1. Introduction
Initial and boundary value problems of ordinary differential equations play an important role in many fields. Various applications of boundary to physical, biological, chemical, and other branches of applied mathematics are well documented in the literature. The main idea of this paper is to present a new algorithm for computing the solutions of singular second-order initial/boundary value problems (IBVPs) of the form:
{
p
(
x
)
u
″
(
x
)
+
q
(
x
)
u
′
(
x
)
+
r
(
x
)
u
(
x
)
=
F
(
x
,
u
)
,
a
1
u
(
0
)
+
b
1
u
′
(
0
)
+
c
1
u
(
1
)
=
0
,
a
2
u
(
1
)
+
b
2
u
′
(
1
)
+
c
2
u
′
(
0
)
=
0
,
where u
(
x
)
∈
W
2
3
[
0
,
1
]
, for x ∈ [0, 1], p ≠ 0, p(x), q(x), r(x) ∈ C[0, 1]. a1, b1,c1, a2, b2, c2 arc real constants and satisfy that a1 u(0) + b1 u'(0) + c1 u (1) and a2 u(1) + b2u'(1) + c2u'(0) are linear independent. F(x, u) is continuous.
Remark 1.1. We find that if
b
1
=
c
1
=
b
2
=
c
2
=
0
,
a
1
≠
0
,
a
2
≠
0
,
the problems are two-point BVPs; if
b
1
=
c
1
=
a
2
=
b
2
=
0
,
a
1
≠
0
,
c
2
≠
0
,
the problems are initial value problems; if
b
1
=
a
2
=
0
,
a
1
=
c
1
≠
0
,
b
2
=
c
2
≠
0
,
the problems are periodic BVPs; if
b
1
=
a
2
=
0
,
a
1
=
-
c
1
≠
0
,
b
2
=
-
c
2
≠
0
,
the problems are anti-periodic BVPs.
Such problems have been investigated in many researches. Specially, the existence and uniqueness of the solution of (1.1) have been discussed in 12345. And in recent years, there are also a large number of special-purpose methods are proposed to provide accurate numerical solutions of the special form of (1.1), such as collocation methods 6, finite-element methods 7, Galerkin-wavelet methods 8, variational iteration method 9, spectral methods 10, finite difference methods 11, etc.
On the other hands, reproducing kernel theory has important applications in numerical analysis, differential equation, probability and statistics, machine learning and precessing image. Recently, using the reproducing kernel method, Cui and Geng 1213141516 have make much effort to solve some special boundary value problems.
According to our method, which is presented in this paper, some reproducing kernel Hilbert spaces have been presented in the first step. And in the second step, the homogeneous IBVPs is deal with in the RKHS. Finally, one analytic approximation of the solutions of the second-order BVPs is given by reproducing kernel method under the assumption that the solution to (1.1) is unique.
2. Some RKHS
In this section, we will introduce the RKHS
W
2
1
[
0
,
1
]
and
W
2
3
[
0
,
1
]
. Then we will construct a RKHS
H
2
3
[
0
,
1
]
, in which every function satisfies the boundary condition of (1.1).
2.1. The RKHS W21[0,1]
Inner space W21[0,1]
is defined as
W
2
1
[
0
,
1
]
=
{
u
(
x
)
|
u
is absolutely continuous real valued functions, u' ∈ L2[0, 1]}. The inner product in W21[0,1]
is given by
(
f
,
h
)
W
2
1
=
f
(
0
)
h
(
0
)
+
∫
0
1
f
′
(
t
)
h
′
(
t
)
d
t
,
f
,
h
∈
W
2
1
[
0
,
1
]
and the norm |
|
u
|
|
W
2
1
is denoted by |
|
u
|
|
W
2
1
=
(
u
,
u
)
W
2
1
. From 1718, W21[0,1]
is a reproducing kernel Hilbert space and the reproducing kernel is
K
1
(
t
,
s
)
=
1
+
min
{
t
,
s
}
2.2. The RKHS W23[0,1]
Inner space W23[0,1]
is defined as
W
2
3
[
0
,
1
]
=
{
u
(
x
)
|
u
,
u
′
,
u
″
is absolutely continuous real valued functions, u"' ∈ L2[0, 1]}.
From 15171819, it is clear that W23[0,1]
become a reproducing kernel Hilbert space if we endow it with suitable inner product.
Zhang and Lu 18 and Long and Zhang 19 give us a clue to relate the inner product with the boundary conditions (1.1). Set L = D3, and
γ
1
f
=
a
1
f
(
0
)
+
b
1
f
′
(
0
)
+
c
1
f
(
1
)
,
γ
2
f
=
a
2
f
(
1
)
+
b
2
f
′
(
1
)
+
c
2
f
′
(
0
)
,
γ
3
f
=
a
3
f
(
0
)
+
b
3
f
′
(
0
)
+
c
3
f
″
(
0
)
,
where a3, b3, c3 is random but satisfying that γ3 is linearly independent of γ1 and γ2.
It is easy to know that γ1, γ2, γ3 are linearly independent in Ker L. Then from 1819, it is easy to know one of the inner products of W23[0,1]
(
f
,
h
)
W
2
3
=
∑
i
=
1
3
γ
i
f
γ
i
h
+
∫
0
1
f
‴
(
t
)
h
‴
(
t
)
d
t
,
f
,
h
∈
W
2
3
[
0
,
1
]
and its corresponding reproducing kernel K2(t, s).
2.3. The RKHS H23[0,1]
Inner space H23[0,1]
is defined as
H
2
3
[
0
,
1
]
=
{
u
(
x
)
|
u
,
u
′
,
u
″
are absolutely continuous real valued functions, u"' ∈ L2[0, 1], and, a1 u(0) + b1 u'(0) + c1 u(1) = 0, a2 u(1) + b2u'(1) + c2u'(0) = 0}.
It is clear that H23[0,1]
is the complete subspace of W23[0,1]
, so H23[0,1]
is a RKHS. If P, which is the orthogonal projection from W23[0,1]
to H23[0,1]
, is found, we can get the reproducing kernel of H23[0,1]
obviously. Under the assumptions of Section 2, note
P
f
(
t
)
=
(
γ
3
f
)
e
3
(
t
)
+
∫
0
1
G
(
t
,
τ
)
⋅
f
‴
(
τ
)
d
τ
,
∀
f
∈
W
2
3
[
0
,
1
]
Theorem 2.1. Under the assumptions above, P is the orthogonal projection from
W
2
3
[
a
,
b
]
to H23[0,1]
.
Proof. For all f
∈
W
2
3
[
0
,
1
]
, We have
(
γ
1
(
P
f
)
)
(
t
)
=
(
γ
2
(
P
f
)
)
(
t
)
=
0
That means P
f
∈
H
2
3
[
0
,
1
]
. At the same time, for any f
,
h
∈
W
2
3
[
0
,
1
]
(
P
f
,
h
)
=
(
γ
3
f
)
e
3
(
t
)
+
∫
0
1
G
(
t
,
τ
)
⋅
L
f
(
τ
)
d
τ
,
h
=
(
γ
3
f
)
(
γ
3
h
)
+
∫
0
1
L
∫
0
1
G
(
t
,
τ
)
⋅
L
f
(
τ
)
d
τ
⋅
L
h
(
t
)
d
t
=
(
γ
3
f
)
(
γ
3
h
)
+
∫
0
1
L
f
(
t
)
⋅
L
h
(
t
)
d
t
(
f
,
P
h
)
=
f
,
(
γ
3
h
)
e
3
(
t
)
+
∫
0
1
G
(
t
,
τ
)
⋅
L
h
(
τ
)
d
τ
=
(
γ
3
f
)
(
γ
3
h
)
+
∫
0
1
L
f
(
t
)
⋅
L
∫
0
1
G
(
t
,
τ
)
⋅
L
h
(
τ
)
d
τ
d
t
=
(
γ
3
f
)
(
γ
3
h
)
+
∫
0
1
L
f
(
t
)
⋅
L
h
(
t
)
d
t
P is self-conjugate. And
P
(
P
f
)
=
P
(
γ
3
f
)
e
3
(
t
)
+
∫
0
1
G
(
t
,
τ
)
⋅
L
f
(
τ
)
d
τ
=
(
γ
3
f
)
e
3
(
t
)
+
∫
0
1
G
(
t
,
τ
)
⋅
L
(
γ
3
f
)
e
3
(
τ
)
+
∫
0
1
G
(
τ
,
s
)
⋅
L
f
(
s
)
d
s
,
d
τ
=
(
γ
3
f
)
e
3
(
t
)
+
∫
0
1
G
(
t
,
τ
)
⋅
L
f
(
τ
)
d
τ
=
P
f
P is idempotent.
So P is the orthogonal projection from W23[0,1]
to H23[0,1]
.
The proof of the Theorem 2.1 is complete.
Now, H23[0,1]
is a RKHS if endowed the inner product with the inner product below
(
f
,
h
)
H
2
3
=
γ
3
f
γ
3
h
+
∫
0
1
f
‴
(
t
)
⋅
h
‴
(
t
)
d
t
and the corresponding reproducing kernel K3(t, s) is given in Appendix 4.
3. The reproducing kernel method
In this section, the representation of analytical solution of (1.1) is given in the reproducing kernel space H23[0,1]
.
Note Lu = p(x)u"(x) + q(x)u'(x) + r(x)u(x) in (1.1). It is clear that L
:
H
2
3
[
0
,
1
]
→
W
2
1
[
0
,
1
]
is a bounded linear operator.
Put φi(x) = K1(xi, x), Ψi(x) = L*φi(x), where L* is the adjoint operator of L. Then
Ψ
i
(
x
)
=
(
L
*
φ
i
(
y
)
,
K
3
(
x
,
y
)
)
=
(
φ
i
(
y
)
,
L
y
K
3
(
x
,
y
)
)
=
(
L
y
K
3
(
x
,
y
)
,
φ
i
(
x
)
)
¯
=
L
y
K
3
(
x
,
y
)
|
y
=
x
i
Lemma 3.1. Under the assumptions above, if
{
x
i
}
i
=
1
∞
is dense on [0, 1] then
{
Ψ
i
(
x
)
}
i
=
1
∞
is the complete basis H23[0,1]
.
The orthogonal system
{
Ψ
i
¯
(
x
)
}
i
=
1
∞
of H23[0,1]
can be derived from Gram-Schmidt orthogonalization process of {Ψi(x)}i=1∞
, and
Ψ
i
¯
(
x
)
=
∑
j
=
1
i
β
i
j
Ψ
j
(
x
)
.
Then
Theorem 3.1. If {xi}i=1∞
is dense on [0, 1] and the solution of (1.1) is unique, the solution can be expressed in the form
u
(
x
)
=
∑
i
=
1
∞
∑
k
=
1
i
β
i
k
F
(
x
k
,
u
(
x
k
)
)
Ψ
i
(
x
)
Proof. From Lemma 3.1, {Ψi(x)}i=1∞
is the complete system of H23[0,1]
. Hence we have
u
(
x
)
=
∑
i
=
1
∞
(
u
(
x
)
,
Ψ
¯
i
(
x
)
)
Ψ
¯
i
(
x
)
=
∑
i
=
1
∞
∑
k
=
1
i
β
i
j
(
u
(
x
)
,
Ψ
i
(
x
)
)
Ψ
¯
i
(
x
)
=
∑
i
=
1
∞
∑
k
=
1
i
β
i
k
(
u
(
x
)
,
L
*
φ
k
(
x
)
)
Ψ
¯
i
(
x
)
=
∑
i
=
1
∞
∑
k
=
1
i
β
i
k
(
L
u
(
x
)
,
φ
k
(
x
)
)
Ψ
¯
i
(
x
)
=
∑
i
=
1
∞
∑
k
=
1
i
β
i
k
(
F
(
x
,
u
(
x
)
)
,
φ
k
(
x
)
)
Ψ
¯
i
(
x
)
=
∑
i
=
1
∞
∑
k
=
1
i
β
i
k
F
(
x
k
,
u
(
x
k
)
)
Ψ
¯
i
(
x
)
and the proof is complete.
The approximate solution of the (1.1) is
u
n
(
x
)
=
∑
i
=
1
n
∑
k
=
1
i
β
i
k
F
(
x
k
,
u
(
x
k
)
)
Ψ
¯
i
(
x
)
If (1.1) is linear, that is F(x, u(x)) = F(x), then the approximate solution of (1.1) can be obtained directly from (3.3). Else, the approximate process could be modified into the following form:
{
u
0
(
x
)
=
0
u
n
+
1
(
x
)
=
∑
i
=
1
n
+
1
B
i
Ψ
¯
i
(
x
)
where
B
i
=
∑
k
=
1
i
β
i
k
F
(
x
k
,
u
n
(
x
k
)
)
.
Next, the convergence of un(x) will be proved.
Lemma 3.2. There exists a constant M, satisfied |
u
(
x
)
|
≤
M
|
|
u
|
|
H
2
3
, for all u
(
x
)
∈
H
2
3
[
0
,
1
]
.
Proof. For all x ∈ [0, 1] and u
∈
H
2
3
[
0
,
1
]
, there are
|
u
(
x
)
|
=
|
(
u
(
⋅
)
,
K
3
(
⋅
,
x
)
)
|
≤
|
|
K
3
(
⋅
,
x
)
|
|
H
2
3
⋅
|
|
u
|
|
H
2
3
.
Since
K
3
(
⋅
,
x
)
∈
H
2
3
[
0
,
1
]
, note
M
=
max
x
∈
[
0
,
1
]
|
|
K
3
(
⋅
,
x
)
|
|
H
2
3
.
That is, |u(x)|≤M||u||H23
.
By Lemma 3.2, it is easy to obtain the following lemma.
Lemma 3.3. If
u
n
→
|
|
⋅
|
|
ū
(
n
→
∞
)
, ||un|| is bounded, xn → y(n → ∞) and F(x, u(x)) is continuous, then F
(
x
n
,
u
n
-
1
(
x
n
)
)
→
F
(
y
,
ū
(
y
)
)
.
Theorem 3.2. Suppose that ||un || is bounded in (3.3) and (1.1) has a unique solution. If {xi}i=1∞
is dense on [0, 1], then the n-term approximate solution un(x) derived from the above method converges to the analytical solution u(x) of (1.1).
Proof. First, we will prove the convergence of un (x).
From (3.4), we infer that
u
n
+
1
(
x
)
=
u
n
(
x
)
+
B
n
+
1
Ψ
¯
n
+
1
(
x
)
.
The orthonormality of
{
Ψ
i
¯
}
i
=
1
∞
yield that
|
|
u
n
+
1
|
|
2
=
|
|
u
n
|
|
2
+
(
B
n
+
1
)
2
=
⋯
=
∑
i
=
1
n
+
1
(
B
i
)
2
.
That means ||un+1|| ≥ ||un||. Due to the condition that ||un|| is bounded, ||un|| is convergent and there exists a constant ℓ such that
∑
i
=
1
∞
(
B
i
)
2
=
ℓ
.
If m > n, then
|
|
u
m
-
u
n
|
|
2
=
|
|
u
m
-
u
m
-
1
+
u
m
-
1
-
u
m
-
2
+
⋯
+
u
n
+
1
-
u
n
|
|
2
.
In view of (um - um-1) ⊥ (um-1 - um-2) ⊥ ··· ⊥ (un+1 - un), it follows that
|
|
u
m
-
u
n
|
|
2
=
|
|
u
m
-
u
m
-
1
|
|
2
+
|
|
u
m
-
1
-
u
m
-
2
|
|
2
+
⋯
+
|
|
u
n
+
1
-
u
n
|
|
2
=
∑
i
=
n
+
1
m
(
B
i
)
2
→
0
as
n
→
∞
The completeness of H23[0,1]
shows that un → ū as n → ∞ in the sense of |
|
⋅
|
|
H
2
3
.
Secondly, we will prove that ū is the solution of (1.1).
Taking limits in (3.2), we get
u
¯
(
x
)
=
∑
i
=
1
∞
B
i
Ψ
¯
i
(
x
)
.
So
L
ū
(
x
)
=
∑
i
=
1
∞
B
i
L
Ψ
¯
i
(
x
)
and
(
L
ū
)
(
x
)
=
∑
i
=
1
∞
B
i
(
L
Ψ
¯
i
,
φ
n
)
=
∑
i
=
1
∞
B
i
(
Ψ
¯
i
,
L
*
φ
n
)
=
∑
i
=
1
∞
B
i
(
Ψ
¯
i
,
Ψ
n
)
.
Therefore,
∑
i
=
1
n
β
n
j
(
L
υ
̄
)
(
x
n
)
=
∑
i
=
1
∞
B
i
Ψ
¯
i
,
∑
i
=
1
n
β
n
j
Ψ
j
=
∑
i
=
1
∞
B
i
(
Ψ
¯
i
,
Ψ
¯
n
)
=
B
n
.
If n = 1, then
L
ū
(
x
1
)
=
F
(
x
1
,
u
0
(
x
1
)
)
.
If n = 2, then
β
21
L
ū
(
x
1
)
+
β
22
L
ū
(
x
2
)
=
β
21
F
(
x
1
,
u
0
(
x
1
)
)
+
β
22
F
(
x
2
,
u
1
(
x
2
)
)
.
It is clear that
(
L
ū
)
(
x
2
)
=
F
(
x
2
,
u
1
(
x
2
)
)
.
Moreover, it is easy to see by induction that
(
L
ū
)
(
x
j
)
=
F
(
x
j
,
u
j
-
1
(
x
j
)
)
,
j
=
1
,
2
,
…
Since {xi}i=1∞
is dense on [0, 1], for all Y ∈ [0, 1], there exists a subsequence
{
x
n
j
}
j
=
1
∞
such that
x
n
j
→
Y
as
j
→
∞
.
It is easy to see that
(
L
ū
)
(
x
n
j
)
=
F
(
x
n
j
,
u
n
j
-
1
(
x
n
j
)
)
. Let j → ∞, by the continuity of F(x, u(x)) and Lemma 3.3, we have
(
L
ū
)
(
Y
)
=
F
(
Y
,
ū
(
Y
)
)
.
At the same time, ū
∈
H
2
3
[
0
,
1
]
. Clearly, u satisfies the boundary conditions of (1.1).
That is, ū is the solution of (1.1).
The proof is complete.
In fact, un(x) is just the orthogonal projection of exact solution ū(x) onto the space
Span
{
Ψ
̄
i
}
i
=
1
n
¯
.