservation form. This family, together with Green’s theorem, yield the global relation.
The above family also gives rise to a certain closed differential form. The spectral analysis of this form gives rise to a scalar Riemann–Hilbert problem, which consequently yields an integral representation of the solution. This representation involves integral transforms of all the boundary values, and since some of them are not prescribed as boundary conditions, this form of solution is not yet effective.
The explicit solution of the problem is derived by determining the contribution of the unknown boundary values to the integral representation. This can be achieved by using the global relation, as well as equations obtained from the global relation through certain invariant transformations.
The global relation has had important analytical and numerical implications: first, it has led to novel analytical formulations of a variety of important physical problems from water waves [
20,
21,
22,
23,
24,
25,
26] to three-dimensional layer scattering [
27]. Second, it has led to the development of new techniques for the Laplace, modified Helmholtz, Helmholtz, biharmonic equations, both analytical [
28,
29,
30,
31,
32,
33,
34,
35] and numerical [
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47].
The above analytical solutions are given in terms of infinite series; this is to be contrasted to other techniques based on the eigenvalues of the Laplace operator that yield the solution as a bi-infinite series. The eigenvalues of the Laplace operator for the Dirichlet, Neumann and Robin problems in the interior of an equilateral triangle were first obtained by Lamé in 1833 [
48]; these results have also been derived using the Fokas method [
49]. Completeness for the associated expansions for the Dirichlet and Neumann problems was obtained in [
50,
51,
52,
53] using group theoretic techniques. McCartin rederived these results [
54,
55] and studied the connection of the eigen-structure of the equilateral triangle with that of the regular hexagon [
56]. The above remarks indicate that the existing literature is based on an implicit way for deriving the solution of specific BVPs of the regular hexagon in terms of bi-infinite series. This is to be contrasted with our work which presents a direct approach for deriving explicit integral representations of the solution of a special BVP on the regular hexagon; the extension of the current methodology to more general problems is under investigation.
Organisation of the Paper
In
Section 2 we implement the four steps discussed above for solving the symmetric Dirichlet problem of the modified Helmholtz equation in a regular hexagon. The main achievement of this work is presented in
Section 3 and concerns the fourth step: our analysis yields the solution for the case of odd symmetric Dirichlet data in the closed form (
34). We study the case of even symmetric data in
Section 4, where we derive the expression (
37); this expression in addition to known terms also involves an unknown term. In
Section 5, Figures 1 and 2 depict the numerical verification of the main result of
Section 3; also, Figures 7 and 8 indicate that the unknown term in the expression (
37) is exponentially small in the high frequency limit, and hence this result provides an excellent approximation for this physically significant limit.
2. The Basic Elements
The equation investigated here is the modified Helmholtz equation in the interior of the regular hexagon,
D, namely,
where
is a real valued function and
.
Using complex coordinates,
Equation (
1) becomes
2.1. The Global Relation and the Integral Representation of the Solution in the Interior of a Convex Polygon
We first derive the global relation:
The formal adjoint also satisfies the modified Helmholtz equation
Multiplying Equation (
2) by
, Equation (
3) by
q and subtracting, we find
or equivalently
Using in (
5) the special solution
and employing Green’s theorem, we obtain
where
W is defined by
Suppose that
is the polygon defined via the points
. Then (
6) gives the following global relation for the modified Helmholtz in this polygon:
where
are defined by
or equivalently (in local coordinates) by
In Equation (
10) we have used the identity
where
s is the arclength on the boundary
of the polygon and
denotes the derivative in the outward normal direction to the boundary of the polygon.
In order to derive the integral representation of the solution one has to implement the spectral analysis of the differential form
This procedure yields the following theorem, proven in [
6]:
Theorem 1. Let Ω be the interior of a convex closed polygon in the complex z-plane, with corners . Assume that there exists a solution of the modified Helmholtz equation, i.e., of Equation (2) with , valid on Ω, and suppose that this solution has sufficient smoothness on the boundary of the polygon. Then, q can be expressed in the formwhere are defined by (10), and are the rays in the complex k-planeoriented from zero to infinity. Observe that the solution given in (
12) is given in terms of
which involve integral transforms of both
q and
on the boundary, i.e., both known and unknown functions.
2.2. The Dirichlet Problem on a Regular Hexagon
Let
be the interior of a regular hexagon with vertices
,
where
l is the length of the side and
. The sides
,
will be referred to as sides
.
For the sides
the following parametrizations will be used:
The general Dirichlet problem can be uniquely decomposed to 6 simpler Dirichlet problems, by employing the decomposition
indeed the determinant of the matrix
is non-zero (Its value is
, and for the general case
).
The existence and uniqueness of the solution of the modified Helmholtz equation shows that it is sufficient to solve each one of the above Dirichlet problems. The first of them is the symmetric Dirichlet problem, where the value is prescribed on each side. This symmetric problem is analysed in the next section.
2.3. The Symmetric Dirichlet Problem
The problem analysed in this subsection is the symmetric Dirichlet problem for the modified Helmholtz equation in the regular hexagon (
). Let
be a real function with sufficient smoothness and compatibility at the vertices of the hexagon, i.e.,
. We prescribe the boundary conditions
The above ‘symmetry’ property also holds for the Neumann boundary values. This fact is the consequence of the following three observations:
The modified Helmholtz operator is invariant under the transformation , namely under rotation of . Since the Dirichlet data are invariant under this rotation, then the (unique) solution of the Helmholtz equation is also invariant under this rotation.
If
q is invariant under this transformation, then the differential form
is also invariant under the transformation
:
Evaluating the above differential form on each side we obtain
where the second equality is a direct consequence of the fact that the Dirichlet data are invariant under this rotation.
Applying the parametrization of the regular hexagon on Equation (
10) we obtain:
with
where
,
and
are defined by
The function is known, whereas the unknown function contains the unknown Neumann boundary value .
Using (
15), the global relation (
8) takes the form
where the known function
is defined by
The integral representation (
12) of the solution takes the form
where
are the rays defined by
oriented from zero to infinity. The principal arguments of
are
, respectively.
Since the function can be uniquely written as a sum of an odd and an even function, we will only consider two particular cases:
- (i)
the odd case, ;
- (ii)
the even case .
The solution and the Neumann boundary values inherit the analogous properties:
- (i)
in the odd case, , which yields ;
- (ii)
in the even case, , which yields for all .
3. Derivation of the Solution for the Symmetric Odd Case
In what follows we will show that the contribution of the unknown functions
to the solution representation (
19) can be computed explicitly.
Applying the condition
in (
17) we obtain the equation
where
is given in (
18) and
is defined by
Solving (
21) for
and substituting the resulting expression in (
15) we find
The functions
can be obtained from (
22) by replacing
k with
for
.
Regarding the integral representation of the solution, we restrict our attention to the first integral of (
19), namely the integral along
(the negative imaginary axis).
Let
Solving (
21) for
and substituting the resulting expression in the first integral of (
19) we find that the known part of this integral is given by the expression
The unknown part involves the functions
and
and is given by
In what follows we will show that the contribution of the unknown functions, namely of the sum , can be computed in terms of the given boundary conditions.
The first integral in the rhs of
can be deformed from
to
, where
is a ray with
; choosing
we obtain
where
and
The above deformation is justified, since it can be shown that the integrand of
is bounded and analytic in the region where
: letting
, we can rewrite the first term of the integrand of
in the form
We observe the following:
The zeros of occur when , thus .
The function is bounded and analytic for .
Indeed, if , then . Thus, if , it follows that . Hence, .
Therefore, the exponentials and are bounded.
The function is bounded and analytic for , namely in the region where .
Indeed, this expression involves the exponentials and , which are bounded in this region, since .
The function
is bounded and analytic for
.
Indeed, since
k is at the lower half plane, then
which is bounded if
.
If , then , which yields .
Similar considerations apply to the second term of the integrand of
; this term can be rewritten in the form
We observe the following:
The function is bounded and analytic for .
The function is bounded and analytic for , namely in the region where .
Thus, it is bounded and analytic for .
Using the underlined symmetries, we can express the integral representation of the solution in the form
where
and
are given by applying in (
23) and (
24) the following rotations:
We define
, where we employ the notation
. Then, we rewrite the expression in (
25) in the form
Thus, it is sufficient to compute the contribution
. In this direction we find (via rotation) that
Using that
and
the above expression is simplified to
Employing the global relation (
21) we obtain
In summary, the solution takes the form
where
is defined by
and
is defined by
Note also that the integrals of
can be deformed on a sector of angle
. For example, in
the ray
can be deformed in a ray
in the sector
; analogous results are valid for the remaining
.
Observing that
, Equation (
29) can be further simplified to
In order to write the integral representation in a more compact form we make the change of variables in the integrals in and . In this procedure:
- 1.
the fraction remains invariant;
- 2.
the rays become ;
- 3.
the exponent becomes ;
- 4.
the remaining integrands are equal to the corresponding integrands in and .
We make the change of variables
in the integrand of (
33), so that the contour of integration transforms from the negative imaginary axis
to the real imaginary axis, and we summarize the above result in the form of a proposition.
Proposition 1. Let q satisfy the modified Helmholtz Equation (2) in the interior of a regular hexagon defined in (13). Assume that on each side of this hexagon an odd symmetric Dirichlet boundary condition is prescribed, namely,with and . The solution q can be computed in closed form:where are defined as follows: 4. The Symmetric Even Case
Applying the condition
in (
17) we obtain the following equation
where
and
is known and given in (
18).
Following the same stems used in
Section 3 we derive the analogue of (
28), which yields the following formula for
:
where in addition to the known part which involves
, there now exists an unknown part which involves
.
Thus, the analogue of (
29) now takes the form
where
is known function defined by
is also known and defined by
whereas
is the unknown function defined by
It can be shown that each of
decays exponentially fast as
. The rigorous proof of this statement will be presented elsewhere. In the next section, this fact will be indicated via the numerical evaluation of each of the terms appearing in Equation (
37).
6. Conclusions
In this work we have presented the explicit solution of a particular boundary value problem for the modified Helmholtz equation in a regular hexagon: we have solved the case where the same Dirichlet datum
is prescribed in all sides of the hexagon, and this function is odd. This explicit solution is described in Proposition 1. We have also obtained an approximate analytical representation for the solution for the case that
is even. The exact representation is given by Equation (
37), where the terms
and
are given in terms of
, but the terms
involve the unknown Neumann boundary value. However, these terms are exponentially small as
. Thus, for the case of large
, Equation (
37) provides the solution to this problem with an exponentially small error. The above analytical results were verified numerically in
Section 5. The rigorous investigation on the analytical and numerical accuracy of the latter approximate formula will be presented in future work.
It should be noted that the arbitrary Dirichlet problem can be decomposed into 6 separate and simpler Dirichlet BVPs, which are defined in
Section 2.3; the first of these BVPs is the symmetric Dirichlet problem. The analysis of the remaining problems is a work in progress.