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

and Applied Analysis 3 2. A Series Solution for Nonhomogeneous Problem (13)–(16) under Hypotheses (7)–(9). Convergence We suppose that the hypotheses (7)–(9) hold.Wewill find the solution of nonhomogeneous problem with homogeneous boundary conditions (13)–(16) where we will suppose that the vector valued functionG(x, t) satisfies the conditions that we will indicate to ensure the convergence of the solution proposal. We will suppose that the vector valued function G(x, t) satisfies conditions (8) replacing f(x) by G(x, t), and, therefore, G(x, t) admits a series expansion of Sturm-Liouville eigenfunctions which are given by G (x, t) = X 0 (x) T 0 (t) + ∑ λ n ∈F X n (x) T n (t) , (22) where the set of eigenvaluesF are given by equation (27) of [15], with the positive roots λ k ∈ (kπ, (k + 1)π), k ≥ 1, of equation (16) of [15], to which is added the eigenvalue λ 0 ∈ (0, π) if (1 − b 2 + ρ 0 b 1 b 1 )(1 − ρ 0 b 1 )/b 1 < 1, and, by equation (35) of [15], the eigenvalue 0 is also added if 1 ∈ σ(−A 2 A 1 ), and the eigenfunctions are given by X 0 (x) = α ((1 − ρ 0 b 1 ) x − b 1 ) , α = { 1, 0 ∈ F 0, 0 ∉ F X n (x) = (1 − ρ 0 b 1 ) sen (λ n x) − b 1 λ n cos (λ n x) (23)

(1) The matrix coefficient  is a matrix which satisfies the following condition: Re () > 0, ∀ ∈  () , where () denotes the set of all the eigenvalues of a matrix  in C × .Thus,  is a positive stable matrix (where Re() denotes the real part of  ∈ C).
This paper deals with the construction of the exact series solution of the nonhomogeneous problem   (, ) −   (, ) =  (, ) , 0 <  < 1,  > 0 (13) 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 C × , its 2-norm denoted by ‖‖ is defined by [19, page 56] where, for a vector  in C  , ‖‖ If   () is a polynomial of degree , then by fórmula 2.323 of [20, page 92], one gets We need to recall two well-known inequalities [21]: (i) The Schwarz inequality: Let ,  ∈ R so that  ≤ ; if  and  are continuous functions on [, ], then (ii) The Hölder inequality: If we consider the convergent series of positive terms ∑ ≥0   and ∑ ≥0   , 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.

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  (, ) =  0 ()  0 () + ∑ where the set of eigenvalues F are given by equation ( 27) of [15], with the positive roots   ∈ (, ( + 1)),  ≥ 1, of equation ( 16) of [15], to which is added the eigenvalue  0 ∈ (0, ) if (1 −  2 +  0  1  1 )(1 −  0  1 )/ 1 < 1, and, by equation (35) of [15], the eigenvalue 0 is also added if 1 ∈ (− Ã2 Ã1 ), and the eigenfunctions are given by and coefficients fulfilling the Bessel inequality; see [11, page 223] and [22]: We know that the positive roots   ,  ≥ 1 fulfill Lemma 1 of [15]; then, lim and taking into account that   ∈ (, ( + 1)),  ≥ 1, then the numerical series ∑ ≥1 1/ 2  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 V(, ) 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  2−2 () is a polynomial of degree 2−2 with positive coefficients.Thus, Performing the change of variable V =  2  ( − ) and thaking into account (19), we can write expression (39) in the form where and taking into account that the coefficients of  () 2−2 () and  ()  2−2 (0) are positive, one gets from (39) and (40) that Now, one gets that where V(, ) is a solution of problem (34), whose convergence has been studied in [15]; we will study the convergence using (26) one gets that lim then there exists a positive integer  0 ∈ N so that, for all index  so that   ∈ F and  >  0 , one gets that and taking into account that To check that solution (, ) given in ( 35) is a solution of problem ( 13)-( 16), it is sufficient to show that the series is uniformly convergent.To prove this, note that (, ) satisfies the boundary condition ( 14) and ( 15); then,  15) is given by (, ) = V(, ) + (, ).

A Series Solution for Nonhomogeneous
Problem (13)-( 16) under Hypotheses (10)- (12).Convergence We will suppose that the vector valued function (, ) satisfies conditions (10) replacing () by (, ), and, therefore, (, ) admits a series expansion of Sturm-Liouville eigenfunctions which is given by  (, ) =  0 () T0 () + ∑ where Using again the eigenfunction method, we will construct a formal solution of the problem ( 13)-( 15) in the form where Nothe that, as in Section 2, from (61) it follows that  (, ) =  0 () R0 () where V(, ) is a solution of homogeneous problem with homogeneous boundary values conditions (34), whose convergence has been studied in [16]; we will study the convergence of (, ) solution of problem (36), defined by but this can be considered a special case of the one studied in Section 2 taking  1 = 0. Thus, we have the following Theorem.

Algorithm and Examples
We can summarize the process to calculate the solution of the problem ( 13)-( 15) from Theorems 1 and 2 in Algorithm 1.
(1) If we consider the associated problem (34), it is easy to check that conditions ( 7)-( 9) hold.In fact, this problem was solved in Example 3.1 of [15].Using Algorithm 1 of [15], we can obtain the solution V(, ) Consider the associated problem (34) and check the following options: Case 1. Conditions ( 7)-( 9) holds.Continue using Algorithm 1 of [15] to obtain a solution V(, ) of problem (34).Once obtained, continue with Algorithm 2. Case 2. Conditions ( 10)-( 12) holds.Continue using Algorithm 1 of [16] to obtain a solution V(, ) of problem (34).Once obtained, continue with Algorithm 3. Case 3. If these conditions are not satisfied, algorithm stop because we can not obtain the solution of problem ( 13)- (15) with the given data.
The solution of problem (34) is given by After obtaining the solution of the homogeneous problem with homogeneous conditions (34), we continue with Algorithm 2.
We will follow Algorithm 2 step by step.
Example 4. We consider problem ( 13)-( 15) where matrix  ∈ C 4×4 is given by and the matrices   ,   ,  ∈ {1, 2} are given by (68).The vectorial valued function () is defined by and the vectorial valued function (, ) is given by We will follow Algorithm 1 step by step.
After obtaining the solution of the homogeneous problem with homogeneous conditions (34), we continue with Algorithm 3.
We will follow Algorithm 3 step by step.