OJMSiOpen Journal of Modelling and Simulation2327-4018Scientific Research Publishing10.4236/ojmsi.2015.31001OJMSi-52188ArticlesPhysics&Mathematics On the Construction of Analytic-Numerical Approximations for a Class of Coupled Differential Models in Engineering milioDefez1*VicenteSoler2*RobertoCapilla3*Departamento de Ingeniera Electrónica, Universitat Politècnica de València, Valencia, SpainInstituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, SpainDepartamento de Matemática Aplicada, Universitat Politècnica de València, Valencia, Spain* E-mail:edefez@imm.upv.es(MD);vsoler@mat.upv.es(VS);rcapilla@eln.upv.es(RC);0912201403011187 October 20141 November 2014 3 December 2014© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

In this paper, a method to construct an analytic-numerical solution for homogeneous parabolic coupled systems with homogeneous boundary conditions of the type u t = Au xx, A 1 u(o, t) + B 1 u x(o, t) = 0, A 2 u(1, t) + B 2 u x(1, t) = 0, o t>0, u (x,0) = f(x), where A is a positive stable matrix and A 1, B 1, B 1, B 2, are arbitrary matrices for which the block matrix is non-singular, is proposed.

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  -  , in Chemical Physics  -  , coupled diffusion problems  -  , modelling of coupled thermoelastoplastic response of clays subjected to nuclear waste heat  , etc. The solution of these problems has motivated the study of vector and matrix Sturm- Liouville problems, see  -  for example.

Recently   , an exact series solution for the homogeneous initial-value problem

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

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

and also that the matrix pencil

Condition (7) is well known in the literature of singular systems of differential equations, see  , 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

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  for details) and we can

introduce matrices and defined by

that satisfy the conditions.

Under the above assumptions, the homogeneous problem (1)-(4) was solved in   in two different cases:

(a) If we consider the following hypotheses:

Then, if the vector valued function satisfies hypotheses

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.  .

(b) If we consider the following hypotheses:

Then, if the vector valued function satisfies the hypotheses

then we can construct an exact series solution of homogeneous problem (1)-(4). This construction was made in Ref.  .

Observe that under the different hypotheses (a) and (b), the exact solution of problem (1)-(1) is given by the series

where, under hypothesis (a), the value of is given by

and is the set of eigenvalues, where is the solution of the equation

and under hypothesis (b), the value of is given by

and is the set of eigenvalues, where is the solution of the equation

Under both hypotheses (a) and (b), the value of, and are given by

and

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    .

(c) Other problem is the calculation of the matrix exponential, which may present difficulties, see   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   . If is a matrix in, its 2-norm denoted by is defined by (  , p. 56)

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

Let us introduce the notation

and by (  , p. 556) it follows that

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  and (  , p. 223), one gets that

Thus, we can take a positive constant, defined by

satisfying

Moreover, by (23), we have

If we define by

we have that

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

where, as, , we have for:

where

Observe that for a fixed the numerical series is convergent, because using Lemma 1 of Ref.  if hypothesis (a) holds, or Lemma 2 of Ref.  if hypothesis (b) holds, one gets, , and by application of D’Alembert’s criterion for series:

then

Taking into account that and, , it follows that

and by (34) there is a positive integer so that

Using (29), (31), (32) and (36), if, we have

As eigenvalues, then, for it follows that

Taking into account that, from (37) one gets that

We take the first positive integer so that

We define the vector valued function as

Using (38) one gets that

thus

Remark 1. Note that to determine the positive integer we need to check condition (36), which requires

knowledge the exact eigenvalues. From Ref.   it is well know that, then

and by (35), we can replace condition (36) by take the first positive integer satisfying

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

where

with defined by (25). Note that

It is easy to see that

and

Replacing in (47) and taking norms, one gets

We define for by

by applying the Cauchy-Schwarz inequality for integrals and (28), one gets:

We have

Taking satisfying

it follows that

Moreover, working component by component:

Applying the Cauchy-Schwarz inequality for integrals again:

and

By (55) and taking into account (57) and (58):

Note that from the definition of, (52), it follows that

then, replacing in (60) one gets

We take

then, if we define

from (54) we have that

and from (62) and (53):

Using the 2-norm properties, from (66) we have

By other hand, we can write

where taking norm, applying (32) and (33) together the mean value theorem, under the hypothesis, one gets

where

Replacing in (51) we obtain

where

Given and, consider approximations of for satisfiying

then

and therefore

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

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,   , based on Hermite matrix polynomials,  , and other existing methods in the literature, see   for example. Suppose we take the matrix as an approximation of matrix, so that

We define the approximation by:

and from (65), (64) and (45) one gets that

We take

and suppose we make the approximation accurate enough satisfying condition

Thus, if satisfies (77) it follows that

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.  . 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.  . 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

and the matrices, are

Also, the vectorial valued function will be defined as

This is precisely the example 1 of Ref.  whose exact solution is given by:

We will follow algorithm 1 step by step:

1. Hypothesis (a) holds with. Note that although is singular, taking, the matrix pencil

is regular. Therefore, we take.

2. Performing calculations similar to those made in Ref.  , 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 ( , p. 571), we can determine the exact value of given by:

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.

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