On Exact Series Solution for Strongly Coupled Mixed Parabolic Boundary Value Problems

and Applied Analysis 3 (iii) For b 2 = −1 see Ker (B 2 + I) =⟨ { { {

(i) The matrix coefficient  is a matrix which satisfies the following condition: Re () > 0 for all eigenvalues  of , and thus  is a positive stable matrix (where Re() denotes the real part of  ∈ C).
In order to construct the eigenfunctions in [11], the following matrices Ã1 and B1 were defined by thus satisfying the condition where matrix  denotes, as usual, the identity matrix.Under hypothesis (3), matrix  2 − ( 2 +  0  2 ) B1 is regular; see [11, page 431], and Ã2 and B2 are the matrices defined by exist where () denotes the set of all the eigenvalues of a matrix  in C × .These eigenfunctions introduced in [11] were also used in [13] to construct a series solution of the initial-value problem: where () = ( 1 (),  2 (), . . .,   ())  is an -dimensional vector, under the additional hypothesis: is an invariant subspace with respect to matrix , where a subspace  of C  is invariant by the matrix  ∈ C × , if () ⊂ .It is not difficult to show examples where this assumption (9) is held but ( 14) is not held.Let us consider the following example: Example 1.We will consider the homogeneous parabolic problem with homogeneous conditions (10)- (13), where matrix  ∈ C 4×4 is chosen: and the 4 × 4 matrices   ,   ,  ∈ {1, 2}, are Due to ( 5)-( 7) we obtain It is easy to verify that ( B1 ) = {0,1} and ( B2 ) = {1, 0, −1}.If we take  1 = 1, one gets Taking into account that ( B2 ), we will have three possible values for  2 .
(i) For  2 = 1 there is and then condition (9) is not fulfilled.
Observe that, in this example, hypothesis ( 9) is satisfied but in (14) it is not satisfied.Thus, the method proposed in [13] cannot be used to solve this problem.This paper deals with the construction of eigenfunctions of problem ( 10)-( 12) by assuming hypotheses (2), (3), and (4) but not hypothesis (9).This set of eigenfunctions allows us to construct a series solution of the problem ( 10)-( 13).We provide conditions for the function () and the matrix coefficients, in order to ensure the existence of a series solution of the problem.
The paper is organized as follows: in Section 2 a set of eigenfunctions will be constructed under a new condition, different from condition (9); in Section 3 a series solution for the problem is presented.In Section 4 we will introduce an algorithm and give an illustrative example.
Throughout this paper we will assume the results and nomenclature given in [11].If  is a matrix in C × , we denote by  † its Moore-Penrose pseudoinverse [12].A collection of examples, properties, and applications of this concept may be found in [14], and  † can be efficiently computed with the MATLAB and Mathematica computer algebra systems.

The New Conditions
In Section 2 of [11] the eigenfunctions of problem ( 10)- (12) were constructed by using a matrix variable separation technique.We can repeat the calculations in this section to reach condition (44): Instead of ( 9), we will assume the following new condition:  From relation (6) Ã1  =  is obtained (because, obviously, B1  = 0).Considering (8) B2  =  is obtained.Thus (− Ã2 Ã1 −  2 B2 B1 ) = − 2 ; that is, − 2 is the eigenvalue which will be equal to cot () in (26): Let us assume that  2 given in (28) satisfies We will observe that, under hypothesis (29), we have guaranteed the existence of the solutions for the equation Equation (30) has a unique solution   in each interval (, (+1)) for  ≥ 1, as seen in Figure 1.Also, the following lemma is easily demonstrated.
In order to ensure that (  ) ̸ = 0 fulfils (37) we have and under condition (40), the solution of (37) is given by The eigenfunctions associated to the problem ( 10)-( 12) are then given by Working in a similar form to that in [11, page 433], we can show that also  = 0 is an eigenvalue of problem (10), if Under hypothesis (43), if we denote one gets that function is an eigenfunction of problem ( 10) associated to eigenvalue  = 0.As a conclusion, the following theorem has been demonstrated.
We can summarize the results in the following theorem.