^{1}

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^{2}

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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
*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 partial differential systems with coupled boundary-value conditions are frequent in different areas of science and technology, as in scattering problems in Quantum Mechanics [

Recently [

where

tructed under the following hypotheses and notation:

1. The matrix coefficient

where

2. Matrices

and also that the matrix pencil

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

Using condition (7) we can introduce the following matrices

which satisfy the condition

introduce matrices

that satisfy the conditions

Under the above assumptions, the homogeneous problem (1)-(4) was solved in [

(a) If we consider the following hypotheses:

Then, if the vector valued function

with the additional condition:

where a subspace

(b) If we consider the following hypotheses:

Then, if the vector valued function

under the additional condition:

then we can construct an exact series solution

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

and

with an additional solution

and under hypothesis (b), the value of

and

with an additional solution

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

and

taking

The series solution of problem (1)-(4) given in (16) presents some computational difficulties:

(a) The infiniteness of the series.

(b) Eigenvalues

(c) Other problem is the calculation of the matrix exponential, which may present difficulties, see [

For this reason we propose in this paper to solve the following problem:

Given an admissible error

This paper deals with the construction of analytic-numerical solutions of problem (1)-(4) in a subdomain

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 [

where for a vector

Let us introduce the notation

and by ( [

Let

Thus, we can take a positive constant

satisfying

Moreover, by (23), we have

If we define

we have that

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

where, as

where

Observe that for a fixed

then

Taking into account that

and by (34) there is a positive integer

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

As eigenvalues

Taking into account that

We take the first positive integer

We define the vector valued function

Using (38) one gets that

thus

Remark 1. Note that to determine the positive integer

knowledge the exact eigenvalues

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

Approximation

where

with

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:

We have

Taking

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

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

where

Replacing in (51) we obtain

where

Given

then

and therefore

Remark 2. From (61), and taking into account the definition of

so that, if

Similarly, can be taken in practice

instead of the definition (63).

Approximation

We define the approximation

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

We take

and suppose we make the approximation accurate enough satisfying condition

Thus, if

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

Let

Suppose that the approximations

Theorem 2. We consider problem (1)-(4) satisfying hypotheses (5), (6) and (7). Let

defined by (20), (26), (28) and (68) respectively. Let

the interval

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

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

Example 1. We will construct an approximate solution in the subdomain

and the

Also, the vectorial valued function

This is precisely the example 1 of Ref. [

We will follow algorithm 1 step by step:

1. Hypothesis (a) holds with

is regular. Therefore, we take

2. Performing calculations similar to those made in Ref. [

3. It is easy to calculate

4. Note that

Then, by (43):

then we take

5. We have

then we can take

6. We need to determinate the

We can solve exactly this equation,

and then

In summary,

7. We calculate

the smallest of them is

8. We have that

9. We have that

10. To be applicable the algorithm 1, the approximations

As the roots were calculated with 50 decimal accurate, we accept these approximations of the roots.

11. We have to take

12. We have to compute approximations

then, we can obtain

13. Functions

14. Vectors

We don’t compute

15. Compute

where

and our approximation satisfies

As an example, consider the point

It is easy to check that, from (82), one gets

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.