On the Construction of Analytic-Numerical Approximations for a Class of Coupled Differential Models in Engineering ()
1. Introduction
Coupled partial differential systems with coupled boundary-value conditions are frequent in different areas of science and technology, as in scattering problems in Quantum Mechanics [1] - [3] , in Chemical Physics [4] - [6] , coupled diffusion problems [7] - [9] , modelling of coupled thermoelastoplastic response of clays subjected to nuclear waste heat [10] , etc. The solution of these problems has motivated the study of vector and matrix Sturm- Liouville problems, see [11] - [14] for example.
Recently [15] [16] , an exact series solution for the homogeneous initial-value problem
(1)
(2)
(3)
(4)
where
and
are a
-dimensional vectors, was cons-
tructed under the following hypotheses and notation:
1. The matrix coefficient
is a matrix which satisfies the following condition
(5)
where
denotes the set of all the eigenvalues of a matrix
in
. Thus,
is a positive stable matrix (where
denotes the real part of
).
2. Matrices
, are
complex matrices, and we assume that the block matrix
(6)
and also that the matrix pencil
(7)
Condition (7) is well known in the literature of singular systems of differential equations, see [17] , and involves the existence of some
so that matrix
is invertible. In this case, matrix
is invertible with the possible exception of at most a finite number of complex numbers
. In particular, we may assume that
.
Using condition (7) we can introduce the following matrices
and
defined by
(8)
which satisfy the condition
, where matrix
denotes, as usual, the identity matrix. Under hypothesis (6), is it easy to show that matrix
is regular (see [18] for details) and we can
introduce matrices
and
defined by
(9)
that satisfy the conditions
.
Under the above assumptions, the homogeneous problem (1)-(4) was solved in [15] [16] in two different cases:
(a) If we consider the following hypotheses:
(10)
Then, if the vector valued function
satisfies hypotheses
(11)
with the additional condition:
(12)
where a subspace
of
is invariant by the matrix
if
, we can construct an exact series solution
of homogeneous problem (1)-(4). This construction was made in Ref. [15] .
(b) If we consider the following hypotheses:
(13)
Then, if the vector valued function
satisfies the hypotheses
(14)
under the additional condition:
(15)
then we can construct an exact series solution
of homogeneous problem (1)-(4). This construction was made in Ref. [16] .
Observe that under the different hypotheses (a) and (b), the exact solution of problem (1)-(1) is given by the series
(16)
where, under hypothesis (a), the value of
is given by
(17)
and
is the set of eigenvalues
, where
is the solution of the equation
(18)
with an additional solution
if
(19)
and under hypothesis (b), the value of
is given by
(20)
and
is the set of eigenvalues
, where
is the solution of the equation
(21)
with an additional solution
if
(22)
Under both hypotheses (a) and (b), the value of
,
and
are given by
(23)
(24)
and
(25)
taking
in Formulaes (23)-(25) if we consider hypothesis
.
The series solution of problem (1)-(4) given in (16) presents some computational difficulties:
(a) The infiniteness of the series.
(b) Eigenvalues
are not exactly computable because Equation (18) (or Equation (21) under hypothesis
holds) is not solvable in a closed form, although well known and efficient algorithms for approximation, see references [13] [19] [20] .
(c) Other problem is the calculation of the matrix exponential, which may present difficulties, see [21] [22] for example.
For this reason we propose in this paper to solve the following problem:
Given an admissible error
and a bounded subdomain
,
. How do we construct an approximation that avoids the above-quoted difficulties and whose error with respect to the exact solution (16) is less than
uniformly in
?
This paper deals with the construction of analytic-numerical solutions of problem (1)-(4) in a subdomain
,
, with a priori error
. The work is organized as follows: in Section 2 we
construct the approximate solution. In Section 3 we will introduce an algorithm and give an illustrative example.
Throughout this paper we will assume the results and nomenclature given in [15] [16] . If
is a matrix in
, its 2-norm denoted by
is defined by ( [23] , p. 56)
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where for a vector
in
,
is the usual euclidean norm of
, and the 2-norm satisfies
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Let us introduce the notation
(26)
and by ( [23] , p. 556) it follows that
(27)
2. The Proposed Approximation
Let
,
, be and we take an admissible error
. Observe first that given (24), using Parseval’s identity for scalar Sturm-Liouville problems, see [24] and ( [11] , p. 223), one gets that
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Thus, we can take a positive constant
, defined by
(28)
satisfying
(29)
Moreover, by (23), we have
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If we define
by
(30)
we have that
(31)
On the other hand, we know from (27) that
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where, as
,
, we have for
:
(32)
where
(33)
Observe that for a fixed
the numerical series
is convergent, because using Lemma 1 of Ref. [15] if hypothesis (a) holds, or Lemma 2 of Ref. [16] if hypothesis (b) holds, one gets
,
, and by application of D’Alembert’s criterion for series:
![]()
then
(34)
Taking into account that
and
,
, it follows that
(35)
and by (34) there is a positive integer
so that
(36)
Using (29), (31), (32) and (36), if
, we have
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As eigenvalues
, then, for
it follows that
(37)
Taking into account that
, from (37) one gets that
(38)
![]()
![]()
(39)
We take the first positive integer
so that
(40)
We define the vector valued function
as
(41)
Using (38) one gets that
![]()
thus
(42)
Remark 1. Note that to determine the positive integer
we need to check condition (36), which requires
knowledge the exact eigenvalues
. From Ref. [15] [16] it is well know that
, then
![]()
and by (35), we can replace condition (36) by take the first positive integer
satisfying
(43)
Approximation
defined by (41) involves computation of the exact eigenvalues
,
which is not easy in practice. Now we study the admissible tolerance when one considers approximate eigen- values
,
in expression (41), taking
(44)
where
(45)
(46)
with
defined by (25). Note that
(47)
It is easy to see that
(48)
(49)
and
(50)
Replacing in (47) and taking norms, one gets
(51)
We define
for
by
(52)
by applying the Cauchy-Schwarz inequality for integrals and (28), one gets:
![]()
We have
![]()
Taking
satisfying
(53)
it follows that
(54)
Moreover, working component by component:
(55)
![]()
![]()
(56)
Applying the Cauchy-Schwarz inequality for integrals again:
(57)
and
(58)
![]()
(59)
By (55) and taking into account (57) and (58):
(60)
Note that from the definition of
, (52), it follows that
(61)
then, replacing in (60) one gets
(62)
We take
(63)
then, if we define
(64)
from (54) we have that
(65)
and from (62) and (53):
(66)
Using the 2-norm properties, from (66) we have
(67)
By other hand, we can write
![]()
where taking norm, applying (32) and (33) together the mean value theorem, under the hypothesis
, one gets
![]()
where
(68)
Replacing in (51) we obtain
(69)
where
(70)
Given
and
, consider approximations
of
for
satisfiying
(71)
then
![]()
and therefore
(72)
Remark 2. From (61), and taking into account the definition of
and
given in (64), it follows that
![]()
so that, if
is enough small, it can take
in the computation of
.
Similarly, can be taken in practice
(73)
instead of the definition (63).
Approximation
need to compute the exact value of the matrix exponential
. However, the approximate calculation of the exponential matrix
can be performed by methods such as those based on the Taylor series, [25] [26] , based on Hermite matrix polynomials, [27] , and other existing methods in the literature, see [22] [23] for example. Suppose we take the matrix
as an approximation of matrix
, so that
(74)
We define the approximation
by:
(75)
and from (65), (64) and (45) one gets that
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We take
(76)
and suppose we make the approximation accurate enough satisfying condition
(77)
Thus, if
satisfies (77) it follows that
(78)
and from (42), (72) and (78):
![]()
Summarizing, the following results has been established:
Theorem 1. We consider problem (1)-(4) satisfying hypotheses (5), (6) and (7). Let
,
. Suppose that the hypothesis (a) is verified, this ensures that there is an exact solution
of problem (1)-(4), see Ref. [15] . Let
,
,
,
and
be the constant defined by (17), (26), (28), (30) and (68) respectively. Let
and
be positive integers satisfying conditions (43) and (40). Let
be the
-first approximate roots of the equation (18), each one in the interval
,
, and let
be the approximation of the additional solution
to be consider if condition (19) holds.
Let
be satisfying (53) and let
,
,
and
be the positive constants defined by (63), (64) and (68) respectively. Suppose that the approximations
satisfy (71), where
is the constant defined by (70).
Suppose that the approximations
of matrices
, for
satisfy that the approximation error is less than
, where
is a positive constant which satisfies (77). Consider the functions
,
defined by (45) and vectors
,
, defined by (46), joint the vector
defined by (24) if
. Then, the vector valued function
defined by (75) satisfies
![]()
Theorem 2. We consider problem (1)-(4) satisfying hypotheses (5), (6) and (7). Let
, and we consider the subdomain
. Suppose that the hypothesis (b) is verified, this ensures that there is an exact solution
of problem (1)-(4), see Ref. [16] . Let
,
,
and
be the constant
defined by (20), (26), (28) and (68) respectively. Let
and
be positive integers satisfying conditions (43) and (40). Take
and
. Let
be the
-first approximate roots of the equation (21), each one in
the interval
,
, and let
be the approximation of the additional solution
to be consider if condition (22) holds. Let
be satisfying (53) and let
,
,
and
be the positive constants defined by (63), (64) and (68) respectively. Suppose that the approximations
satisfy (71), where
is the constant defined by (70). Suppose that the approximations
of matrices
, for
satisfy that the approximation error is less than
, where
is a positive constant which satisfies (77). Consider the functions
,
defined by (45) and vectors
,
, defined by (46), joint the vector
defined by (24) if
. Then, the vector valued function
defined by (75) satisfies
![]()
3. Algorithm 1, Algorithm 2 and Example
We can give the following algorithms, according to the hypothesis (a) or (b) is satisfied, to construct the approximation
.
Algorithm 1. Construction of the analytic-numerical solution of problem (1)-(4) under hypotheses (a) in the subdomain
,
, with a priori error bound
.
Algorithm 2. Construction of the analytic-numerical solution of problem (1)-(4) under hypotheses (b) in the subdomain
,
, with a priori error bound
.
Example 1. We will construct an approximate solution in the subdomain
, with a priori error bound
, of the homogeneous parabolic problem with homogeneous conditions (1)-(4), where the matrix
is chosen
(79)
and the
matrices
, are
(80)
Also, the vectorial valued function
will be defined as
(81)
This is precisely the example 1 of Ref. [15] whose exact solution is given by:
(82)
We will follow algorithm 1 step by step:
1. Hypothesis (a) holds with
. Note that although
is singular, taking
, the matrix pencil
(83)
is regular. Therefore, we take
.
2. Performing calculations similar to those made in Ref. [15] , one gets that
,
and
.
3. It is easy to calculate
,
, thus
. Similarly
,
and
.
4. Note that
![]()
Then, by (43):
![]()
then we take
.
5. We have
![]()
then we can take
.
6. We need to determinate the
-first roots of equation
![]()
We can solve exactly this equation,
,
, with an additional solution
, be- cause
![]()
and then
.
In summary,
,
,
,
,
,
. We take the approximate values (50 exact decimal)
![]()
7. We calculate
for
:
![]()
the smallest of them is
, as
, we take
.
8. We have that
,
,
and
.
9. We have that
.
10. To be applicable the algorithm 1, the approximations
may satisfy:
![]()
As the roots were calculated with 50 decimal accurate, we accept these approximations of the roots.
11. We have to take
satisfying (77). In our case
![]()
12. We have to compute approximations
of matrices
, for
with a maxi- mum error
. In this case, using minimal theorem ([28] , p. 571), we can determine the exact value of
given by:
(84)
then, we can obtain
for
replacing in (84).
13. Functions
,
, defined by (45) are given by:
![]()
14. Vectors
,
, defined by (46) are given by:
![]()
We don’t compute
defined by (25) because
.
15. Compute
defined by (75), obtaining:
![]()
where
![]()
and our approximation satisfies
![]()
As an example, consider the point
. We have the approximation
![]()
It is easy to check that, from (82), one gets
![]()
4. Conclusion
In this paper, a method to construct an analytic-numerical solution for homogeneous parabolic coupled systems with homogeneous boundary conditions of the type (1)-(4) has been presented. An algorithm with an illustrative example is given.