Received 7 October 2014; revised 1 November 2014; accepted 3 December 2014

ABSTRACT

In this paper, a method to construct an analytic-numerical solution for homogeneous parabolic coupled systems with homogeneous boundary conditions of the type, , , , , , where is a positive stable matrix and, , , are arbitrary matrices for which the block matrix is non-singular, is proposed.

Keywords:

Coupled Diffusion Problems, Coupled Boundary Conditions, Vector Boundary-Value Differential Systems, Sturm-Liouville Vector Problems, Analytic-Numerical Solution

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)

where for a vector in, is the usual euclidean norm of, and the 2-norm satisfies

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

Thus, we can take a positive constant, defined by

(28)

satisfying

(29)

Moreover, by (23), we have

If we define by

(30)

we have that

(31)

On the other hand, we know from (27) that

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

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

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.

References