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

Authors: Vicente Soler; Emilio Defez; Roberto Capilla; José Antonio Verdoy
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

An exact series solution for nonhomogeneous parabolic coupled systems of the typeu
t
-
A
u
x
x
=
G
x
,
t
,
A
1
u
0
,
t
+
B
1
u
x
0
,
t
=
0
,
A
2
u
l
,
t
+
B
2
u
x
l
,
t
=
0
,
0
<
x
<
1
,
t
>
0
,
u
x
,
0
=
f
x,
where A
1
,
A
2
,
B
1, and B
2 are arbitrary matrices for which the block matrix A
1
B
1
A
2
B
2 is nonsingular, and A is a positive stable matrix, is constructed.


---

## Body

## 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
(1)
u
t
(
x
,
t
)
-
A
u
x
x
(
x
,
t
)
=
0
,
0
<
x
<
1
,
t
>
0
A
1
u
(
0
,
t
)
+
B
1
u
x
(
0
,
t
)
=
0
,
t
>
0
A
2
u
(
1
,
t
)
+
B
2
u
x
(
1
,
t
)
=
0
,
t
>
0
u
(
x
,
0
)
=
f
(
x
)
,
0
≤
x
≤
1
,
where u
=
(
u
1
,
u
2
,
…
,
u
m
)
T and f
(
x
)
=
(
f
1
(
x
)
,
f
2
(
x
)
,
…
,
f
m
(
x
)
)
T are m-dimensional vectors, was constructed under the following hypotheses and notations.(1)
The matrix coefficientA is a matrix which satisfies the following condition:
(2)
Re
(
z
)
>
0
,
∀
z
∈
σ
(
A
)
,
whereσ
(
C
) denotes the set of all the eigenvalues of a matrix C in C
m
×
m. Thus, A is a positive stable matrix (where Re
(
z
) denotes the real part of z
∈
C).
(2)
MatricesA
i
,
B
i
,
i
=
1,2, are m
×
m complex matrices, and we assume that the block matrix
(3)
(
A
1
B
1
A
2
B
2
)
is
regular
and also that the matrix pencil(4)
A
1
+
ρ
0
B
1
is
regular
that is, condition (4) involves the existence of some ρ
0
∈
C, matrix A
1
+
ρ
0
B
1 being invertible; see [17]. Using condition (4), we can introduce the following matrices A
~
1 and B
~
1 defined by
(5)
A
~
1
=
(
A
1
+
ρ
0
B
1
)
-
1
A
1
,
B
~
1
=
(
A
1
+
ρ
0
B
1
)
-
1
B
1
,
which satisfy the condition A
~
1
+
ρ
0
B
~
1
=
I, where matrix I denotes, as usual, the identity matrix. Under hypothesis (3), it is easy to show that matrix B
2
-
(
A
2
+
ρ
0
B
2
)
B
~
1 is regular (see [18] for details) and we can introduce matrices A
~
2 and B
~
2 defined by
(6)
A
~
2
=
[
B
2
-
(
A
2
+
ρ
0
B
2
)
B
~
1
]
-
1
A
2
,
B
~
2
=
[
B
2
-
(
A
2
+
ρ
0
B
2
)
B
~
1
]
-
1
B
2
,
that satisfy the conditions B
~
2
-
(
A
~
2
+
ρ
0
B
~
2
)
B
~
1
=
I
,
B
~
2
A
~
1
-
A
~
2
B
~
1
=
I. Under the above assumptions, in [15], we have consider the following essential hypothesis:
(7)
exist
b
1
∈
σ
(
B
~
1
)
-
{
0
}
,
b
2
∈
σ
(
B
~
2
)
,
v
∈
C
m
-
{
0
}
,
such
that
(
B
~
1
-
b
1
I
)
v
=
(
B
~
2
-
b
2
I
)
v
=
0
,
if the vector valued function f
(
x
) satisfies hypotheses
(8)
f
∈
C
2
(
[
0,1
]
)
(
1
-
ρ
0
b
1
)
f
(
0
)
+
b
1
f
′
(
0
)
=
0
-
(
1
-
b
2
+
ρ
0
b
1
b
2
b
1
)
f
(
1
)
+
b
2
f
′
(
1
)
=
0
,
under the additional hypothesis
(9)
f
(
x
)
∈
Ker
⁡
(
B
~
1
-
b
1
I
)
∩
Ker
⁡
(
B
~
2
-
b
2
I
)
,
0
≤
x
≤
1
,
Ker
⁡
(
B
~
1
-
b
1
I
)
∩
Ker
⁡
(
B
~
2
-
b
2
I
)
is
an
invariant
subspace
with
respect
to
matrix
A
,
where a subspace E of C
m is invariant by the matrix A
∈
C
m
×
m, if A
(
E
)
⊂
E, in order to construct an exact series solution u
(
x
,
t
) of homogeneous problem (1).Moreover, in [16], under the above assumptions and replacing the condition (7) by the following hypothesis
(10)
0
∈
σ
(
B
~
1
)
,
a
2
∈
σ
(
A
~
2
)
,
and
we
have
w
∈
C
m
-
{
0
}
,
so
that
B
~
1
w
=
(
A
~
2
-
a
2
I
)
w
=
0
,if the vector valued functionf
(
x
) satisfies the new hypotheses
(11)
f
∈
C
2
(
[
0,1
]
)
f
(
0
)
=
0
a
2
f
(
1
)
+
f
′
(
1
)
=
0
,under the additional hypothesis(12)
f
(
x
)
∈
Ker
⁡
(
B
~
1
)
∩
Ker
⁡
(
A
~
2
-
a
2
I
)
,
0
≤
x
≤
1
,
Ker
⁡
(
B
~
1
)
∩
Ker
⁡
(
A
~
2
-
a
2
I
)
is
an
invariant
subspace
respect
to
matrix
A
,
then an exact series solution u
(
x
,
t
) of homogeneous problem (1) is constructed, see [16].This paper deals with the construction of the exact series solution of the nonhomogeneous problem(13)
u
t
(
x
,
t
)
-
A
u
x
x
(
x
,
t
)
=
G
(
x
,
t
)
,
0
<
x
<
1
,
t
>
0
(14)
A
1
u
(
0
,
t
)
+
B
1
u
x
(
0
,
t
)
=
0
,
t
>
0
(15)
A
2
u
(
1
,
t
)
+
B
2
u
x
(
1
,
t
)
=
0
,
t
>
0
(16)
u
(
x
,
0
)
=
f
(
x
)
,
0
≤
x
≤
1
.We provide conditions for the vector valued functionG
(
x
,
t
) 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 B is a matrix in C
m
×
m, its 2-norm denoted by ∥
B
∥ is defined by [19, page 56]
(17)
∥
B
∥
=
sup
⁡
z
≠
0
∥
B
z
∥
2
∥
z
∥
2
,
where, for a vector y in C
m, ∥
z
∥
2 is the usual euclidean norm of y. Let us introduce the notation α
(
C
)
=
max
⁡
{
Re
(
z
)
;
z
∈
σ
(
C
)
}. By [19, page 556], it follows that
(18)
∥
e
t
B
∥
≤
e
α
(
B
)
t
∑
k
=
0
m
‍
∥
m
B
∥
k
t
k
k
!
.IfP
m
(
x
) is a polynomial of degree m, then by fórmula 2.323 of [20, page 92], one gets
(19)
∫
‍
P
m
(
x
)
e
a
x
d
x
=
e
a
x
a
∑
k
=
0
m
‍
(
-
1
)
k
P
m
(
k
)
(
x
)
a
k
.We need to recall two well-known inequalities [21]:(i)
The Schwarz inequality: Let a
,
b
∈
R so that a
≤
b; if f and g are continuous functions on [
a
,
b
], then
(20)
∫
a
b
‍
f
(
x
)
g
(
x
)
d
x
≤
(
∫
a
b
‍
f
(
x
)
2
d
x
)
1
/
2
(
∫
a
b
‍
g
(
x
)
2
d
x
)
1
/
2
.
(ii)
The Hölder inequality: If we consider the convergent series of positive terms ∑
n
≥
0
‍
a
n and ∑
n
≥
0
‍
b
n, then
(21)
∑
n
≥
0
‍
a
n
1
/
2
b
n
1
/
2
≤
(
∑
n
≥
0
‍
a
n
)
1
/
2
(
∑
n
≥
0
‍
b
n
)
1
/
2
.The paper is organized as follows. In Section2, 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 G
(
x
,
t
) satisfies the conditions that we will indicate to ensure the convergence of the solution proposal.We will suppose that the vector valued functionG
(
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
(22)
G
(
x
,
t
)
=
X
0
(
x
)
T
0
(
t
)
+
∑
λ
n
∈
F
‍
X
n
(
x
)
T
n
(
t
)
,
where the set of eigenvalues F are given by equation (27) of [15], with the positive roots λ
k
∈
(
k
π
,
(
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
(23)
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
)
and coefficients
(24)
T
n
(
t
)
=
∫
0
1
‍
G
(
x
,
t
)
X
n
(
x
)
d
x
∫
0
1
‍
X
n
2
(
x
)
d
x
,
T
0
(
t
)
=
∫
0
1
‍
G
(
x
,
t
)
X
0
(
x
)
d
x
∫
0
1
‍
X
0
2
(
x
)
d
x
,
fulfilling the Bessel inequality; see [11, page 223] and [22]:
(25)
∑
λ
n
∈
F
‍
∥
T
n
(
t
)
∥
2
∥
X
n
(
x
)
∥
2
=
∑
λ
n
∈
F
‍
∥
T
n
(
t
)
∥
2
∫
0
1
‍
X
n
2
(
x
)
d
x
≤
∫
0
1
‍
∥
G
(
x
,
t
)
∥
2
d
x
.
We know that the positive roots λ
k
,
k
≥
1 fulfill Lemma 1 of [15]; then,
(26)
lim
⁡
k
→
∞
λ
k
=
∞
,
and taking into account that λ
k
∈
(
k
π
,
(
k
+
1
)
π
)
,
k
≥
1, then the numerical series ∑
k
≥
1
‍
1
/
λ
k
2 is convergent.Using the eigenfunction method, we will construct a formal solution of the problem (13)–(15) in the form
(27)
u
(
x
,
t
)
=
X
0
(
x
)
R
0
(
t
)
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
R
n
(
t
)
,
where
(28)
B
n
(
t
)
=
∫
0
t
‍
e
A
λ
n
2
s
T
n
(
s
)
d
s
,
B
0
(
t
)
=
∫
0
t
‍
T
0
(
s
)
d
s
,
(29)
R
n
(
t
)
=
B
n
(
t
)
+
D
n
,
R
0
(
t
)
=
B
0
(
t
)
+
D
0
.Taking into account thatu
(
x
,
t
) have to satisfy the initial condition (16), one gets that
(30)
u
(
x
,
0
)
=
f
(
x
)
=
X
0
(
x
)
R
0
(
0
)
+
∑
λ
n
∈
F
‍
X
n
(
x
)
R
n
(
0
)
;
thus, as f
(
x
) satisfies (8), then it also admits a series expansion of Sturm-Liouville eigenfunctions:
(31)
R
n
(
0
)
=
D
n
=
∫
0
1
‍
f
(
x
)
X
n
(
x
)
d
x
∫
0
1
‍
X
n
2
(
x
)
d
x
,
R
0
(
0
)
=
D
0
=
∫
0
1
‍
f
(
x
)
X
0
(
x
)
d
x
∫
0
1
‍
X
0
2
(
x
)
d
x
.
Note that we can write
(32)
u
(
x
,
t
)
=
X
0
(
x
)
R
0
(
t
)
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
R
n
(
t
)
=
X
0
(
x
)
D
0
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
D
n
+
X
0
(
x
)
B
0
(
t
)
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
B
n
(
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
)
,
where
(33)
v
(
x
,
t
)
=
X
0
(
x
)
D
0
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
D
n
is a solution of the homogeneous problem with homogeneous boundary conditions:
(34)
v
t
(
x
,
t
)
-
A
v
x
x
(
x
,
t
)
=
0
,
0
<
x
<
1
,
t
>
0
A
1
v
(
0
,
t
)
+
B
1
v
x
(
0
,
t
)
=
0
,
t
>
0
A
2
v
(
1
,
t
)
+
B
2
v
x
(
1
,
t
)
=
0
,
t
>
0
v
(
x
,
0
)
=
f
(
x
)
,
t
>
0
,
the convergence of v
(
x
,
t
) has been studied previously in [15], and
(35)
w
(
x
,
t
)
=
X
0
(
x
)
B
0
(
t
)
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
B
n
(
t
)
is a solution of the nonhomogeneous problem with homogeneous boundary conditions:
(36)
w
t
(
x
,
t
)
-
A
w
x
x
(
x
,
t
)
=
G
(
x
,
t
)
,
0
<
x
<
1
,
t
>
0
A
1
w
(
0
,
t
)
+
B
1
w
x
(
0
,
t
)
=
0
,
t
>
0
A
2
w
(
1
,
t
)
+
B
2
w
x
(
1
,
t
)
=
0
,
t
>
0
w
(
x
,
0
)
=
0
,
t
>
0
.
Now, we will study the convergence of the formal solution obtained in (27). Previously, we need to find a bound to the integral
(37)
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
2
d
s
.
Using (18), one gets that
(38)
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
2
≤
e
-
2
α
(
A
)
λ
n
2
(
t
-
s
)
(
∑
k
=
0
m
-
1
‍
∥
m
A
∥
k
(
λ
n
2
(
t
-
s
)
)
k
k
!
)
2
=
e
-
2
α
(
A
)
λ
n
2
(
t
-
s
)
P
2
m
-
2
(
λ
n
2
(
t
-
s
)
)
,
where P
2
m
-
2
(
x
) is a polynomial of degree 2
m
-
2 with positive coefficients. Thus,
(39)
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
2
d
s
≤
∫
0
t
‍
e
-
2
α
(
A
)
λ
n
2
(
t
-
s
)
P
2
m
-
2
(
λ
n
2
(
t
-
s
)
)
d
s
.
Performing the change of variable v
=
λ
n
2
(
t
-
s
) and thaking into account (19), we can write expression (39) in the form
(40)
∫
0
t
‍
e
-
2
α
(
A
)
λ
n
2
(
t
-
s
)
P
2
m
-
2
(
λ
n
2
(
t
-
s
)
)
d
s
=
e
-
2
α
(
A
)
2
λ
n
2
α
(
A
)
(
L
-
∑
k
=
0
2
m
-
2
‍
P
2
m
-
2
(
k
)
(
λ
n
2
t
)
(
2
α
(
A
)
)
k
)
,
where
(41)
L
=
∑
k
=
0
2
m
-
2
‍
P
2
m
-
2
(
k
)
(
0
)
(
2
α
(
A
)
)
k
,
and taking into account that the coefficients of P
2
m
-
2
(
k
)
(
x
) and P
2
m
-
2
(
k
)
(
0
) are positive, one gets from (39) and (40) that
(42)
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
2
d
s
≤
L
2
λ
n
2
α
(
A
)
,
t
≥
0
.
Now, one gets that
(43)
u
(
x
,
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
)
,
where v
(
x
,
t
) is a solution of problem (34), whose convergence has been studied in [15]; we will study the convergence of w
(
x
,
t
), solution of problem (36), defined by (35), where B
n
(
t
) are defined by (28). Taking norm and using (20), one gets that
(44)
∥
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
B
n
(
t
)
∥
2
=
∥
∑
λ
n
∈
F
‍
∫
0
t
‍
e
-
A
λ
n
2
(
t
-
s
)
T
n
(
s
)
X
n
(
x
)
d
s
∥
2
≤
(
∑
λ
n
∈
F
‍
∥
∫
0
t
‍
e
-
A
λ
n
2
(
t
-
s
)
T
n
(
s
)
X
n
(
x
)
d
s
∥
)
2
≤
(
∑
λ
n
∈
F
‍
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
T
n
(
s
)
X
n
(
x
)
d
s
∥
)
2
≤
(
∑
λ
n
∈
F
‍
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
∥
T
n
(
s
)
X
n
(
x
)
∥
d
s
)
2
≤
(
∑
λ
n
∈
F
‍
(
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
2
d
s
)
1
/
2
m
m
m
m
×
(
∫
0
t
‍
∥
T
n
(
s
)
∥
2
|
X
n
(
x
)
|
2
d
s
)
1
/
2
∑
λ
n
∈
F
‍
)
2
.
We define a
n
=
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
2
d
s, using inequality (42); it follows that
(45)
a
n
=
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
2
d
s
≤
L
2
λ
n
2
α
(
A
)
and as series ∑
λ
n
∈
F
‍
(
1
/
λ
n
2
) is convergent, then series ∑
λ
n
∈
F
‍
a
n is also convergent. We define b
n
=
∫
0
t
‍
∥
T
n
(
s
)
∥
2
|
X
n
(
x
)
|
2
d
s; it follows that
(46)
∥
T
n
(
s
)
∥
2
|
X
n
(
x
)
|
2
=
∥
T
n
(
s
)
∥
2
∥
X
n
(
x
)
∥
2
|
X
n
(
x
)
|
2
∥
X
n
(
x
)
∥
2
;
(47)
|
X
n
(
x
)
|
2
∥
X
n
(
x
)
∥
2
=
|
X
n
(
x
)
|
2
∫
0
1
‍
X
n
2
(
x
)
d
x
≤
(
λ
n
2
b
1
2
+
|
1
-
ρ
0
b
1
|
2
+
2
|
b
1
|
|
1
-
ρ
0
b
1
|
λ
n
)
×
(
λ
n
2
b
1
2
2
-
b
1
(
1
-
ρ
0
b
1
)
2
+
(
1
-
ρ
0
b
1
)
2
2
mmmmm
+
λ
n
2
b
1
2
-
(
1
-
ρ
0
b
1
)
2
4
λ
n
sin
(
2
λ
n
)
mmmmm
+
b
1
(
1
-
ρ
0
b
1
)
2
cos
⁡
(
2
λ
n
)
λ
n
2
b
1
2
2
)
-
1
=
A
(
λ
n
)
,
using (26) one gets that
(48)
lim
⁡
n
→
∞
A
(
λ
n
)
=
2
,
then there exists a positive integer n
0
∈
N so that, for all index n so that λ
n
∈
F and n
>
n
0, one gets that
(49)
|
X
n
(
x
)
|
2
∥
X
n
(
x
)
∥
2
<
3
,
and replacing in (46),
(50)
∥
T
n
(
s
)
∥
2
X
n
2
(
x
)
≤
3
∥
T
n
(
s
)
∥
2
∥
X
n
(
x
)
∥
2
,
f
f
f
f
n
∈
{
m
∈
N
:
λ
m
∈
F
,
m
>
n
0
}
.
Applying Bessel’s inequality (25), it follows that
(51)
∑
λ
n
∈
F
‍
∥
T
n
(
s
)
∥
2
|
X
n
(
x
)
|
2
≤
3
∑
λ
n
∈
F
‍
∥
T
n
(
s
)
∥
2
∥
X
n
(
x
)
∥
2
≤
3
∫
0
1
‍
∥
G
(
x
,
s
)
∥
2
d
x
.
This ensures that the series ∑
λ
n
∈
F
‍
∥
T
n
(
s
)
∥
2
|
X
n
(
x
)
|
2 is uniformly convergent and integrating in the interval [
0
,
t
]
,
t
≥
0; therefore,
(52)
∑
λ
n
∈
F
‍
∫
0
t
‍
∥
T
n
(
s
)
∥
2
|
X
n
(
x
)
|
2
d
s
≤
3
∫
0
t
‍
∫
0
1
‍
∥
G
(
x
,
s
)
∥
2
d
x
d
s
,
where, for a fixed value of t
∈
[
c
,
d
], the positive terms series b
n has the partial sum bounded if we suppose that vector valued function G
(
x
,
t
) satisfies the following condition:
(53)
sup
⁡
t
>
0
∫
0
1
‍
∥
G
(
x
,
t
)
∥
2
d
x
=
M
<
∞
.
If condition (53) holds, series ∑
n
/
λ
n
∈
F
‍
b
n is convergent. Using (21), (42), and (52) in (44), it follows that
(54)
∥
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
B
n
(
t
)
∥
2
≤
(
∑
λ
n
∈
F
‍
∫
0
t
‍
∥
e
-
A
λ
n
2
(
t
-
s
)
∥
2
d
s
)
×
(
∑
λ
n
∈
F
‍
∫
0
t
‍
∥
T
n
(
s
)
∥
2
X
n
2
(
x
)
d
s
)
≤
3
L
2
α
(
A
)
(
∫
0
t
‍
∫
0
1
‍
∥
G
(
x
,
s
)
∥
2
d
x
d
s
)
∑
λ
n
∈
F
‍
1
λ
n
2
,
and taking into account that ∑
λ
n
∈
F
‍
(
1
/
λ
n
2
) is convergent, series ∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
B
n
(
t
) is uniformly convergent on any domain [
0,1
]
×
[
c
,
d
].To check that solutionw
(
x
,
t
) given in (35) is a solution of problem (13)–(16), it is sufficient to show that the series
(55)
∑
λ
n
∈
F
‍
λ
n
2
e
-
λ
n
2
A
t
X
n
(
x
)
B
n
(
t
)
is uniformly convergent. To prove this, note that G
(
x
,
t
) satisfies the boundary condition (14) and (15); then,
(56)
∫
0
1
‍
∂
2
∂
x
2
(
G
(
x
,
t
)
)
X
n
(
x
)
d
x
=
-
λ
n
2
∫
0
1
‍
G
(
x
,
t
)
X
n
(
x
)
d
x
,
And, by (24), one gets that
(57)
λ
n
2
T
n
(
t
)
∥
X
n
(
x
)
∥
2
=
-
∫
0
1
‍
G
x
x
(
x
,
t
)
X
n
(
x
)
d
x
;
Then, if the following condition
(58)
sup
⁡
t
>
0
∫
0
1
‍
∥
G
x
x
(
x
,
t
)
∥
2
d
x
=
N
<
∞
holds, the convergence of the series (55) can be derived in the same way as the convergence of the series ∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
X
n
(
x
)
B
n
(
t
) has been deduced, and, thus, series (55) is uniformly convergent on any domain [
0,1
]
×
[
c
,
d
].Summarizing, the following result has been established.Theorem 1.
Consider a nonhomogeneous problem with homogeneous boundary values conditions (13)–(15) which satisfies conditions (7)–(9). Suppose that hypotheses of Theorem 2 of [15] hold, then we can construct a solution v
(
x
,
t
) of homogeneous problem with homogeneous boundary values conditions (34). Suppose that G
(
x
,
t
) satisfies conditions (8) replacing f
(
x
) by G
(
x
,
t
) and satisfies conditions (53) and (58). Then, w
(
x
,
t
), defined by (35), is a solution of nonhomogeneous problem with homogeneous boundary values conditions (36), and the solution of problem (13)–(15) is given by u
(
x
,
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
).


## 3. A Series Solution for Nonhomogeneous Problem(13)–(16) under Hypotheses(10)–(12): Convergence

We will suppose that the vector valued functionG
(
x
,
t
) satisfies conditions (10) replacing f
(
x
) by G
(
x
,
t
), and, therefore, G
(
x
,
t
) admits a series expansion of Sturm-Liouville eigenfunctions which is given by
(59)
G
(
x
,
t
)
=
X
0
(
x
)
T
~
0
(
t
)
+
∑
λ
n
∈
F
‍
sin
(
λ
n
x
)
T
~
n
(
t
)
,
where
(60)
T
~
n
(
t
)
=
∫
0
1
‍
G
(
x
,
t
)
sin
(
λ
n
x
)
d
x
∫
0
1
‍
sin
2
(
λ
n
x
)
d
x
,
T
~
0
(
t
)
=
∫
0
1
‍
G
(
x
,
t
)
x
d
x
∫
0
1
‍
x
2
d
x
.
Using again the eigenfunction method, we will construct a formal solution of the problem (13)–(15) in the form
(61)
u
(
x
,
t
)
=
X
0
(
x
)
R
~
0
(
t
)
+
∑
λ
n
∈
F
‍
e
-
A
λ
n
2
t
sin
(
λ
n
x
)
R
~
n
(
t
)
,
where
(62)
B
~
n
(
t
)
=
∫
0
t
‍
e
A
λ
n
2
s
T
~
n
(
s
)
d
s
,
B
~
0
(
t
)
=
∫
0
t
‍
T
~
0
(
s
)
d
s
,
(63)
R
~
n
(
t
)
=
B
~
n
(
t
)
+
D
~
n
,
R
~
0
(
t
)
=
B
~
0
(
t
)
(
t
)
+
D
~
0
,
and as f
(
x
) satisfies (10), one gets
(64)
D
~
n
=
∫
0
1
‍
f
(
x
)
sin
(
λ
n
x
)
d
x
∫
0
1
‍
sin
2
(
λ
n
x
)
d
x
,
D
~
0
=
∫
0
1
‍
f
(
x
)
x
d
x
∫
0
1
‍
x
2
d
x
.
Nothe that, as in Section 2, from (61) it follows that
(65)
u
(
x
,
t
)
=
X
0
(
x
)
R
~
0
(
t
)
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
sin
(
λ
n
x
)
R
~
n
(
t
)
=
X
0
(
x
)
D
~
0
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
sin
(
λ
n
x
)
D
~
n
+
X
0
(
x
)
B
~
0
(
t
)
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
sin
(
λ
n
x
)
B
~
n
(
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
)
,
where v
(
x
,
t
) 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 w
(
x
,
t
) solution of problem (36), defined by
(66)
w
(
x
,
t
)
=
X
0
(
x
)
B
~
0
(
t
)
+
∑
λ
n
∈
F
‍
e
-
λ
n
2
A
t
sin
(
λ
n
x
)
B
~
n
(
t
)
,
but this can be considered a special case of the one studied in Section 2 taking b
1
=
0. Thus, we have the following Theorem.Theorem 2.
Consider a nonhomogeneous problem with homogeneous boundary values conditions (13)–(15) which satisfies conditions (10)–(12). Suppose that hypotheses of Theorem 3.1 of [16] hold; then, we can construct a solution v
(
x
,
t
) of homogeneous problem with homogeneous boundary values conditions (34). Suppose that vector valued function G
(
x
,
t
) satisfies conditions (10) replacing f
(
x
) by G
(
x
,
t
) and satisfies conditions (53) and (58). Then, w
(
x
,
t
), defined by (35), is a solution of nonhomogeneous problem with homogeneous boundary values conditions (36), and the solution of problem (13)–(15) is given by u
(
x
,
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
).


## 4. 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.Algorithm 1:Solution of problem (13)–(15).
u
t
(
x
,
t
)
-
A
u
x
x
(
x
,
t
)
=
G
(
x
,
t
), 0
<
x
<
1,  t
>
0
A
1
u
(
0
,
t
)
+
B
1
u
x
(
0
,
t
)
=
0, t
>
0
A
2
u
(
1
,
t
)
+
B
2
u
x
(
1
,
t
)
=
0, t
>
0
u
(
x
,
0
)
=
f
(
x
), 0
≤
x
≤
1
(1) 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
(
x
,
t
) 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
(
x
,
t
) 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.Example 3.
We consider problem (13)–(15) where matrix A
∈
C
4
×
4 is given by
(67)
A
=
(
2
0
0
-
1
1
2
1
-
2
-
1
0
2
1
0
0
0
1
)
,
and the matrices A
i
,
B
i
,
i
∈
{
1,2
} are
(68)
A
1
=
(
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
)
,
A
2
=
(
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
)
B
1
=
(
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
)
,
B
2
=
(
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
)
;
the vectorial valued function f
(
x
) will be defined as
(69)
f
(
x
)
=
(
0
x
2
-
1
0
0
)
,
and the vectorial valued function G
(
x
,
t
) is given by
(70)
G
(
x
,
t
)
=
(
0
(
x
-
1
)
2
x
3
e
-
t
0
0
)
.We will follow Algorithm1 step by step.(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
(
x
,
t
) of problem (34) with the values ρ
0
=
1
,
b
1
=
1
,
b
2
=
0. The solution of problem (34) is given by
(71)
v
(
x
,
t
)
=
(
∑
n
≥
0
‍
-
32
(
-
1
)
n
e
-
(
1
/
2
)
(
π
+
2
n
π
)
2
t
cos
⁡
(
(
1
/
2
)
(
π
+
2
n
π
)
x
)
π
3
(
2
n
+
1
)
3
)
×
(
0
1
0
0
)
,
with the eigenvalues setF
=
{
λ
k
=
π
/
2
+
k
π
:
k
≥
0
} and eigenfunctions
(72)
X
k
(
x
)
=
-
(
π
2
+
k
π
)
cos
⁡
(
(
π
2
+
k
π
)
x
)
,
k
≥
0
.
After obtaining the solution of the homogeneous problem with homogeneous conditions (34), we continue with Algorithm 2.Algorithm 2:Algorithm to compute the solution of problem (13)–(15) when conditions (7)–(9) holds.
(1)  Check that vector valued functionG
(
x
,
t
) satisfies conditions (53), (58) and (8)  replacing f
(
x
) by
G
(
x
,
t
). If these conditions are not satisfied algorithm stop because we can not obtain the solution of
problem (13)–(15) with the given data.
(2) Determine coefficientsT
n
(
t
), n
≥
0 defined by (24).
(3) Determine coefficientsB
n
(
t
), n
≥
0 defined by (28), where D
n, n
≥
0 are defined by (31).
(4) Determinew
(
x
,
t
) defined by (35), solution of problem (36).
(5) The solution of problem (13)–(15) is given by u
(
x
,
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
).We will follow Algorithm2 step by step.(1)
It is trivial to check that, for fixedt,
(73)
G
(
x
,
t
)
∈
C
2
(
[
0,1
]
)
.
Therefore,(74)
(
1
-
ρ
0
b
1
)
G
(
0
,
t
)
+
b
1
G
x
(
0
,
t
)
=
(
0
0
0
0
)
,
-
(
1
-
b
2
+
ρ
0
b
1
b
2
b
1
)
G
(
1
,
t
)
+
b
2
G
x
(
1
,
t
)
=
(
0
0
0
0
)
;
then, the vector valued functionG
(
x
,
t
) satisfies condition (8) replacing f
(
x
) by G
(
x
,
t
). By other hand, one gets that
(75)
∫
0
1
‍
∥
G
(
x
,
t
)
∥
2
d
x
=
∫
0
1
‍
(
e
-
2
t
(
-
1
+
x
)
4
x
6
)
d
x
=
e
-
2
t
2310
,
and, thereby,sup
⁡
t
>
0
∫
0
1
‍
∥
G
(
x
,
t
)
∥
2
d
x
≤
1
/
2310
=
M and condition (53) holds. Similarly,
(76)
∫
0
1
‍
∥
G
x
x
(
x
,
t
)
∥
2
d
x
=
∫
0
1
‍
(
(
6
e
-
t
(
-
1
+
x
)
2
x
+
12
e
-
t
(
-
1
+
x
)
x
2
+
2
e
-
t
x
3
)
2
)
d
x
=
12
e
-
2
t
35
,
and, thereby,sup
⁡
t
>
0
∫
0
1
‍
∥
G
x
x
(
x
,
t
)
∥
2
d
x
≤
12
/
35
=
N and condition (58) holds.
(2)
Forn
≥
0, coefficients T
n
(
t
) defined by (24) are given by
(77)
T
n
(
t
)
=
(
(
(
2
n
+
1
)
7
π
7
)
-
1
(
(
2
n
+
1
)
7
π
7
)
-
1


(
64
e
-
t
(
(
-
1
)
n
(
2
n
+
1
)
π
(
(
2
n
+
1
)
2
π
2
-
144
)
mmmmmmm
-
6
(
(
2
n
+
1
)
2
π
2
-
80
)
)
)
×
(
(
2
n
+
1
)
7
π
7
)
-
1
)
×
(
0
1
0
0
)
.
(3)
Forn
≥
0, coefficients B
n
(
t
) defined by (28) are given by
(78)
B
n
(
t
)
=
(
(
(
2
n
+
1
)
7
π
7
(
-
2
+
(
2
n
+
1
)
2
π
2
)
)
-
1
(
(
2
n
+
1
)
7
π
7
(
-
2
+
(
2
n
+
1
)
2
π
2
)
)
-
1

(
128
(
e
(
-
2
+
(
2
n
+
1
)
2
π
2
)
t
/
2
-
1
)
m
m
m
m
×
(
(
-
1
)
n
(
2
n
+
1
)
π
(
(
2
n
+
1
)
2
π
2
-
144
)
mmmmim
-
6
(
(
2
n
+
1
)
2
π
2
-
80
)
)
)
×
(
(
2
n
+
1
)
7
π
7
(
-
2
+
(
2
n
+
1
)
2
π
2
)
)
-
1
)
×
(
0
1
0
0
)
.
(4)
The solutionw
(
x
,
t
) of problem (36) defined by (35) is given by
(79)
w
(
x
,
t
)
=
∑
n
≥
0
‍
-
(
m
m
m
m
m
m
×
(
(
2
n
+
1
)
6
π
6
(
-
2
+
(
2
n
+
1
)
2
π
2
)
)
-
1
(
(
(
2
n
+
1
)
π
x
2
)
64
e
-
(
2
n
+
1
)
2
π
2
t
/
2
(
e
(
-
2
+
(
2
n
+
1
)
2
π
2
)
t
/
2
-
1
)
m
m
m
m
m
m
m
×
A
(
n
)
cos
⁡
(
(
2
n
+
1
)
π
x
2
)
)
m
m
m
m
m
m
×
(
(
2
n
+
1
)
6
π
6
(
-
2
+
(
2
n
+
1
)
2
π
2
)
)
-
1
(
(
2
n
+
1
)
π
x
2
)
)
×
(
0
1
0
0
)
,
where(80)
A
(
n
)
=
(
480
+
(
2
n
+
1
)
π
(
-
144
(
-
1
)
n
mmmmmmmmiim
+
(
2
n
+
1
)
π
(
(
-
1
)
n
(
2
n
+
1
)
π
-
6
)
)
)
.
(5)
The solution of problem (13)–(15) is given by u
(
x
,
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
).Example 4.
We consider problem (13)–(15) where matrix A
∈
C
4
×
4 is given by
(81)
A
=
(
2
0
0
1
1
2
0
-
2
-
1
0
2
1
0
0
0
1
)
,
and the matrices A
i
,
B
i
,
i
∈
{
1,2
} are given by (68). The vectorial valued function f
(
x
) is defined by
(82)
f
(
x
)
=
(
0
0
x
2
-
2
x
0
)
,
and the vectorial valued function G
(
x
,
t
) is given by
(83)
G
(
x
,
t
)
=
(
0
0
x
(
x
-
1
)
2
e
-
t
0
)
.We will follow Algorithm1 step by step.(1)
If we consider the associated problem (34), it is easy to check that conditions (10)–(12) hold. In fact, this problem was solved in Example 4.1 of [16]. Using Algorithm 1 of [16] we can obtain the solution v
(
x
,
t
) of problem (34) with the values ρ
0
=
1
,
a
2
=
0. The solution of problem (34) is given by
(84)
v
(
x
,
t
)
=
∑
n
≥
0
‍
-
32
e
-
(
1
/
2
)
(
π
+
2
n
π
)
2
t
sin
(
(
1
/
2
)
(
1
+
2
k
)
π
x
)
π
3
(
2
k
+
1
)
3
×
(
0
0
1
0
)
,
with the eigenvalues setF
=
{
λ
k
=
π
/
2
+
k
π
:
k
≥
0
} and eigenfunctions
(85)
X
k
(
x
)
=
sin
(
(
π
2
+
k
π
)
x
)
,
k
≥
0
.
After obtaining the solution of the homogeneous problem with homogeneous conditions (34), we continue with Algorithm 3.Algorithm 3:Algorithm to compute the solution of problem (13)–(15) when conditions (10)–(12) holds.
(1)  Check that vector valued functionG
(
x
,
t
) satisfies conditions (53), (58) and (10) replacing f
(
x
) by
G
(
x
,
t
). If these conditions are not satisfied algorithm stop because we can not obtain the solution of
problem (13)–(15) with the given data.
(2) Determine coefficientsT
~
n
(
t
), n
≥
0 defined by (60).
(3) Determine coefficientsB
~
n
(
t
), n
≥
0 defined by (62), where D
~
n, n
≥
0 are defined by (64).
(4) Determinew
(
x
,
t
) defined by (66), solution of problem (36).
(5) The solution of problem (13)–(15) is given by u
(
x
,
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
).We will follow Algorithm3 step by step.(1)
We will check that the vector valued functionG
(
x
,
t
) satisfies conditions (10) replacing f
(
x
) by G
(
x
,
t
).
It is trivial to check that, for fixedt,
(86)
G
(
x
,
t
)
∈
C
2
(
[
0,1
]
)
.
Therefore,(87)
G
(
0
,
t
)
=
(
0
0
0
0
)
,
a
2
G
(
1
,
t
)
+
G
x
(
1
,
t
)
=
(
0
0
0
0
)
.
Then, conditions (10) hold. Furthermore, one gets that
(88)
∫
0
1
‍
∥
G
(
x
,
t
)
∥
2
d
x
=
∫
0
1
‍
(
e
-
2
t
x
2
(
x
2
-
1
)
4
)
d
x
=
128
e
-
2
t
3465
;
thus,sup
⁡
t
>
0
∫
0
1
‍
∥
G
(
x
,
t
)
∥
2
d
x
≤
128
/
3465
=
M
<
∞ and condition (53) holds.
Similarly,(89)
∫
0
1
‍
∥
G
x
x
(
x
,
t
)
∥
2
d
x
=
∫
0
1
‍
(
(
8
e
-
t
x
(
x
2
-
1
)
+
e
-
t
x
(
8
x
2
+
4
(
x
2
-
1
)
)
)
2
)
d
x
=
64
e
-
2
t
7
;
thus,sup
⁡
t
>
0
∫
0
1
‍
∥
G
x
x
(
x
,
t
)
∥
2
d
x
≤
64
/
7
=
N
<
∞ and condition (58) holds.
(2)
Forn
≥
0, coefficients T
~
n
(
t
) defined by (60) are given by
(90)
T
~
n
(
t
)
=
-
1536
(
-
1
)
n
e
-
t
(
-
10
+
(
2
n
+
1
)
2
π
2
)
(
2
n
+
1
)
6
π
6
(
0
0
1
0
)
.
(3)
Forn
≥
0, coefficients B
~
n
(
t
) defined by (62) are given by
(91)
B
~
n
(
t
)
=
-
3072
(
-
1
)
n
(
e
(
1
/
2
)
(
-
2
+
(
2
n
+
1
)
2
π
2
)
t
-
1
)
(
(
2
n
+
1
)
2
π
2
-
10
)
(
2
n
+
1
)
6
π
6
(
-
2
+
(
2
n
+
1
)
2
π
2
)
×
(
0
0
1
0
)
.
(4)
The solutionw
(
x
,
t
) of problem (36), defined by (66), is given by
(92)
w
(
x
,
t
)
=
∑
n
≥
0
‍
-
(
(
(
2
n
+
1
)
6
π
6
(
-
2
+
(
2
n
+
1
)
2
π
2
)
)
-
1
(
(
(
2
n
+
1
)
π
x
2
)
(
(
2
n
+
1
)
π
x
2
)
3072
(
-
1
)
n
e
-
(
2
n
+
1
)
2
π
2
t
/
2
(
e
(
-
2
+
(
2
n
+
1
)
2
π
2
)
t
/
2
-
1
)
m
m
m
m
m
m
m
m
×
(
(
2
n
+
1
)
2
π
2
-
10
)
sin
(
(
2
n
+
1
)
π
x
2
)
)
×
(
(
2
n
+
1
)
6
π
6
(
-
2
+
(
2
n
+
1
)
2
π
2
)
)
-
1
(
(
2
n
+
1
)
π
x
2
)
(
(
(
2
n
+
1
)
π
x
2
)
3072
(
-
1
)
n
e
-
(
2
n
+
1
)
2
π
2
t
/
2
(
e
(
-
2
+
(
2
n
+
1
)
2
π
2
)
t
/
2
-
1
)
)
×
(
0
0
1
0
)
.
(5)
The solution of problem (13)–(15) is given by u
(
x
,
t
)
=
v
(
x
,
t
)
+
w
(
x
,
t
).


## 5. Conclusions

In this paper, the construction of the exact series solution of the nonhomogeneous problem (13)–(16) has been presented. Conditions for the vector valued function G
(
x
,
t
) in order to ensure the existence and convergence of a series solution of the proposed problem have been presented. An algorithm with two illustrative examples was given.



---

*Source: 826860-2014-10-14.xml*