On the Construction of Analytic-Numerical Approximations for a Class of Coupled Differential Models in Engineering

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


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 ( ) ( ) , , 0, 0 1, 0 ( ) ( ) ( ) ( ) 1, 1, 0, 0 ( ) ( ) where , , , , , ,  are a m -dimensional vectors, was constructed under the following hypotheses and notation: 1.The matrix coefficient A is a matrix which satisfies the following condition where ( ) C σ denotes the set of all the eigenvalues of a matrix C in m m ×  .Thus, A is a positive stable matrix (where

( )
Re z denotes the real part of z ∈  ).

Matrices , , 1, 2 i i
A B i = , are m m × complex matrices, and we assume that the block matrix and also that the matrix pencil 1 1 is regular.
Condition (7) is well known in the literature of singular systems of differential equations, see [17], and involves the existence of some 0 ρ ∈  so that matrix 1 invertible with the possible exception of at most a finite number of complex numbers ρ .In particular, we may assume that 0 ρ ∈  .
Using condition (7) we can introduce the following matrices 1 A  and 1 B  defined by ( ) ( ) which satisfy the condition 1 0 1 , where matrix I 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 2 A  and 2 B  defined by ( ) ( ) 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: exist 0 , , and 0 , such that 0 Then, if the vector valued function ( ) with the additional condition: Ker Ker , 0 1 and Ker Ker is an invariant subspace with respect to matrix , where a subspace E of m  is invariant by the matrix A E E ⊂ , we can construct an exact series solution ( ) , u x t of homogeneous problem (1)-( 4).This construction was made in Ref. [15].(b) If we consider the following hypotheses: , and we have 0 , so that 0 Then, if the vector valued function ( ) under the additional condition: Ker Ker , 0 1 and Ker Ker is an invariant subspace respect to matrix , then we can construct an exact series solution ( ) , u x t 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 where, under hypothesis (a), the value of α is given by with an additional solution and under hypothesis (b), the value of α is given by , where n λ is the solution of the equation ( ) with an additional solution ( ) Under both hypotheses (a) and (b), the value of ( ) ( ) taking 1 0 b = in Formulaes ( 23)-( 25) if we consider hypothesis ( ) b .The series solution of problem ( 1)-(4) given in ( 16) presents some computational difficulties: (a) The infiniteness of the series.(b) Eigenvalues n λ are not exactly computable because Equation (18) (or Equation (21) under hypothesis ( ) b 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 0 ε > and a bounded subdomain [ ] [ ] [ ] , 0 0 t > .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 ε uniformly in This paper deals with the construction of analytic-numerical solutions of problem (1)-(4) in a subdomain , 0 0 t > , with a priori error 0 ε > .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

The Proposed Approximation
, 0 0 t > , be and we take an admissible error 0 ε > .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 0 M > , defined by ( ) , Moreover, by (23), we have we have that On the other hand, we know from ( 27) that where Observe that for a fixed 0 m ≥ the numerical series 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: Taking into account that ( ) and by (34) there is a positive integer 0 n so that Using ( 29), (31), ( 32) and (36), if Taking into account that We take the first positive integer 1 n so that We define the vector valued function ( ) 1 , , Using (38) one gets that Remark 1.Note that to determine the positive integer 0 n we need to check condition (36), which requires knowledge the exact eigenvalues n λ .From Ref. [15] [16] it is well know that ( ) where with ( ) defined by (25).Note that It is easy to see that and Replacing in (47) and taking norms, one gets We define ( ) by applying the Cauchy-Schwarz inequality for integrals and (28), one gets: 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 ( ) then, replacing in (60) one gets  ( then, if we define ( ) 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  1 Replacing in (51) we obtain ( ) Similarly, can be taken in practice instead of the definition (63).

u x t x t n u x t u x t n u x t n u x t n u x t n x t n u x t u x t n u x t n u x t n u x t n x t n
ε ε ε Summarizing, the following results has been established: Theorem 1.We consider problem (1)-( 4) satisfying hypotheses (5), ( 6) and (7).
Let  n λ be the 1 n -first approximate roots of the equation (18), each one in the interval ( ) and let  0 λ be the approximation of the additional solution Let 0 γ > be satisfying (53) and let Λ ,  ,  and L  be the positive constants defined by (63), ( 64) and (68) respectively.Suppose that the approximations  n λ satisfy (71), where S is the constant defined by (70).

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 ( )  (10).Compute constant α defined by (17).
2. Performing calculations similar to those made in Ref. [15], one gets that 1 1 b = , 2 0 b = and 0 α = . 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 37) one gets that 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 

≤
approximation error is less than  , where  is a positive constant which satisfies (77).Consider the functions , defined by (46), joint the vector ( ) be consider if condition(22) holds.Let 0 γ > be satisfying (53) and let Λ ,  ,  and L  be the positive constants defined by (63), (64) and (68) respectively.Suppose that the approximations  n λ satisfy (71), where S is the constant defined by (70).Suppose that the approximations  approximation error is less than  , where  is a positive constant which satisfies (77).Consider the functions

Algorithm 2 .
Construction of the analytic-numerical solution of problem (1)-(4) under hypotheses (b) in the subdomain

4 : 5 : 6 :Example 1 .
Determine the first positive integer 0 n which satisfies (43).Determine the first positive integer 1 n which satisfies (40).Determine approaches  n λ of the 1 n -first roots of Equation (21) each one in the interval We will construct an approximate solution in the subdomain homogeneous parabolic problem with homogeneous conditions (1)-(4),

6 .
We need to determinate the 1 n -first roots of equation error  .In this case, using minimal theorem ([28], p. 571), we can determine the exact value of e As given by: n =  , defined by (46) are given by: From (61), and taking into account the definition of  and  given in (64), it follows that Construction of the analytic-numerical solution of problem (1)-(4) under hypotheses (a) in the subdomain