Abstract

An exact series solution for nonhomogeneous parabolic coupled systems of the type , where , and are arbitrary matrices for which the block matrix is nonsingular, and A is a positive stable matrix, is constructed.

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], thermo-elastoplastic modelling [10], and so forth. The solution of these problems has motivated the study of vector and matrix Sturm-Liouville problems; see [11–14], for example.

Recently, see [15, 16], an exact series solution for the homogeneous initial-value problem where and are -dimensional vectors, was constructed under the following hypotheses and notations.(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  that is, condition (4) involves the existence of some , matrix being invertible; see [17]. Using condition (4), we can introduce the following matrices and defined by which satisfy the condition , where matrix denotes, as usual, the identity matrix. Under hypothesis (3), it is easy to show that matrix is regular (see [18] for details) and we can introduce matrices and defined by that satisfy the conditions . Under the above assumptions, in [15], we have consider the following essential hypothesis: if the vector valued function satisfies hypotheses under the additional hypothesis where a subspace of is invariant by the matrix , if , in order to construct an exact series solution of homogeneous problem (1).

Moreover, in [16], under the above assumptions and replacing the condition (7) by the following hypothesis

if the vector valued function satisfies the new hypotheses

under the additional hypothesis then an exact series solution of homogeneous problem (1) is constructed, see [16].

This paper deals with the construction of the exact series solution of the nonhomogeneous problem

We provide conditions for the vector valued function in order to ensure the existence and convergence of a series solution of the problem (13)–(16).

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 [19, page 56] where, for a vector in , is the usual euclidean norm of . Let us introduce the notation . By [19, page 556], it follows that

If is a polynomial of degree , then by fórmula . of [20, page 92], one gets

We need to recall two well-known inequalities [21]:(i)The Schwarz inequality: Let so that ; if and are continuous functions on , then (ii)The Hölder inequality: If we consider the convergent series of positive terms and , then

The paper is organized as follows. In Section 2, the solution of (13)–(16) is obtained under hypothesis (7)–(9), and the convergence of the series solution for the problem, under these hypotheses (7)–(9), is studied. In Section 3, the solution of (13)–(16) is obtained under hypotheses (10)–(12) and the convergence of the series solution for the problem, under these hypotheses (10)–(12), is also studied. In Section 4, we will introduce an algorithm and give two illustrative examples. Conclusions are given in Section 5.

2. A Series Solution for Nonhomogeneous Problem (13)–(16) under Hypotheses (7)–(9): Convergence

We suppose that the hypotheses (7)–(9) hold. We will find the solution of nonhomogeneous problem with homogeneous boundary conditions (13)–(16) where we will suppose that the vector valued function satisfies the conditions that we will indicate to ensure the convergence of the solution proposal.

We will suppose that the vector valued function satisfies conditions (8) replacing by , and, therefore, admits a series expansion of Sturm-Liouville eigenfunctions which are given by where the set of eigenvalues are given by equation (27) of [15], with the positive roots , of equation (16) of [15], to which is added the eigenvalue if , and, by equation (35) of [15], the eigenvalue is also added if , and the eigenfunctions are given by and coefficients fulfilling the Bessel inequality; see [11, page 223] and [22]: We know that the positive roots fulfill Lemma 1 of [15]; then, and taking into account that , then the numerical series is convergent.

Using the eigenfunction method, we will construct a formal solution of the problem (13)–(15) in the form where

Taking into account that have to satisfy the initial condition (16), one gets that thus, as satisfies (8), then it also admits a series expansion of Sturm-Liouville eigenfunctions: Note that we can write where is a solution of the homogeneous problem with homogeneous boundary conditions: the convergence of has been studied previously in [15], and is a solution of the nonhomogeneous problem with homogeneous boundary conditions: Now, we will study the convergence of the formal solution obtained in (27). Previously, we need to find a bound to the integral Using (18), one gets that where is a polynomial of degree with positive coefficients. Thus, Performing the change of variable and thaking into account (19), we can write expression (39) in the form where and taking into account that the coefficients of and are positive, one gets from (39) and (40) that Now, one gets that where is a solution of problem (34), whose convergence has been studied in [15]; we will study the convergence of , solution of problem (36), defined by (35), where are defined by (28). Taking norm and using (20), one gets that We define , using inequality (42); it follows that and as series is convergent, then series is also convergent. We define ; it follows that using (26) one gets that then there exists a positive integer so that, for all index so that and , one gets that and replacing in (46), Applying Bessel’s inequality (25), it follows that This ensures that the series is uniformly convergent and integrating in the interval ; therefore, where, for a fixed value of , the positive terms series has the partial sum bounded if we suppose that vector valued function satisfies the following condition: If condition (53) holds, series is convergent. Using (21), (42), and (52) in (44), it follows that and taking into account that is convergent, series is uniformly convergent on any domain .