Calculating heat and wave propagation from lateral Cauchy data
UDC 519.6 We give an overview of recent methods based on semi-discretisation in time for the inverse ill-posed problem of calculating the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are given on a portion of the lateral boundary...
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Institute of Mathematics, NAS of Ukraine
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| author | Chapko, R. Johansson, B. T. Chapko, R. Chapko, R. Johansson, B. T. |
| author_facet | Chapko, R. Johansson, B. T. Chapko, R. Chapko, R. Johansson, B. T. |
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We give an overview of recent methods based on semi-discretisation in time for the inverse ill-posed problem of calculating the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data is to be generated on the remaining lateral part of the cylinder for either the heat or wave equation. The semi-discretisation in time constitutes of applying the Laguerre transform or the Rothe method (finite difference approximation), and has the feature that the similar sequence of elliptic problems is obtained for both the heat and wave equation, only the values of certain parameters change. The elliptic equations are solved numerically by either a boundary integral approach involving the Nystreom method or a method of fundamental solutions (MFS). Theoretical properties are stated together with discretisation strategies in space. Systems of linear equations are obtained for finding values of densities or coefficients. Tikhonov regularization is incorporated for the stable solution of the linear equations. Numerical results included show that the proposed strategies give good accuracy with an economical computational cost. |
| doi_str_mv | 10.37863/umzh.v74i2.6880 |
| first_indexed | 2026-03-24T03:30:37Z |
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DOI: 10.37863/umzh.v74i2.6880
UDC 519.6
R. Chapko (Ivan Franko Nat. Univ. Lviv, Ukraine),
B. T. Johansson (Linköping Univ., Sweden)
CALCULATING HEAT AND WAVE PROPAGATION
FROM LATERAL CAUCHY DATA
РОЗРАХУНОК ПОШИРЕННЯ ТЕПЛА I ХВИЛЬ
ЗА ДАНИМИ КОШI НА БIЧНIЙ МЕЖI
We give an overview of recent methods based on semi-discretization in time for the inverse ill-posed problem of calculating
the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are
given on a portion of the lateral boundary of a space-time cylinder and the corresponding data is to be generated on the
remaining lateral part of the cylinder for either the heat or wave equation. The semi-discretization in time constitutes of
applying the Laguerre transform or the Rothe method (finite difference approximation), and has the feature that the similar
sequence of elliptic problems is obtained for both the heat and wave equation, only the values of certain parameters change.
The elliptic equations are solved numerically by either a boundary integral approach involving the Nyström method or a
method of fundamental solutions. Theoretical properties are stated together with discretization strategies in space. Systems
of linear equations are obtained for finding values of densities or coefficients. Tikhonov regularization is incorporated for
the stable solution of the linear equations. Numerical results included show that the proposed strategies give good accuracy
with an economical computational cost.
Наведено огляд останнiх методiв, що ґрунтуються на частковiй дискретизацiї за часом, для обернених некоректних
задач обчислення розв’язку еволюцiйних рiвнянь за нестацiонарними даними Кошi. Зокрема, значення функцiї та її
нормальної похiдної заданi на частинi бiчної межi просторово-часового цилiндра i необхiдно згенерувати вiдповiднi
данi на рештi бiчної межi для випадкiв рiвняння теплопровiдностi та хвильового рiвняння. Часткова дискретизацiя
за часом полягає у застосуваннi перетворення Лагерра або методу Роте (скiнченнорiзницева апроксимацiя) i має ту
особливiсть, що для рiвняння теплопровiдностi i хвильового рiвняння отримано однаковi послiдовностi елiптичних
задач, якi вiдрiзняються лише значеннями певних параметрiв. Елiптичнi рiвняння розв’язано чисельно за допомогою
граничних iнтегральних рiвнянь методом Нистрьома або методом фундаментальних розв’язкiв. Теоретичнi власти-
востi викладенi разом iз стратегiями дискретизацiї за просторовими змiнними. Отримано системи лiнiйних рiвнянь
для знаходження значень густин або коефiцiєнтiв. Для одержання стiйкого розв’язку лiнiйних рiвнянь застосовано
регуляризацiю Тихонова. Наведенi числовi результати показують, що запропонованi пiдходи дають хорошу точнiсть
при економних обчислювальних затратах.
1. Introduction. We consider the lateral Cauchy problem for the heat equation:
1
c
\partial u
\partial t
= \Delta u in D \times (0, T ),
u = f2 on \Gamma 2 \times (0, T ),
\partial u
\partial \nu
= g2 on \Gamma 2 \times (0, T ),
u(x, 0) = 0 for x \in D,
(1.1)
with c > 0 a given constant specifying the heat diffusivity, together with the lateral Cauchy problem
for the wave equation
c\bigcirc R. CHAPKO, B. T. JOHANSSON, 2022
274 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
CALCULATING HEAT AND WAVE PROPAGATION FROM LATERAL CAUCHY DATA 275
1
a2
\partial 2u
\partial t2
= \Delta u in D \times (0, T ),
u = f2 on \Gamma 2 \times (0, T ),
\partial u
\partial \nu
= g2 on \Gamma 2 \times (0, T ),
\partial u
\partial t
(x, 0) = u(x, 0) = 0 for x \in D,
(1.2)
where a > 0 is the given constant speed of sound. In both problems, \Gamma 2 is a portion of the boundary
of the bounded domain D, and the final time T > 0.
We assume that the given lateral data f2 and g2 are sufficiently smooth and compatible such that
there exists a solution. For the heat equation, the solution is unique by the Holmgren uniqueness
theorem. However, for the wave equation, uniqueness is more subtle due to finite speed of pro-
pagation. The solution can be shown to be unique in the region described by (x, t) \in D \times (0, T )
with \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t} (x,\Gamma 2) < T - t (geodesic distance). For both equations, the respective solution does not
in general depend continuously on the data, that is both these problems are ill-posed. For proof
of uniqueness and additional properties of lateral Cauchy problems, see [18] (Chapt. 3) and [22]
(Chapt. 4) and references therein (for overview and references to other inverse ill-posed problems for
parabolic and hyperbolic equations, see, for example, [3, 15, 16, 21]).
In applications, lateral Cauchy problems occur for example when a part of the boundary is
inaccessible for measurements (this part can be too hot to place sensors on or too risky to approach
like in measurements of heart activity). To model a typical situation, for the remaining part of this
work, let D be the annular region between two bounded simply connected domains D1 and D2,
with \=D1 \subset D2, in \mathrm{I}\mathrm{R}d, d = 2, 3. The boundary of D1 is denoted by \Gamma 1 and the boundary of D2
by \Gamma 2. It is assumed that each boundary part is a simple closed sufficiently smooth surface (curve
when d = 2). The aim is then to calculate the solution to (1.1) or (1.2) and, in particular, to find the
corresponding data on the inner inaccessible lateral boundary \Gamma 1 \times (0, T ).
In [4, 5, 7, 10], numerical methods are derived, based on various time-transformations, for the
stable reconstructions of the solution to the respective lateral Cauchy problem. We shall survey these
methods here and present the main findings together with some additional numerical results.
We begin in Section 2 by presenting the two semidiscretizations in time; the Laguerre transform
respectively the Rothe method (finite difference approximation). Both these transformations applied
to either the heat or wave equation render the similar sequence of elliptic equations. Included in
Section 2 is the definition and explicit representation of what is known as a fundamental sequence of
the obtained elliptic equations. In Section 3, it is outlined how to generate a numerical approximation
to the sequence of elliptic equations based on integral equations and the Nyström method. As
an alternative, in Section 4, a method of fundamental solutions (MFS) is given for the numerical
solution of the elliptic equations. The discretization strategies in space in combination with the
semidiscretization in time render explicit expressions for the sought Cauchy data on the inner lateral
boundary part. Numerical results are presented in Section 5.
2. Semidiscretization in time of (1.1) and (1.2). 2.1. The Laguerre transform. The Laguerre
transformation with respect to the time-variable of an element u(x, t) has the following representa-
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
276 R. CHAPKO, B. T. JOHANSSON
tion:
u(x, t) = \kappa
\infty \sum
p=0
\widetilde up(x)Lp(\kappa t), (2.1)
where Lp(t) =
\sum p
k=0
\biggl(
p
k
\biggr)
( - t)k
k!
is the Laguerre polynomial of order p [1] (Chapt. 22), \kappa > 0 is a
given constant and the Fourier – Laguerre coefficients \widetilde up are defined as
\widetilde up(x) = \infty \int
0
e - \kappa tLp(\kappa t)u(x, t) dt, p = 0, 1, 2, . . . . (2.2)
Assuming that the solution to (1.1) and (1.2) has been extended into the time interval (0,\infty ), then
applying the transform (2.2) with respect to the time-variable, we obtain (details are given in [7, 13])
the following theorem.
Theorem 2.1. The function u defined in (2.1) is a solution of the lateral Cauchy problem for the
heat equation (1.1) respectively the wave equation (1.2) when T = \infty provided that the Fourier –
Laguerre coefficients \widetilde up, p = 0, 1, 2, . . . , are the solution of the following sequence of elliptic Cauchy
problems:
\Delta \widetilde up - \gamma 0\widetilde up = p - 1\sum
m=0
\gamma p - m\widetilde um in D,
\widetilde up = \widetilde f2,p on \Gamma 2, (2.3)
\partial \widetilde up
\partial \nu
= \widetilde g2,p on \Gamma 2,
where
\widetilde f2,p(x) = \infty \int
0
e - \kappa tLp(\kappa t)f2(x, t) dt, p = 0, 1, 2, . . . ,
\widetilde g2,p(x) = \infty \int
0
e - \kappa tLp(\kappa t)g2(x, t) dt, p = 0, 1, 2, . . . ,
with the coefficients \gamma p being in the case of the heat equation: \gamma p =
\kappa
c
, p = 0, 1, 2, . . . , and in the
case of the wave equation: \gamma p =
\kappa 2
a2
(p+ 1), p = 0, 1, 2, . . . .
We remark that when T = \infty uniqueness of a solution to the lateral Cauchy problem (1.2) can
be shown by time-transformation in combination with uniqueness of a solution to elliptic equations
with Cauchy data, for assumptions and details see [19].
2.2. The Rothe method. We present an alternative to the time-transformation of the previous
section, which operates directly on the time interval (0, T ) that is without any time extension of the
solution.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
CALCULATING HEAT AND WAVE PROPAGATION FROM LATERAL CAUCHY DATA 277
The time derivatives in the heat equation (1.1) and the wave equation (1.2) are discretized by a
finite difference approximation [12]. Thus, on the equidistant mesh\bigl\{
tp = (p+ 1)ht, p = - 1, . . . , Nt - 1, ht = T/Nt, Nt \in \mathrm{I}\mathrm{N}
\bigr\}
we approximate the solution u by the sequence \^up \approx u(\cdot , tp), p = 0, . . . , Nt - 1; the elements of
this sequence satisfy the equations
\Delta \^up - \alpha 0\^up = \alpha 2\^up - 1 + \alpha 1\^up - 2 in D
with the coefficients \alpha p being: in the case of the heat equation: \alpha 0 =
1
cht
, \alpha 2 = - 1
cht
, \alpha 1 = 0 and
in the case of the wave equation: \alpha 0 =
1
a2h2t
, \alpha 2 = - 2
a2h2t
and \alpha 1 =
1
a2h2t
.
Note that other higher order finite difference approximations of the time derivatives can also be
applied (such as [17] applied, for example, in [5] and [6]).
2.3. A fundamental sequence. Interestingly, the described semidiscretization approaches (the
Laguerre transform respectively the Rothe method) for the lateral Cauchy problems for the heat and
wave equation, all lead to stationary elliptic problems that can be written into the following form:
\Delta up - \gamma 2up =
p - 1\sum
m=0
\beta p - mum in D, (2.4)
up = f2,p on \Gamma 2,
\partial up
\partial \nu
= g2,p on \Gamma 2, (2.5)
with given functions f2,p and g2,p, p = 0, . . . , N, N \in \mathrm{I}\mathrm{N} and with the constants \gamma 2 and \beta i being
explicitly known with their values depending on the type of semidiscretization used together with the
type of the underlying governing partial differential equation (heat or wave equation).
In order to generate solutions to (2.4), (2.5), we shall need what is known as a fundamental
sequence.
Definition 2.1. The sequence of functions \{ \Phi p\} Np=0 is a fundamental sequence for (2.3) provided
that
\Delta x\Phi p(x, y) - \gamma 2\Phi p(x, y) -
p - 1\sum
m=0
\beta p - m\Phi m(x, y) = \delta (x - y),
where \delta is the Dirac delta function.
It is possible to derive explicit expressions for the elements in this fundamental sequence in \mathrm{I}\mathrm{R}d,
details can be found in [7, 8, 10] and we recall the result.
Theorem 2.2. The functions \Phi p specified by:
a) in the two-dimensional case (d = 2)
\Phi p(x, y) = K0(\gamma | x - y| )vp(| x - y| ) +K1(\gamma | x - y| )wp(| x - y| ), x \not = y; (2.6)
b) in the three-dimensional case (d = 3)
\Phi p(x, y) =
e - \gamma | x - y|
| x - y|
\widetilde vp(| x - y| ), x \not = y (2.7)
for p = 0, 1, 2, . . . , N, constitute a fundamental sequence of the elliptic equations (2.3) in the sense
of Definition 2.1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
278 R. CHAPKO, B. T. JOHANSSON
The elements K0 and K1 are what is known as modified Bessel functions [1] (Chapt. 9.6 – 9.11),
which for \ell = 0, 1, 2, . . . , have the following representation:
K\ell (z) =
1
2
\Bigl( z
2
\Bigr) - \ell
\ell - 1\sum
k=0
(\ell - k - 1)!
k!
\biggl(
- z
2
4
\biggr) k
+ ( - 1)\ell +1 \mathrm{l}\mathrm{n}
\Bigl( z
2
\Bigr)
I\ell (z)+
+
( - 1)\ell
2
\Bigl( z
2
\Bigr) \ell
\infty \sum
k=0
[\psi (k + 1) + \psi (\ell + k + 1)]
\biggl(
z2
4
\biggr) k
k!(\ell + k)!
,
I\ell (z) =
\Bigl( z
2
\Bigr) \ell
\infty \sum
k=0
\biggl(
z2
4
\biggr) k
k!\Gamma (\ell + k + 1)
,
\psi (1) = - \zeta , \psi (\ell ) = - \zeta +
\ell - 1\sum
k=1
1
k
, \ell = 2, 3, . . . ,
with \Gamma (\ell ) the gamma function and \zeta \approx 0.57721 the Euler constant.
The polynomials vp and wp for p = 0, 1, . . . , N are given by
vp(r) =
[ p2 ]\sum
m=0
ap,2mr
2m and wp(r) =
[ p - 1
2 ]\sum
m=0
ap,2m+1r
2m+1, w0 = 0,
with [q] being the largest integer not greater than q. The coefficients ap for p = 0, 1, . . . , N are
obtained from the recurrence relations
ap,0 = 1,
ap,p = - 1
2\gamma p
\beta 1ap - 1,p - 1,
ap,k =
1
2\gamma k
\Biggl\{
4
\biggl[
k + 1
2
\biggr] 2
ap,k+1 -
p - 1\sum
m=k - 1
\beta p - mam,k - 1
\Biggr\}
, k = p - 1, . . . , 1.
The polynomials \widetilde vp for p = 0, 1, . . . are given by
\widetilde vp(r) = p\sum
m=0
\widetilde ap,mrm,
where the coefficients \widetilde ap for p = 0, 1, . . . are obtained from the recurrence relations\widetilde ap,0 = 1,
\widetilde ap,p = - 1
2\gamma p
\beta 1 \widetilde ap - 1,p - 1,
\widetilde ap,k =
1
2\gamma k
\Biggl\{
k(k + 1)\widetilde ap,k+1 -
p - 1\sum
m=k - 1
\beta p - m \widetilde am,k - 1
\Biggr\}
, k = p - 1, . . . , 1.
We then turn to the numerical solution of (2.4), (2.5).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
CALCULATING HEAT AND WAVE PROPAGATION FROM LATERAL CAUCHY DATA 279
3. Numerical solution of the stationary problems via a boundary integral equation method
(BIEM). Following the integral approach [9, 11] for the Cauchy problem for the Laplace equation,
we search for the solution of the Cauchy problem (2.4), (2.5) in the following potential-layer form:
up(x) =
1
\pi
2\sum
\ell =1
p\sum
m=0
\int
\Gamma \ell
q\ell m(y)\Phi p - m(x, y) ds(y), x \in D, (3.1)
with the unknown densities q1m and q2m, m = 0, . . . , N, defined on the two boundary parts \Gamma 1
and \Gamma 2, respectively, and \Phi p is given by (2.6) or (2.7).
The boundary integral operators in (3.1) have the similar jump properties as the classical single-
layer operator for the Laplace equation. This can be verified by noticing from the above expansion
of the elements K0 and K1 that the functions in the fundamental sequence each have at most a
logarithmic singularity in the 2-dimensional case, and in the 3-dimensional case from (2.7) we see
the presence of a weak singularity. Therefore, matching (3.1) against the data (2.5) and employing
the corresponding jump properties, we obtain the following system of boundary integral equations:
1
\pi
2\sum
\ell =1
\int
\Gamma \ell
q\ell p(y)\Phi 0(x, y) ds(y) = Fp(x), x \in \Gamma 2,
q2p(x) +
1
\pi
2\sum
\ell =1
\int
\Gamma \ell
q\ell p(y)
\partial \Phi 0(x, y)
\partial \nu (x)
ds(y) = Gp(x), x \in \Gamma 2,
(3.2)
for p = 0, . . . , N, with the right-hand sides
Fp(x) = f2,p(x) -
1
\pi
2\sum
\ell =1
p - 1\sum
m=0
\int
\Gamma \ell
q\ell m(y)\Phi p - m(x, y) ds(y)
and
Gp(x) = g2,p(x) -
p - 1\sum
m=0
q2m(x) - 1
\pi
2\sum
\ell =1
p - 1\sum
m=0
\int
\Gamma \ell
q\ell m(y)
\partial \Phi p - m(x, y)
\partial \nu (x)
ds(y).
The following is shown in [7].
Theorem 3.1. The system (3.2) has a unique solution for a dense set of data Fp and Gp with
the solution and data in corresponding L2-spaces on the boundary.
The full discretization of the sequence of systems (3.2) of ill-posed integral equations can be
realized by a Nyström method based on trigonometrical quadratures when the dimension d = 2 and
by a discrete projection method with spherical harmonics as basis functions when d = 3. In both
cases, parametric representations of the given boundary parts are needed. Additionally, when d = 3,
it is assumed that the boundary surfaces \Gamma 1 and \Gamma 2 can each be smoothly mapped bijectively onto
the unit sphere (for details we refer to [8], Sect. 4). In [10], for 3-dimensional domains, discretization
using the boundary element method is instead used.
The discretization renders a set of linear equations to solve for the values of the densities at a
finite number of points on the respective boundary part. Due to ill-posedness of the lateral Cauchy
problems, Tikhonov regularization is applied to obtain a stable solution of the linear equations.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
280 R. CHAPKO, B. T. JOHANSSON
Using (3.1) together with the obtained values of the densities, explicit formulas can be given for
the numerical approximation of the sought Cauchy data on the interior lateral boundary \Gamma 1 \times (0, T ),
see [7] (Sect. 4.2).
4. Numerical solution of the stationary problems via a MFS. The method of fundamental
solutions has become a popular choice for approximating solutions to elliptic equations having an
explicitly known fundamental solution, see [14, 20]. As an alternative to the BIEM of the previous
section, an MFS can be derived for (2.4), (2.5).
Following [4], the function up solving (2.3) is approximated by the element up,n, where
up(x) \approx up,n(x) =
p\sum
m=0
n\sum
k=1
\alpha mk\Phi p - m(x, yk), x \in D, (4.1)
for n > 0 with \Phi p given by (2.6) for 2-dimensional domains and by (2.7) for 3-dimensional domains,
and with the coefficients \alpha mk \in \BbbR , k = 1, 2, . . . , n, m = 0, 1, . . . , p, to be determined. The so-
called source points yk, k = 1, 2, . . . , n, are located outside of the domain D (on what is known as
artificial boundaries).
The coefficients \alpha mk in (4.1) is determined by collocating on the boundary of the solution domain
D using a set of so-called collocation points. To select source and collocation points in an efficient
way, we assume that the boundaries of the domain D have the following parametrisation:
\Gamma \ell = \{ x\ell (s) = (x1\ell (s), x2\ell (s)), s \in [0, 2\pi ]\} , \ell = 1, 2,
in the 2-dimensional case and
\Gamma \ell = \{ x\ell (\theta , \phi ) = \rho \ell (\theta , \phi )(\mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s}\phi , \mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{s}\mathrm{i}\mathrm{n}\phi , \mathrm{c}\mathrm{o}\mathrm{s} \theta ), \theta \in [0, \pi ], \phi \in [0, 2\pi ]\} , \ell = 1, 2,
in the 3-dimensional case.
Note that since D is an annular domain, source points have to be placed both in the unbounded
exterior region of D and in the bounded region enclosed by \Gamma 1. We construct an artificial boundary
curve (surface when the dimension d = 3) in each of these two regions, and place evenly distributed
source points yk on these boundaries. For 2-dimensional domains, source points are distributed
according to the rule
yk =
\left\{ 2x2 (\widetilde sk) for even k,
0.5x1 (\widetilde sk) for odd k,
(4.2)
where
\widetilde sk =
2\pi
n
k for k = 1, . . . , n. (4.3)
For 3-dimensional domains, source points are distributed as
yk =
\left\{ 2x2
\Bigl( \widetilde \theta k, \widetilde \phi k\Bigr) for even k,
0.5x1
\Bigl( \widetilde \theta k, \widetilde \phi k\Bigr) for odd k,
(4.4)
where
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
CALCULATING HEAT AND WAVE PROPAGATION FROM LATERAL CAUCHY DATA 281
\widetilde \theta k = \pi
\biggl\{
k - 1\widetilde n
\biggr\}
+ 1
\widetilde n+ 1
, \widetilde \phi k = \pi
\biggl[
k - 1\widetilde n
\biggr]
+ 1
\widetilde n+ 1
,
\widetilde n =
\sqrt{}
n
2
, with \{ q\} = q - [q] for k = 1, . . . , n.
In two dimensions, we assume that n is even, n = 2\xi , and correspondingly in three dimensions,
n = 2\xi 2, where \xi \in \BbbN .
The collocation points are generated in a similar way but with the constants 2 and 0.5 in (4.2) and
(4.4) both replaced by unity. The approximation (4.1) is assumed to satisfy the boundary conditions in
(2.3) at the collocation points on the outer boundary part \Gamma 2. This in combination with the observation
that it is only the coefficients in front of \Phi 0 in up,n in (4.1) which are not present in up - 1,n p > 0,
render the following recursive system to determine the coefficients \alpha mk :
n\sum
k=1
\alpha pk\Phi 0 (\widetilde xj , yk) = \widetilde f2,p (\widetilde xj) - p - 1\sum
m=0
n\sum
k=1
\alpha mk\Phi p - m (\widetilde xj , yk) ,
n\sum
k=1
\alpha pk
\partial \Phi 0
\partial \nu (x)
(\widetilde xj , yk) = \widetilde g2,p (\widetilde xj) - p - 1\sum
m=0
n\sum
k=1
\alpha mk
\partial \Phi p - m
\partial \nu (x)
(\widetilde xj , yk) ,
(4.5)
where the collocation points \widetilde xj are given by when d = 2:
\widetilde xj = x2(sj), sj =
4\pi
n+ 1
j for j = 1, . . . , n/2,
and when d = 3:
\widetilde xj = x2(\theta j , \phi j), \widetilde \theta j = \pi
\biggl\{
2j - 1\widetilde n
\biggr\}
+ 1
\widetilde n+ 1
, \widetilde \phi j = \pi
\biggl[
2j - 1\widetilde n
\biggr]
+ 1
\widetilde n+ 1
for j = 1, . . . , n/2.
Note that when the parameter p = 0, the sums in the right-hand side of (4.5) are set to zero.
The system (4.5) is ill-conditioned due to the ill-posedness of the Cauchy problem (2.3) and,
therefore, in order to obtain a stable solution, we apply Tikhonov regularization.
The following is shown in [4] building in particular on results in [2] and gives a theoretical
justification of the derived MFS.
Theorem 4.1. Let yk be a dense set of source points distributed evenly over the artificial boun-
dary parts. Then the corresponding basis elements used in the described MFS is a linearly inde-
pendent and dense set on \Gamma 1 respectively on \Gamma 2 in the L2-sense. The same holds for the normal
derivatives of the basis elements on those two boundary parts.
Combining the MFS approximation with either the Laguerre expansion or the Rothe method,
explicit approximation formulations are obtained for the sought Cauchy data on the inner lateral
boundary part \Gamma 1 \times (0, T ), see [4] (Sect. 3.2) and [5] (Sect. 4), respectively.
5. Numerical experiments. In this section, we illustrate the considered approaches for the
lateral Cauchy problems in a 2-dimensional doubly connected domain for both the parabolic and
hyperbolic cases. Let the domain D \subset \mathrm{I}\mathrm{R}2 be bounded by the inner boundary curve
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
282 R. CHAPKO, B. T. JOHANSSON
\Gamma 1 = \{ x1(s) = (0.6 \mathrm{c}\mathrm{o}\mathrm{s} s, 0.5 \mathrm{s}\mathrm{i}\mathrm{n} s), s \in [0, 2\pi ]\}
and outer boundary curve
\Gamma 2 =
\bigl\{
x2(s) =
\bigl(
\mathrm{c}\mathrm{o}\mathrm{s} s, \mathrm{s}\mathrm{i}\mathrm{n} s - 0.5 \mathrm{c}\mathrm{o}\mathrm{s}2 s
\bigr)
, s \in [0, 2\pi ]
\bigr\}
.
As a semidiscretization approach in time, we use the Laguerre transform. Keeping with the above
notation, clearly, the approximation of the Cauchy data have then the form
uN,n(x1(s), t) = \kappa
N\sum
p=0
\~up,n(x1(s))Lp(\kappa t)
and
\partial uN,n
\partial \nu
(x1(s), t) = \kappa
N\sum
p=0
\partial \~up,n
\partial \nu
(x1(s))Lp(\kappa t),
where the values \~up,n and
\partial \~up,n
\partial \nu
on \Gamma 1 can be calculated by the MFS or by the suggested integral
equation method. The space discretization parameter n, which correspond to the number of source
points for the MFS and to the number of quadrature points in the boundary integral equation approach
is taken as n = 32. We consider the case when the given Cauchy data on the outer boundary \Gamma 2
has no noise as well as when some noise are added. In the case of noisy data, noise is added to the
known function g2 to render g\delta 2 whilst the function f2 is defined exactly. The noise is such that\bigm\| \bigm\| \bigm\| g2 - g\delta 2
\bigm\| \bigm\| \bigm\|
L2(\Gamma 2\times (0,\infty ))
\leq \delta ,
where \delta is the noise level.
Example 1. We consider the parabolic Cauchy problem (1.1) with c = 1. As the exact solution,
we use the restriction of the fundamental solution for the heat equation
uex(x, t) =
100
4\pi t
e -
| x - x\ast | 2
4t , (x, t) \in D \times (0,\infty ), x\ast = (0, 4).
Then the Cauchy data on the boundary \Gamma 2 is
f2(x, t) = uex(x, t), g2(x, t) =
\partial uex
\partial \nu (x)
(x, t) with (x, t) \in D \times (0,\infty ).
Note that in this case the exact solutions of the sequence of Cauchy problems (2.4), (2.5) have the
form
uexp (x) =
100
2\pi
\Phi p (x, x
\ast ) , x \in D. (5.1)
The relative L2-errors of the reconstruction of the Cauchy data on the inner boundary \Gamma 1 for
exact and 5% noisy data in (2.4), (2.5) are given in Table 1 for various p (compared against the
corresponding exact data obtained from (5.1)). The columns ep contains the error for function values
and the columns qp are for normal derivatives. The corresponding relative L2-errors e and q of the
reconstruction of the Cauchy data on \Gamma 1 \times (0, T ] with T = 5 to the parabolic equation (1.1) are
also presented. All integrals in these errors are calculated using the trapezoidal quadrature rule. The
regularization parameters were chosen by trial and error: we calculated the numerical solutions for
\alpha = 10 - \ell with \ell = 1, . . . , 15 and used the value giving the most accurate result.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
CALCULATING HEAT AND WAVE PROPAGATION FROM LATERAL CAUCHY DATA 283
Table 1
Exact data 5% noise
p MFS BIEM MFS BIEM
ep qp ep qp ep qp ep q
0 1.93E–1 2.16E–1 1.18E–4 9.54E–4 8.87E–2 4.88E–1 7.10E–2 1.86E–1
5 3.34E–2 2.33E–1 1.36E–4 6.38E–4 1.10E+0 3.93E+0 1.29E–1 3.16E–1
10 3.32E–2 3.98E–1 1.56E–4 1.63E–3 1.29E+0 1.02E+1 2.77E–1 1.88E+0
15 3.04E–1 1.02E+0 5.39E–4 1.29E–3 1.60E+1 2.90E+1 1.48E+0 3.01E+0
20 1.21E–1 2.35E+0 1.32E–3 1.84E–2 1.30E+1 3.53E+1 1.32E+0 1.71E+1
N e q e q e q e q
20 1.31E–2 8.32E–2 1.27E–2 2.68E–2 4.41E–1 5.87E+0 1.69E–2 3.78E–1
\alpha 1E-10 1E-7 1E-4 1E-2
Example 2. We now solve the hyperbolic lateral Cauchy problem (1.2) with a = 1. The Cauchy
data is generated by first solving the Dirichlet initial boundary-value problem for the wave equation
with boundary functions
f\ell (x, t) = t2e - t+2(x1 + x2), x \in \Gamma \ell , t > 0, \ell = 1, 2,
and then taking restrictions of the solution and its normal derivative on the outer boundary \Gamma 2\times (0, T ).
The L2-errors of the reconstruction of the Cauchy data on the inner boundary \Gamma 1 from (2.4),
(2.5) for exact and 2% noisy data are given in Table 2 for various p together with the corresponding
reconstruction of the Cauchy data in the wave equation (1.2) for T = 5.
Table 2
Exact data 2% noise
p MFS BIEM MFS BIEM
ep qp ep qp ep qp ep q
0 2.13E–2 3.38E–1 4.71E–4 7.41E–3 1.25E–1 1.09E+0 7.10E–2 1.86E–1
5 4.37E–2 8.27E–1 1.83E–4 2.57E–3 9.36E–2 1.15E+0 1.29E–1 3.16E–1
10 3.24E–1 6.10E+0 4.91E–4 7.90E–3 5.35E–1 3.65E+0 2.77E–1 1.88E+0
15 1.26E+0 2.45E+1 1.22E–3 1.80E–2 4.59E+0 3.59E+1 1.48E+0 3.01E+0
20 3.52E+0 7.06E+1 2.93E–3 4.95E–2 2.48E+1 2.19E+2 1.32E+0 1.71E+1
N e q e q e q e q
20 5.24E+0 4.41E+0 1.37E–2 9.56E–2 3.47E+1 1.28E+2 1.69E–2 3.78E–1
\alpha 1E–10 1E–7 1E–4 1E–2
As can be seen from the two tables the results are more accurate for the BIEM than the MFS. This
can to some extent be attributed to the fact that the BIEM involves a rather large amount of analytical
work related to the existing singularities in the kernels, special quadratures, etc. Furthermore, the
MFS results are for a fixed set of source points (4.2) and (4.4), adjusting these the results can most
likely be further improved.
For both methods, it is also seen from the tables that the accuracy of the approximations decreases
with increasing values of p. This is natural since errors are propagating forward in (2.4), (2.5) due
to the recursive structure of that system. However, it is pleasing to see that the decrease in accuracy
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
284 R. CHAPKO, B. T. JOHANSSON
is rather mild, and in total accurate solutions to the time-dependent Cauchy problems are obtained.
In general, it is known that numerical solution to time-dependent lateral Cauchy problems is less
accurate near the final time T. This is not really seen here due to the existence and smoothness of
the solution in time.
We point out that numerical solution of parabolic and hyperbolic Cauchy problems in 3-dimen-
sional doubly connected domains via the presented approaches are considered in [5, 10]. Also,
semidiscretization in time using finite differences is investigated in [5, 7].
6. Conclusion. We summarized in this paper results by the authors related to the numerical
solution of time-dependent lateral Cauchy problems. The general approach consists of the following
steps. Firstly, a semidiscretization in time is carried out (by either the Laguerre transform or the
Rothe method). This leads to a sequence of Cauchy problems for elliptic equations with a recursive
right-hand side. A key property is that the obtained stationary problems are similar for both parabolic
and hyperbolic lateral Cauchy problems only values of some parameters changes. The next step is
the explicit construction of a special sequence of fundamental solutions. This gives the possibility
to apply a standard version of the MFS to the sequence of stationary elliptic Cauchy problems. It
renders a sequence of ill-conditioned linear systems having a recurrent right-hand side for finding the
coefficients in the MFS expansion. Tikhonov regularization is incorporated for the stable solution.
Once these coefficients have been found, the final step is the calculation of the Cauchy data on the
inner boundary. As an alternative to the solution steps for the elliptic equations, a BIEM can be
applied. This method is based on the single-layer approach incorporation the constructed sequence
of fundamental solutions. As a result, the stationary elliptic problems are reduced to a sequence
of boundary integral equations. The full discretization by some suitable projection method leads to
linear systems for identifying values of densities in the integral equations. Due to the ill-posedness
of the Cauchy problem the obtained linear systems are ill-conditioned and also here the Tikhonov
regularization is incorporated for the stable solution. Numerical results confirm the applicability of
the stated steps for the numerical approximation of solutions to lateral Cauchy problems for evolution
equations.
References
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CALCULATING HEAT AND WAVE PROPAGATION FROM LATERAL CAUCHY DATA 285
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Received 19.08.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 2
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| id | umjimathkievua-article-6880 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:30:37Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4f/9855a32296508ee7900c19de13da664f.pdf |
| spelling | umjimathkievua-article-68802025-03-31T08:45:58Z Calculating heat and wave propagation from lateral Cauchy data Calculating heat and wave propagation from lateral Cauchy data Chapko, R. Johansson, B. T. Chapko, R. Chapko, R. Johansson, B. T. parabolic and hyperbolic Cauchy problem; 2- and 3-dimensional doubly connected domains; single-layer potential; boundary integral equations; method of fundamental solutions; trigonometric quadrature method; Tikhonov regularization. UDC 519.6 We give an overview of recent methods based on semi-discretisation in time for the inverse ill-posed problem of calculating the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data is to be generated on the remaining lateral part of the cylinder for either the heat or wave equation. The semi-discretisation in time constitutes of applying the Laguerre transform or the Rothe method (finite difference approximation), and has the feature that the similar sequence of elliptic problems is obtained for both the heat and wave equation, only the values of certain parameters change. The elliptic equations are solved numerically by either a boundary integral approach involving the Nystreom method or a method of fundamental solutions (MFS). Theoretical properties are stated together with discretisation strategies in space. Systems of linear equations are obtained for finding values of densities or coefficients. Tikhonov regularization is incorporated for the stable solution of the linear equations. Numerical results included show that the proposed strategies give good accuracy with an economical computational cost. Сделан обзор последних методов, основанных на частичной дискретизации по времени для обратных некорректных задач вычисления решения эволюционных уравнений по нестационарным данным Коши. В частности, значение функции и ее нормальной производной заданы на части боковой поверхности пространственно-временного цилиндра и необходимо сгенерировать соответствующие данные на остальной боковой поверхности для случаев уравнений теплопроводности и волнового. Частичная дискретизации по времени состоит в применении преобразования Лагерра или метода Роте (конечно-разностная аппроксимация) и имеет ту особенность, что для уравнений теплопроводности и волнового получено одинаковые последовательности эллиптических задач, которые отличаются только значениями определенных параметров. Эллиптические уравнения численно решены с помощью граничных интегральных уравнений методом Ныстрема или методом фундаментальных решений. Теоретические свойства изложены вместе из стратегиями дискретизации по пространственным переменным. Получены системы линейных уравнений для нахождения значений плотностей или коэффициентов. Для получения устойчивого решения линейных уравнений применено регуляризацию Тихонова. Приведенные численные результаты по\-ка\-зы\-ва\-ют, что предлагаемые подходы дают хорошую точность при экономных вычислительных затратах. УДК 519.6Розрахунок поширення тепла i хвиль за даними Кошi на бiчнiй межiНаведено огляд останнiх методiв, що ґрунтуються на частковiй дискретизацiї за часом, для обернених некоректних задач обчислення розв’язку еволюцiйних рiвнянь за нестацiонарними даними Кошi. Зокрема, значення функцiї та її нормальної похiдної заданi на частинi бiчної межi просторово-часового цилiндра i необхiдно згенерувати вiдповiднi данi на рештi бiчної межi для випадкiв рiвняння теплопровiдностi та хвильового рiвняння. Часткова дискретизацiя за часом полягає у застосуваннi перетворення Лагерра або методу Роте (скiнченно-рiзницева апроксимацiя) i має ту особливiсть, що для рiвняння теплопровiдностi i хвильового рiвняння отримано однаковi послiдовностi елiптичних задач, якi вiдрiзняються лише значеннями певних параметрiв. Елiптичнi рiвняння розв’язано чисельно за допомогою граничних iнтегральних рiвнянь методом Нистрьома або методом фундаментальних розв’язкiв. Теоретичнi властивостi викладенi разом iз стратегiями дискретизацiї за просторовими змiнними. Отримано системи лiнiйних рiвнянь для знаходження значень густин або коефiцiєнтiв. Для одержання стiйкого розв’язку лiнiйних рiвнянь застосовано регуляризацiю Тихонова. Наведенi числовi результати показують, що запропонованi пiдходи дають хорошу точнiсть при економних обчислювальних затратах. Institute of Mathematics, NAS of Ukraine 2022-02-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6880 10.37863/umzh.v74i2.6880 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 2 (2022); 274 - 285 Український математичний журнал; Том 74 № 2 (2022); 274 - 285 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6880/9186 Copyright (c) 2022 Роман Степанович Хапко, B. Tomas Johansson |
| spellingShingle | Chapko, R. Johansson, B. T. Chapko, R. Chapko, R. Johansson, B. T. Calculating heat and wave propagation from lateral Cauchy data |
| title | Calculating heat and wave propagation from lateral Cauchy data |
| title_alt | Calculating heat and wave propagation from lateral Cauchy data |
| title_full | Calculating heat and wave propagation from lateral Cauchy data |
| title_fullStr | Calculating heat and wave propagation from lateral Cauchy data |
| title_full_unstemmed | Calculating heat and wave propagation from lateral Cauchy data |
| title_short | Calculating heat and wave propagation from lateral Cauchy data |
| title_sort | calculating heat and wave propagation from lateral cauchy data |
| topic_facet | parabolic and hyperbolic Cauchy problem 2- and 3-dimensional doubly connected domains single-layer potential boundary integral equations method of fundamental solutions trigonometric quadrature method Tikhonov regularization. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6880 |
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