2. Reproducing kernel spaces

In this section, we define some useful reproducing kernel functions [1–23].

Definition 2.1 (reproducing kernel). Let E be a nonempty set. A function K : E � E ! ℂ is called a reproducing kernel of the Hilbert space H if and only if


The last condition is called the reproducing property as the value of the function φ at the point t is reproduced by the inner product of φ with Kð Þ �; t :

Then, we need some notation that we use in the development of this chapter. Next, we define several spaces with inner product over those spaces. Thus, the space defined as

$$W\_2^3[0,1] = \left\{ v[\boldsymbol{\upsilon}, \boldsymbol{\upsilon}', \boldsymbol{\upsilon}'] : [0,1] \to \mathbb{R} \text{ are absolutely continuous}, \boldsymbol{\upsilon}^{(3)} \in L^2[0,1] \right\} \tag{1}$$

is a Hilbert space. The inner product and the norm in W<sup>3</sup> <sup>2</sup>½ � 0; 1 are defined by

$$\begin{aligned} \langle v, g \rangle\_{\mathcal{W}\_2^3} &= \sum\_{i=0}^2 v^{(i)}(0) g^{(i)}(0) + \int\_0^1 v^{(3)}(\mathbf{x}) g^{(3)}(\mathbf{x}) d\mathbf{x}, \quad v, g \in \mathcal{W}\_2^3[0, 1], \\ \|v\|\_{\mathcal{W}\_2^3} &= \sqrt{\langle v, v \rangle\_{\mathcal{W}\_2^3}}, \quad v \in \mathcal{W}\_2^3[0, 1]. \end{aligned} \tag{2}$$

respectively. Thus, the space W<sup>3</sup> <sup>2</sup>½ � 0; 1 is a reproducing kernel space, that is, for each fixed <sup>y</sup><sup>∈</sup> ½ � <sup>0</sup>; <sup>1</sup> and any <sup>v</sup><sup>∈</sup> <sup>W</sup><sup>3</sup> <sup>2</sup>½ � 0; 1 , there exists a function Ry such that

$$
\sigma(y) = \left< v(\mathbf{x}), R\_{\mathcal{Y}}(\mathbf{x}) \right>\_{\mathcal{W}^{3\_{\mathcal{Y}}}\_{2}} \tag{3}
$$

h i v; ɡ <sup>G</sup><sup>1</sup>

<sup>c</sup>1ð Þ¼ <sup>y</sup> <sup>1</sup>, c2ð Þ¼ <sup>y</sup> y, c3ð Þ¼ <sup>y</sup> <sup>y</sup><sup>2</sup>

<sup>120</sup> , d2ð Þ¼ <sup>y</sup> �y<sup>4</sup>

i¼0

v xð Þ;Ryð Þ<sup>x</sup> � �

Note, the property of the reproducing kernel as

through iterative integrations by parts for (11), we have

þ X 2

i¼0

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>R</sup>ð Þ<sup>i</sup>

W<sup>4</sup> 2 <sup>¼</sup> <sup>X</sup> 2

<sup>y</sup> ð Þþ 0

i¼0

v xð Þ;Ryð Þ<sup>x</sup> � �

ð Þ �<sup>1</sup> ð Þ <sup>2</sup>�<sup>i</sup>

respectively. The space G<sup>1</sup>

Theorem 1.1. The space W<sup>3</sup>

<sup>d</sup>1ð Þ¼ <sup>y</sup> <sup>1</sup> <sup>þ</sup> <sup>y</sup><sup>5</sup>

v;Ry � � W<sup>3</sup> 2 <sup>¼</sup> <sup>X</sup> 2

is given by

where

Proof. Since

If

function Qy is given by [1] as

<sup>2</sup> <sup>¼</sup> <sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>ɡ</sup>ð Þ<sup>i</sup> ð Þþ <sup>0</sup>

∥v∥G<sup>1</sup> <sup>2</sup> ¼

Ryð Þ¼ x

ð1 0 v0 ð Þx ɡ<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>G</sup><sup>1</sup> 2

Qy <sup>¼</sup> <sup>1</sup> <sup>þ</sup> x, x <sup>⩽</sup><sup>y</sup>

�

X 6

8 >>>>><

>>>>>:

i¼1

X 6

i¼1

<sup>4</sup> , c4ð Þ¼ <sup>y</sup> <sup>y</sup><sup>2</sup>

<sup>24</sup> <sup>þ</sup> y, d3ð Þ¼ <sup>y</sup> <sup>y</sup><sup>2</sup>

ð1 0

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>R</sup>ð Þ<sup>i</sup>

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>1</sup> <sup>R</sup>ð Þ <sup>5</sup>�<sup>i</sup>

W<sup>3</sup> 2

<sup>y</sup> ð Þþ 1

<sup>v</sup>ð Þ<sup>3</sup> ð Þ<sup>x</sup> <sup>R</sup>ð Þ<sup>3</sup>

1 þ y, x > y:

cið Þ<sup>y</sup> xi�<sup>1</sup>

dið Þ<sup>y</sup> xi�<sup>1</sup>

, v ∈ G<sup>1</sup>

q

ð Þ<sup>x</sup> <sup>d</sup>x, v, <sup>ɡ</sup><sup>∈</sup> <sup>G</sup><sup>1</sup>

<sup>2</sup>½ � 0; 1 ,

<sup>2</sup>½ � 0; 1 is a reproducing kernel space, and its reproducing kernel

<sup>2</sup>½ � 0; 1 is a complete reproducing kernel space whose reproducing kernel Ry

, x ≤ y,

, x > y,

<sup>12</sup> , c5ð Þ¼� <sup>y</sup>

<sup>4</sup> <sup>þ</sup> <sup>y</sup><sup>3</sup>

1 24y

<sup>y</sup> ð Þ<sup>x</sup> <sup>d</sup>x, v, Ry <sup>∈</sup> <sup>W</sup><sup>3</sup>

v xð ÞRð Þ<sup>6</sup>

<sup>y</sup> ð Þ� � <sup>0</sup> ð Þ<sup>1</sup> ð Þ <sup>2</sup>�<sup>i</sup> <sup>R</sup>ð Þ <sup>5</sup>�<sup>i</sup>

ð1 0

h i

<sup>12</sup> , d4ð Þ¼ <sup>y</sup> <sup>d</sup>5ð Þ¼ <sup>y</sup> <sup>d</sup>6ð Þ¼ <sup>y</sup> <sup>0</sup>:

<sup>y</sup> ð Þ0

<sup>y</sup> ð Þx dx:

¼ v yð Þ: (13)

, c6ð Þ¼ y

<sup>2</sup>½ � <sup>0</sup>; <sup>1</sup> � (11)

1 <sup>120</sup> ,

<sup>2</sup>½ � 0; 1 ,

http://dx.doi.org/10.5772/intechopen.75206

Reproducing Kernel Functions

(8)

101

(9)

(10)

(12)

and similarly, we define the space

$$T\_2^3[0,1] = \left\{ \begin{array}{c} \upsilon[\upsilon, \upsilon', \upsilon' \ \vdots \ [0,1] \rightarrow \mathbb{R} \text{ are absolutely continuous}, \\\\ \upsilon' \in L^2[0,1], \upsilon(0) = 0, \upsilon'(0) = 0 \end{array} \right\} \tag{4}$$

The inner product and the norm in T<sup>3</sup> <sup>2</sup>½ � 0; 1 are defined by

$$\begin{aligned} \langle v, g \rangle\_{T\_2^3} &= \sum\_{i=0}^2 v^{(i)}(0) g^{(i)}(0) + \int\_0^1 v'''(t) g'''(t) dt, \quad v, g \in T\_2^3[0, 1], \\ \|v\|\_{T\_2^3} &= \sqrt{\langle v, v \rangle\_{T\_2^3}}, \quad v \in T\_2^3[0, 1]. \end{aligned} \tag{5}$$

respectively. The space T<sup>3</sup> <sup>2</sup>½ � 0; 1 is a reproducing kernel Hilbert space, and its reproducing kernel function rs is given by [1] as

$$r\_s = \begin{cases} \frac{1}{4}s^2t^2 + \frac{1}{12}s^2t^3 - \frac{1}{24}st^4 + \frac{1}{120}t^5, & t \le s, \\\\ \frac{1}{4}s^2t^2 + \frac{1}{12}s^3t^2 - \frac{1}{24}ts^4 + \frac{1}{120}s^5, & t > s, \end{cases} \tag{6}$$

and the space

$$G\_2^1[0,1] = \{ \boldsymbol{\upsilon} | \boldsymbol{\upsilon} : [0,1] \to \mathbb{R} \text{ is absolutely continuous}, \boldsymbol{\upsilon}'(\mathbf{x}) \in L^2[0,1] \},\tag{7}$$

is a Hilbert space, where the inner product and the norm in G<sup>1</sup> <sup>2</sup>½ � 0; 1 are defined by

#### Reproducing Kernel Functions http://dx.doi.org/10.5772/intechopen.75206 101

$$\begin{aligned} \langle v, g \rangle\_{\mathcal{G}\_2^1} &= v^{(i)}(0)g^{(i)}(0) + \int\_0^1 v'(x)g'(x) dx, \quad v, g \in \mathcal{G}\_2^1[0, 1], \\ \|v\|\_{\mathcal{G}\_2^1} &= \sqrt{\langle v, v \rangle\_{\mathcal{G}\_2^1}}, \quad v \in \mathcal{G}\_2^1[0, 1]. \end{aligned} \tag{8}$$

respectively. The space G<sup>1</sup> <sup>2</sup>½ � 0; 1 is a reproducing kernel space, and its reproducing kernel function Qy is given by [1] as

$$Q\_y = \begin{cases} 1+x, & x \leqslant y \\ 1+y, & x > y. \end{cases} \tag{9}$$

Theorem 1.1. The space W<sup>3</sup> <sup>2</sup>½ � 0; 1 is a complete reproducing kernel space whose reproducing kernel Ry is given by

$$R\_y(\mathbf{x}) = \begin{cases} \sum\_{i=1}^6 c\_i(y) \mathbf{x}^{i-1}, & \mathbf{x} \le y\_\prime \\\\ \sum\_{i=1}^6 d\_i(y) \mathbf{x}^{i-1}, & \mathbf{x} > y\_\prime \end{cases} \tag{10}$$

where

Then, we need some notation that we use in the development of this chapter. Next, we define

ð1 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>W</sup><sup>3</sup> 2

<sup>2</sup>½ � 0; 1 , there exists a function Ry such that

v yð Þ¼ v xð Þ;Ryð Þ<sup>x</sup> � �

q

: ½ �! <sup>0</sup>; <sup>1</sup> <sup>R</sup> are absolutely continuous; <sup>v</sup>ð Þ<sup>3</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup>

, v∈ W<sup>3</sup>

W<sup>3</sup> 2

, v<sup>00</sup> : ½ �! 0; 1 R are absolutely continuous,

½ � 0; 1 , vð Þ¼ 0 0, v<sup>0</sup>

, v ∈T<sup>3</sup>

<sup>24</sup> st<sup>4</sup> <sup>þ</sup>

<sup>24</sup> ts<sup>4</sup> <sup>þ</sup>

<sup>2</sup>½ � 0; 1 are defined by

ð1 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>T</sup><sup>3</sup> 2

n o

½ � 0; 1

<sup>2</sup>½ � 0; 1 ,

, (3)

9 >=

>;

<sup>2</sup>½ � 0; 1 ,

<sup>2</sup>½ � 0; 1 are defined by

<sup>v</sup>ð Þ<sup>3</sup> ð Þ<sup>x</sup> <sup>ɡ</sup>ð Þ<sup>3</sup> ð Þ<sup>x</sup> <sup>d</sup>x, v, <sup>ɡ</sup> <sup>∈</sup> <sup>W</sup><sup>3</sup>

<sup>2</sup>½ � 0; 1 ,

<sup>2</sup>½ � 0; 1 is a reproducing kernel space, that is, for each fixed

ð Þ¼ 0 0

<sup>v</sup>000ð Þ<sup>t</sup> <sup>ɡ</sup>000ð Þ<sup>t</sup> <sup>d</sup>t, v, <sup>ɡ</sup><sup>∈</sup> <sup>T</sup><sup>3</sup>

<sup>2</sup>½ � 0; 1 ,

<sup>2</sup>½ � 0; 1 is a reproducing kernel Hilbert space, and its reproducing

1 <sup>120</sup> <sup>t</sup> 5 , t ≤ s,

1 <sup>120</sup> <sup>s</sup> 5

½ � <sup>0</sup>; <sup>1</sup> � �, (7)

, t > s,

ð Þ<sup>x</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup>

<sup>2</sup>½ � 0; 1 are defined by

(1)

(2)

(4)

(5)

(6)

several spaces with inner product over those spaces. Thus, the space defined as

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>ɡ</sup>ð Þ<sup>i</sup> ð Þþ <sup>0</sup>

∥v∥W<sup>3</sup> 2 ¼

v∣v, v<sup>0</sup>

8 ><

>:

i¼0

rs ¼

v00 ∈L<sup>2</sup>

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>ɡ</sup>ð Þ<sup>i</sup> ð Þþ <sup>0</sup>

q

<sup>2</sup>½ �¼ 0; 1 vjv : ½ �! 0; 1 R is absolutely continuous; v<sup>0</sup>

∥v∥T<sup>3</sup> 2 ¼

1 4 s 2 t 2 þ 1 12 s 2 t <sup>3</sup> � <sup>1</sup>

8 >>><

>>>:

1 4 s 2 t 2 þ 1 12 s 3t <sup>2</sup> � <sup>1</sup>

is a Hilbert space, where the inner product and the norm in G<sup>1</sup>

W<sup>3</sup>

<sup>2</sup>½ �¼ 0; 1 vjv; v<sup>0</sup>

100 Differential Equations - Theory and Current Research

h i v; ɡ <sup>W</sup><sup>3</sup> 2 <sup>¼</sup> <sup>X</sup> 2

respectively. Thus, the space W<sup>3</sup>

and similarly, we define the space

T3 <sup>2</sup>½ �¼ 0; 1

The inner product and the norm in T<sup>3</sup>

h i v; ɡ <sup>T</sup><sup>3</sup> 2 <sup>¼</sup> <sup>X</sup> 2

respectively. The space T<sup>3</sup>

G1

and the space

kernel function rs is given by [1] as

<sup>y</sup><sup>∈</sup> ½ � <sup>0</sup>; <sup>1</sup> and any <sup>v</sup><sup>∈</sup> <sup>W</sup><sup>3</sup>

; v00

is a Hilbert space. The inner product and the norm in W<sup>3</sup>

i¼0

$$\begin{aligned} c\_1(y) &= 1, \quad c\_2(y) = y, \quad c\_3(y) = \frac{y^2}{4}, \quad c\_4(y) = \frac{y^2}{12}, \quad c\_5(y) = -\frac{1}{24y}, \quad c\_6(y) = \frac{1}{120}, \\\ d\_1(y) &= 1 + \frac{y^5}{120}, \quad d\_2(y) = \frac{-y^4}{24} + y, \quad d\_3(y) = \frac{y^2}{4} + \frac{y^3}{12}, \quad d\_4(y) = d\_5(y) = d\_6(y) = 0. \end{aligned}$$

Proof. Since

$$\left\langle v, R\_{\mathcal{Y}} \right\rangle\_{\mathcal{W}^3\_2} = \sum\_{i=0}^2 v^{(i)}(0) R\_{\mathcal{Y}}^{(i)}(0) + \int\_0^1 v^{(3)}(\mathbf{x}) R\_{\mathcal{Y}}^{(3)}(\mathbf{x}) d\mathbf{x}, \quad \left\langle v, R\_{\mathcal{Y}} \in \mathcal{W}^3\_2[0, 1] \right\rangle \tag{11}$$

through iterative integrations by parts for (11), we have

$$\begin{split} \left\langle \boldsymbol{v}(\mathbf{x}), \boldsymbol{R}\_{\boldsymbol{y}}(\mathbf{x}) \right\rangle\_{\mathcal{W}^{4}\_{2}} &= \sum\_{i=0}^{2} \boldsymbol{v}^{(i)}(\mathbf{0}) \left[ \boldsymbol{R}\_{\boldsymbol{y}}^{(i)}(\mathbf{0}) - (-1)^{(2-i)} \boldsymbol{R}\_{\boldsymbol{y}}^{(5-i)}(\mathbf{0}) \right] \\ &+ \sum\_{i=0}^{2} (-1)^{(2-i)} \boldsymbol{v}^{(i)}(\mathbf{1}) \boldsymbol{R}\_{\boldsymbol{y}}^{(5-i)}(\mathbf{1}) + \int\_{0}^{1} \boldsymbol{v}(\mathbf{x}) \boldsymbol{R}\_{\boldsymbol{y}}^{(6)}(\mathbf{x}) d\mathbf{x}. \end{split} \tag{12}$$

Note, the property of the reproducing kernel as

$$\left<\boldsymbol{\upsilon}(\mathbf{x}), R\_{\boldsymbol{\mathcal{Y}}}(\mathbf{x})\right>\_{\mathcal{W}\_2^3} = \boldsymbol{\upsilon}(\boldsymbol{y}).\tag{13}$$

If

$$\begin{aligned} R\_y(0) - R\_y^{(5)}(0) &= 0, \\ R\_y'(0) + R\_y^{(4)}(0) &= 0, \\ R\_y''(0) - R\_y^{"\prime}(0) &= 0, \\ R\_y^{(3)}(1) &= 0, \\ R\_y^{(4)}(1) &= 0, \\ R\_y^{(5)}(1) &= 0, \end{aligned} \tag{14}$$

Now, we note that the space given in [1] as

8 >>>>><

>>>>>:

v xð Þ ; <sup>t</sup> <sup>∣</sup> <sup>∂</sup><sup>4</sup>

h i v xð Þ ; <sup>t</sup> ; <sup>ɡ</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>W</sup> <sup>¼</sup> <sup>X</sup>

v ∂x<sup>2</sup>∂t

> ∂6 v ∂x<sup>3</sup>∂t

<sup>3</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup>

2

ð1 0

> ∂j ∂t

> > ∂3 ∂x<sup>3</sup> ∂3 ∂t <sup>3</sup> v xð Þ ; t

Theorem 1.2. The Wð Þ Ω is a reproducing kernel space, and its reproducing kernel function is

v yð Þ¼ ;<sup>s</sup> v xð Þ ; <sup>t</sup> ;Kð Þ <sup>y</sup>;<sup>s</sup> ð Þ <sup>x</sup>; <sup>t</sup> � �

� � (25)

i¼0

þ<sup>X</sup> 2

þ ð1 0 ð1 0

∥v∥<sup>w</sup> ¼

cð Þ¼ Ω v xð Þj ; t v xð Þ ; t is completely continuous in Ω ¼ ½ �� 0; 1 ½ � 0; 1 ;

is a binary reproducing kernel Hilbert space. The inner product and the norm in W

j¼0

<sup>2</sup> , is completely continuous in Ω ¼ ½ �� 0; 1 ½ � 0; 1 ,

∂3 ∂t 3 ∂i <sup>∂</sup>xi <sup>ɡ</sup>ð0; <sup>t</sup><sup>Þ</sup>

" #

∂3 ∂x<sup>3</sup> ∂3 ∂t <sup>3</sup> ɡðx; tÞ

, v∈ Wð Þ Ω ,

W0

� �dxdt,

∂v xð Þ ; 0 <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>0</sup>

> W<sup>3</sup> 2

Kð Þ <sup>y</sup>;<sup>s</sup> ¼ Ryrs (23)

<sup>K</sup>ð Þ <sup>y</sup>;<sup>s</sup> ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>K</sup>ð Þ <sup>x</sup>;<sup>t</sup> ð Þ <sup>y</sup>;<sup>s</sup> : (24)

∂2 v ∂x∂t

∈L<sup>2</sup> ð Þ Ω

cð Þ Ω are defined by

dt

9 >>>>>=

Reproducing Kernel Functions

http://dx.doi.org/10.5772/intechopen.75206

>>>>>;

(21)

103

(22)

ð Þ Ω ,v xð Þ¼ ; 0 0,

is a binary reproducing kernel Hilbert space. The inner product and the norm in Wð Þ Ω are

∂3 ∂t 3 ∂i <sup>∂</sup>xi <sup>v</sup>ð Þ <sup>0</sup>; <sup>t</sup>

<sup>j</sup> v xð Þ ; 0 ;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>W</sup> q

∂j ∂t <sup>j</sup> ɡðx; 0Þ

\* +

Wð Þ¼ Ω

defined by

respectively.

such that for any v ∈ Wð Þ Ω ,

Similarly, the space

W

[1] as

Then by (11), we obtain

$$R\_y^{(6)}(\mathbf{x}) = \delta(\mathbf{x} - \mathbf{y})\_\prime \tag{15}$$

when x 6¼ y,

$$R\_y^{(6)}(\mathbf{x}) = \mathbf{0},\tag{16}$$

therefore,

$$R\_y(\mathbf{x}) = \begin{cases} \sum\_{i=1}^6 c\_i(y) \mathbf{x}^{i-1}, & \mathbf{x} \le y\_\prime \\\\ \sum\_{i=1}^6 d\_i(y) \mathbf{x}^{i-1}, & \mathbf{x} > y\_\prime \end{cases} \tag{17}$$

Since

$$R\_y^{(6)}(\mathbf{x}) = \delta(\mathbf{x} - \mathbf{y})\_\prime \tag{18}$$

we have

$$\begin{aligned} \mathfrak{d}^k R\_{y^+} (y) &= \mathfrak{d}^k R\_{y^-} (y), \quad k = 0, 1, 2, 3, 4, \\ \mathfrak{d}^5 R\_{y^+} (y) - \mathfrak{d}^5 R\_{y^-} (y) &= -1. \end{aligned} \tag{19}$$

From (14) and (19), the unknown coefficients cið Þy and dið Þy ð Þ i ¼ 1; 2;…; 6 can be obtained. Thus, Ry is given by

$$R\_y = \begin{cases} 1 + yx + \frac{1}{4}y^2x^2 + \frac{1}{12}y^2x^3 - \frac{1}{24}yx^4 + \frac{1}{120}x^5, & x \le y \\ 1 + yx + \frac{1}{4}y^2x^2 + \frac{1}{12}y^3x^2 - \frac{1}{24}xy^4 + \frac{1}{120}y^5, & x > y. \end{cases} \tag{20}$$

Now, we note that the space given in [1] as

$$W(\Omega) = \left\{ \begin{aligned} &v(\mathbf{x},t) | \frac{\partial^4 v}{\partial \mathbf{x}^2 \partial t^2}, \text{ is completely continuous in } \Omega = [0,1] \times [0,1], \\ &\frac{\partial^6 v}{\partial \mathbf{x}^3 \partial t^3} \in L^2(\Omega), v(\mathbf{x},0) = 0, \frac{\partial v(\mathbf{x},0)}{\partial t} = 0 \end{aligned} \right\} \tag{21}$$

is a binary reproducing kernel Hilbert space. The inner product and the norm in Wð Þ Ω are defined by

$$\begin{aligned} \langle v(\mathbf{x},t), g(\mathbf{x},t) \rangle\_{\mathcal{W}} &= \sum\_{i=0}^{2} \int\_{0}^{1} \left[ \frac{\partial^{3}}{\partial t^{3}} \frac{\partial^{i}}{\partial \mathbf{x}^{i}} v(\mathbf{0},t) \frac{\partial^{3}}{\partial t^{3}} \frac{\partial^{i}}{\partial \mathbf{x}^{i}} g(\mathbf{0},t) \right] \mathbf{d}t \\ &+ \sum\_{j=0}^{2} \left\langle \frac{\partial^{j}}{\partial t^{j}} v(\mathbf{x},0), \frac{\partial^{j}}{\partial t^{j}} g(\mathbf{x},0) \right\rangle\_{\mathcal{W}^{3}\_{2}} \\ &+ \int\_{0}^{1} \int\_{0}^{1} \left[ \frac{\partial^{3}}{\partial \mathbf{x}^{3}} \frac{\partial^{3}}{\partial \mathbf{t}^{3}} v(\mathbf{x},t) \frac{\partial^{3}}{\partial \mathbf{x}^{3}} \frac{\partial^{3}}{\partial \mathbf{t}^{3}} g(\mathbf{x},t) \right] \mathbf{d}x \mathbf{d}t, \\\\ \|\boldsymbol{v}\|\_{\boldsymbol{w}} &= \sqrt{\langle v, \boldsymbol{v} \rangle\_{\mathcal{W}^{\prime}}} \qquad \boldsymbol{v} \in \mathcal{W}(\Omega), \end{aligned} \tag{22}$$

respectively.

Ryð Þ� <sup>0</sup> <sup>R</sup>ð Þ<sup>5</sup>

<sup>y</sup>ð Þþ <sup>0</sup> <sup>R</sup>ð Þ<sup>4</sup>

<sup>y</sup>ð Þ� 0 R‴

Rð Þ<sup>3</sup> <sup>y</sup> ð Þ¼ 1 0,

Rð Þ<sup>4</sup> <sup>y</sup> ð Þ¼ 1 0,

Rð Þ<sup>5</sup> <sup>y</sup> ð Þ¼ 1 0,

R0

Then by (11), we obtain

102 Differential Equations - Theory and Current Research

when x 6¼ y,

therefore,

Since

we have

Thus, Ry is given by

Ry ¼

8 >><

>>:

R<sup>00</sup>

Rð Þ<sup>6</sup>

Ryð Þ¼ x

∂k

1 þ yx þ

1 þ yx þ

Rð Þ<sup>6</sup>

X 6

8 >>>>><

>>>>>:

Rð Þ<sup>6</sup>

∂5

Ry<sup>þ</sup> ð Þ� <sup>y</sup> <sup>∂</sup><sup>5</sup>

From (14) and (19), the unknown coefficients cið Þy and dið Þy ð Þ i ¼ 1; 2;…; 6 can be obtained.

<sup>x</sup><sup>3</sup> � <sup>1</sup>

<sup>x</sup><sup>2</sup> � <sup>1</sup>

<sup>24</sup> yx<sup>4</sup> <sup>þ</sup>

<sup>24</sup> xy<sup>4</sup> <sup>þ</sup>

1 <sup>120</sup> <sup>x</sup><sup>5</sup>

1 <sup>120</sup> <sup>y</sup><sup>5</sup>

Ry<sup>þ</sup> ð Þ¼ <sup>y</sup> <sup>∂</sup><sup>k</sup>

1 4 y2 <sup>x</sup><sup>2</sup> <sup>þ</sup> 1 <sup>12</sup> <sup>y</sup><sup>2</sup>

1 4 y2 <sup>x</sup><sup>2</sup> <sup>þ</sup> 1 <sup>12</sup> <sup>y</sup><sup>3</sup>

cið Þ<sup>y</sup> xi�<sup>1</sup>

dið Þ<sup>y</sup> xi�<sup>1</sup>

, x ≤ y,

, x > y,

Ry� ð Þy , k ¼ 0; 1; 2; 3; 4,

i¼1

X 6

i¼1

<sup>y</sup> ð Þ¼ 0 0,

<sup>y</sup> ð Þ¼ 0 0,

<sup>y</sup> ð Þ¼ 0 0,

<sup>y</sup> ð Þ¼ x δð Þ x � y , (15)

<sup>y</sup> ð Þ¼ x 0, (16)

<sup>y</sup> ð Þ¼ x δð Þ x � y , (18)

Ry� ð Þ¼� <sup>y</sup> <sup>1</sup>: (19)

, x ≤ y

, x > y:

(14)

(17)

(20)

Theorem 1.2. The Wð Þ Ω is a reproducing kernel space, and its reproducing kernel function is

$$K\_{(y,s)} = R\_y r\_s \tag{23}$$

such that for any v ∈ Wð Þ Ω ,

$$\begin{aligned} \boldsymbol{\upsilon}(\boldsymbol{y},s) &= \langle \boldsymbol{\upsilon}(\boldsymbol{x},t), \boldsymbol{K}\_{(\boldsymbol{y},s)}(\boldsymbol{x},t) \rangle\_{\mathcal{W}'} \\ &\quad \boldsymbol{K}\_{(\boldsymbol{y},s)}(\boldsymbol{x},t) = \boldsymbol{K}\_{(\boldsymbol{x},t)}(\boldsymbol{y},s). \end{aligned} \tag{24}$$

Similarly, the space

$$\widehat{\boldsymbol{W}}(\boldsymbol{\Omega}) = \left\{ \boldsymbol{v}(\mathbf{x},t) | \boldsymbol{v}(\mathbf{x},t) \text{ is completely continuous in } \boldsymbol{\Omega} = [0,1] \times [0,1], \frac{\partial^2 \boldsymbol{v}}{\partial \mathbf{x} \partial t} \in L^2(\boldsymbol{\Omega}) \right\} \tag{25}$$

is a binary reproducing kernel Hilbert space. The inner product and the norm in W cð Þ Ω are defined by [1] as

$$
\langle v(\mathbf{x},t), g(\mathbf{x},t) \rangle\_{\widehat{W}} = \int\_0^1 \left[ \frac{\partial}{\partial t} v(0,t) \frac{\partial}{\partial t} g(0,t) \right] \mathbf{d}t + \langle v(\mathbf{x},0), g(\mathbf{x},0) \rangle\_{W^1\_2}
$$

$$
+ \int\_0^1 \int\_0^1 \left[ \frac{\partial}{\partial \mathbf{x}} \frac{\partial}{\partial t} v(\mathbf{x},t) \frac{\partial}{\partial \mathbf{x}} \frac{\partial}{\partial t} g(\mathbf{x},t) \right] \mathbf{d}x \mathbf{d}t,\tag{26}
$$

$$
\|v\|\_{\widehat{W}} = \sqrt{\langle v, v \rangle\_{\widehat{W}'}} \qquad v \in \widehat{W}(\Omega),
$$

respectively. W cð Þ Ω is a reproducing kernel space, and its reproducing kernel function Gð Þ <sup>y</sup>;<sup>s</sup> is

$$G\_{(y,s)} = Q\_y Q\_s. \tag{27}$$

h i u xð Þ; ɡð Þx <sup>W</sup><sup>1</sup>

and

The space W<sup>1</sup>

Theorem 1.5. The space W<sup>3</sup>

function Ryð Þx can be denoted by

given by

where

<sup>2</sup> ¼ uð Þ0 ɡð Þþ 0

∥u∥W<sup>1</sup> <sup>2</sup> ¼ ð1 0 u0 ð Þx ɡ<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i u; u <sup>W</sup><sup>1</sup> 2

�

X 6

8 >>>>><

>>>>>:

<sup>156</sup> <sup>y</sup><sup>5</sup> � <sup>5</sup>

<sup>624</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup>

<sup>1872</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup>

1 <sup>3744</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup>

<sup>156</sup> <sup>y</sup><sup>5</sup> � <sup>5</sup>

<sup>1872</sup> <sup>y</sup><sup>5</sup> � <sup>5</sup>

1 <sup>3744</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup>

1 <sup>3744</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup>

21 <sup>104</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup>

<sup>624</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup>

cið Þ<sup>y</sup> xi�<sup>1</sup>

dið Þ<sup>y</sup> xi�<sup>1</sup>

<sup>26</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup>

7 <sup>104</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup>

> 5 <sup>624</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup>

<sup>18720</sup> <sup>y</sup><sup>5</sup> � <sup>1</sup>

<sup>26</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup>

<sup>312</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup>

<sup>18720</sup> <sup>y</sup><sup>5</sup> � <sup>1</sup>

5 <sup>624</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup>

21 <sup>104</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup>

<sup>78</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup>

i¼1

X 6

i¼1

, u∈ W<sup>1</sup>

<sup>2</sup>½ � 0; 1 is a reproducing kernel space, and its reproducing kernel function Txð Þy is

<sup>2</sup>½ � 0; 1 is a complete reproducing kernel space, and its reproducing kernel

, x ≤ y,

, x > y,

3 <sup>13</sup> y,

<sup>936</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup>

5 <sup>1872</sup> <sup>y</sup><sup>3</sup> � <sup>1</sup>

<sup>624</sup> <sup>y</sup><sup>2</sup> � <sup>1</sup>

3 <sup>13</sup> y,

<sup>26</sup> y,

<sup>624</sup> <sup>y</sup><sup>2</sup> � <sup>1</sup>

<sup>78</sup> y,

5 <sup>156</sup> y,

<sup>1872</sup> <sup>y</sup><sup>3</sup> :

<sup>936</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup>

5 <sup>1872</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup>

<sup>78</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup>

7 <sup>104</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup>

<sup>26</sup> y,

<sup>78</sup> y,

<sup>104</sup> y,

<sup>156</sup> y,

<sup>1872</sup> <sup>y</sup><sup>3</sup> � <sup>1</sup>

<sup>312</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup>

1 þ x, x ≤ y, 1 þ y, x > y:

q

Txð Þ¼ y

Ryð Þ¼ x

c1ð Þ¼ y 0, c2ð Þ¼ y

c3ð Þ¼ y

c4ð Þ¼ y

c6ð Þ¼ y

d1ð Þ¼ y

d3ð Þ¼ y

d4ð Þ¼ y

d5ð Þ¼� y

d6ð Þ¼� y

d2ð Þ¼� y

c5ð Þ¼� y

5 <sup>516</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup>

5 <sup>624</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup>

5 <sup>1872</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup>

1 120 þ

1 <sup>120</sup> <sup>y</sup><sup>5</sup> ,

5 <sup>624</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup>

5 <sup>1872</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup>

> 5 <sup>3744</sup> <sup>y</sup><sup>4</sup> <sup>þ</sup>

1 <sup>156</sup> <sup>y</sup> <sup>þ</sup>

1 <sup>104</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup>

5 <sup>3744</sup> <sup>y</sup><sup>4</sup> <sup>þ</sup>

> 1 <sup>3744</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup>

ð Þ<sup>x</sup> <sup>d</sup>x, u xð Þ, <sup>ɡ</sup>ð Þ<sup>x</sup> <sup>∈</sup> <sup>W</sup><sup>1</sup>

<sup>2</sup>½ � 0; 1 , (28)

Reproducing Kernel Functions

(30)

105

<sup>2</sup>½ � 0; 1 : (29)

http://dx.doi.org/10.5772/intechopen.75206

### Definition 1.3.

$$\mathcal{W}\_2^3[0,1] = \left\{ \begin{array}{ll} \mu(\mathbf{x})[\mu(\mathbf{x}), \mu'(\mathbf{x}), \mu'(\mathbf{x}), \mu \text{ are absolutely continuous in } [0,1] \\\\ \mu^{(3)}(\mathbf{x}) \in L^2[0,1], \mathbf{x} \in [0,1], \mu(0) = 0, \mu(1) = 0. \end{array} \right\}$$

The inner product and the norm in W<sup>3</sup> <sup>2</sup>½ � 0; 1 are defined, respectively, by

$$\langle u(\mathbf{x}), g(\mathbf{x}) \rangle\_{W^3\_2} = \sum\_{i=0}^2 \mu^{(i)}(0) g^{(i)}(0) + \int\_0^1 \mu^{(3)}(\mathbf{x}) g^{(3)}(\mathbf{x}) d\mathbf{x}, \, u(\mathbf{x}), g(\mathbf{x}) \in W^3\_2[0, 1]$$

and

$$\|\mu\|\_{W\_2^3} = \sqrt{\langle \mu, \mu \rangle\_{W\_2^3}} \qquad \mu \in W\_2^3[0, 1].$$

The space W<sup>3</sup> <sup>2</sup>½ � 0; 1 is a reproducing kernel space, that is, for each fixed y ∈½ � 0; 1 and any u xð Þ <sup>∈</sup> <sup>W</sup><sup>3</sup> <sup>2</sup>½ � 0; 1 , there exists a function Ryð Þx such that

$$
\mu(y) = \left< \mu(\mathbf{x}), R\_{\mathcal{Y}}(\mathbf{x}) \right>\_{W^3\_2}.
$$

Definition 1.4.

$$\mathcal{W}^1\_2[0,1] = \left\{ \begin{array}{ll} \mu(\mathbf{x}) | \mu(\mathbf{x}) \text{, is absolutely continuous in } [0,1] \\\\ \mu'(\mathbf{x}) \in L^2[0,1] \, \mathbf{x} \in [0,1] \end{array} \right\}$$

The inner product and the norm in W<sup>1</sup> <sup>2</sup>½ � 0; 1 are defined, respectively, by

Reproducing Kernel Functions http://dx.doi.org/10.5772/intechopen.75206 105

$$\langle u(\mathbf{x}), g(\mathbf{x}) \rangle\_{W^1\_2} = u(0)g(0) + \int\_0^1 u'(\mathbf{x}) g'(\mathbf{x}) d\mathbf{x}, \\ u(\mathbf{x}), g(\mathbf{x}) \in W^1\_2[0, 1], \tag{28}$$

and

h i v xð Þ ; <sup>t</sup> ; <sup>ɡ</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>W</sup>

104 Differential Equations - Theory and Current Research

respectively. W

Definition 1.3.

and

The space W<sup>3</sup>

Definition 1.4.

u xð Þ <sup>∈</sup> <sup>W</sup><sup>3</sup>

W<sup>3</sup> <sup>2</sup>½ �¼ 0; 1 <sup>c</sup> <sup>¼</sup>

þ ð1 0 ð1 0

∥v∥ W

u xð Þ∣u xð Þ, u<sup>0</sup>

8 ><

>:

<sup>2</sup> <sup>¼</sup> <sup>X</sup> 2

i¼0

∥u∥W<sup>3</sup> <sup>2</sup> ¼

<sup>2</sup>½ � 0; 1 , there exists a function Ryð Þx such that

8 ><

>:

The inner product and the norm in W<sup>3</sup>

W<sup>1</sup> <sup>2</sup>½ �¼ 0; 1

The inner product and the norm in W<sup>1</sup>

h i u xð Þ; ɡð Þx <sup>W</sup><sup>3</sup>

ð Þx , u<sup>00</sup>

<sup>u</sup>ð Þ<sup>3</sup> ð Þ<sup>x</sup> <sup>∈</sup>L<sup>2</sup>

<sup>u</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>ɡ</sup>ð Þ<sup>i</sup> ð Þþ <sup>0</sup>

ð1 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i u; u <sup>W</sup><sup>3</sup> 2

u yð Þ¼ u xð Þ;Ryð Þ<sup>x</sup> � �

u0 ð Þ<sup>x</sup> <sup>∈</sup>L<sup>2</sup>

q

ð1 0

∂ ∂t vð Þ 0; t ∂ ∂t ɡð0; tÞ

> ∂ ∂x ∂ ∂t v xð Þ ; t

<sup>c</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i <sup>v</sup>; <sup>v</sup> <sup>W</sup> c <sup>q</sup> , v<sup>∈</sup> <sup>W</sup>

� �

∂ ∂x ∂ ∂t ɡðx; tÞ

cð Þ Ω is a reproducing kernel space, and its reproducing kernel function Gð Þ <sup>y</sup>;<sup>s</sup> is

� �

dt þ h i v xð Þ ; 0 ; ɡð Þ x; 0 <sup>W</sup><sup>1</sup>

dxdt,

Gð Þ <sup>y</sup>;<sup>s</sup> ¼ QyQs: (27)

cð Þ Ω ,

ð Þx , are absolutely continuous in 0½ � ; 1

<sup>u</sup>ð Þ<sup>3</sup> ð Þ<sup>x</sup> <sup>ɡ</sup>ð Þ<sup>3</sup> ð Þ<sup>x</sup> <sup>d</sup>x, u xð Þ, <sup>ɡ</sup>ð Þ<sup>x</sup> <sup>∈</sup> <sup>W</sup><sup>3</sup>

<sup>2</sup>½ � 0; 1 :

½ � 0; 1 , x∈½ � 0; 1 , uð Þ¼ 0 0, uð Þ¼ 1 0:

<sup>2</sup>½ � 0; 1 are defined, respectively, by

, u∈ W<sup>3</sup>

<sup>2</sup>½ � 0; 1 is a reproducing kernel space, that is, for each fixed y ∈½ � 0; 1 and any

u xð Þ∣u xð Þ, is absolutely continuous in 0½ � ; 1

W<sup>3</sup> 2 :

½ � 0; 1 , x∈ ½ � 0; 1 ,

<sup>2</sup>½ � 0; 1 are defined, respectively, by

2

9 >=

>;

<sup>2</sup>½ � 0; 1

9 >=

>;

(26)

$$\|\|u\|\|\_{W^1\_2} = \sqrt{\langle u, u \rangle\_{W^1\_2}} \quad u \in \mathcal{W}^1\_2[0, 1]. \tag{29}$$

The space W<sup>1</sup> <sup>2</sup>½ � 0; 1 is a reproducing kernel space, and its reproducing kernel function Txð Þy is given by

$$T\_x(y) = \begin{cases} 1+x, & x \le y; \\ 1+y, & x > y. \end{cases} \tag{30}$$

Theorem 1.5. The space W<sup>3</sup> <sup>2</sup>½ � 0; 1 is a complete reproducing kernel space, and its reproducing kernel function Ryð Þx can be denoted by

$$R\_{\mathcal{Y}}(\mathbf{x}) = \begin{cases} \sum\_{i=1}^{6} c\_i(\mathbf{y}) \mathbf{x}^{i-1}, & \mathbf{x} \le \mathbf{y}\_{\prime} \\\\ \sum\_{i=1}^{6} d\_i(\mathbf{y}) \mathbf{x}^{i-1}, & \mathbf{x} > \mathbf{y}\_{\prime} \end{cases}$$

where

c1ð Þ¼ y 0, c2ð Þ¼ y 5 <sup>516</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>156</sup> <sup>y</sup><sup>5</sup> � <sup>5</sup> <sup>26</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup> <sup>78</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup> 3 <sup>13</sup> y, c3ð Þ¼ y 5 <sup>624</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>624</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup> 21 <sup>104</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup> <sup>312</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>26</sup> y, c4ð Þ¼ y 5 <sup>1872</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>1872</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup> 7 <sup>104</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup> <sup>936</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>78</sup> y, c5ð Þ¼� y 5 <sup>3744</sup> <sup>y</sup><sup>4</sup> <sup>þ</sup> 1 <sup>3744</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup> 5 <sup>624</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> 5 <sup>1872</sup> <sup>y</sup><sup>3</sup> � <sup>1</sup> <sup>104</sup> y, c6ð Þ¼ y 1 120 þ 1 <sup>3744</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>18720</sup> <sup>y</sup><sup>5</sup> � <sup>1</sup> <sup>624</sup> <sup>y</sup><sup>2</sup> � <sup>1</sup> <sup>1872</sup> <sup>y</sup><sup>3</sup> � <sup>1</sup> <sup>156</sup> y, d1ð Þ¼ y 1 <sup>120</sup> <sup>y</sup><sup>5</sup> , d2ð Þ¼� y 1 <sup>104</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>156</sup> <sup>y</sup><sup>5</sup> � <sup>5</sup> <sup>26</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup> <sup>78</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup> 3 <sup>13</sup> y, d3ð Þ¼ y 5 <sup>624</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>624</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup> 21 <sup>104</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> 7 <sup>104</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>26</sup> y, d4ð Þ¼ y 5 <sup>1872</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>1872</sup> <sup>y</sup><sup>5</sup> � <sup>5</sup> <sup>312</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup> <sup>936</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>78</sup> y, d5ð Þ¼� y 5 <sup>3744</sup> <sup>y</sup><sup>4</sup> <sup>þ</sup> 1 <sup>3744</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup> 5 <sup>624</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> 5 <sup>1872</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup> 5 <sup>156</sup> y, d6ð Þ¼� y 1 <sup>156</sup> <sup>y</sup> <sup>þ</sup> 1 <sup>3744</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>18720</sup> <sup>y</sup><sup>5</sup> � <sup>1</sup> <sup>624</sup> <sup>y</sup><sup>2</sup> � <sup>1</sup> <sup>1872</sup> <sup>y</sup><sup>3</sup> :

Proof. We have

$$\begin{split} \left< u(\mathbf{x}), R\_{\mathcal{Y}}(\mathbf{x}) \right>\_{\mathcal{W}^{3}\_{2}} &= \sum\_{i=0}^{2} u^{(i)}(\mathbf{0}) R\_{\mathcal{Y}}^{(i)}(\mathbf{0}) \\ &+ \int\_{0}^{1} u^{(3)}(\mathbf{x}) R\_{\mathcal{Y}}^{(3)}(\mathbf{x}) \mathbf{dx}. \end{split} \tag{31}$$

Since

we have

and

Since Ryð Þ<sup>x</sup> <sup>∈</sup> <sup>W</sup><sup>3</sup>

Ryð Þx is given by

5 <sup>516</sup> xy<sup>4</sup> � <sup>1</sup>

8

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

� 5 <sup>312</sup> <sup>x</sup><sup>2</sup>

� <sup>5</sup> <sup>3744</sup> <sup>x</sup><sup>4</sup>

� <sup>1</sup>

5 <sup>516</sup> yx<sup>4</sup> � <sup>1</sup>

� 5 <sup>312</sup> <sup>y</sup><sup>2</sup>

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

� <sup>5</sup> <sup>3744</sup> <sup>x</sup><sup>4</sup>

� <sup>1</sup> <sup>18720</sup> <sup>x</sup><sup>5</sup>

Ryð Þ¼ x

�Rð Þ<sup>6</sup>

Ry<sup>þ</sup> ð Þ¼ <sup>y</sup> <sup>∂</sup><sup>k</sup>

∂5

Ry<sup>þ</sup> ð Þ� <sup>y</sup> <sup>∂</sup><sup>5</sup>

<sup>26</sup> xy<sup>2</sup> � <sup>5</sup>

5 <sup>1872</sup> <sup>x</sup><sup>3</sup>

<sup>y</sup><sup>5</sup> <sup>þ</sup> 5 <sup>624</sup> <sup>x</sup><sup>4</sup>

<sup>y</sup><sup>2</sup> � <sup>1</sup> <sup>1872</sup> <sup>x</sup><sup>5</sup> y3 , x ≤ y

<sup>26</sup> yx<sup>2</sup> � <sup>5</sup>

5 <sup>1872</sup> <sup>y</sup><sup>3</sup>

<sup>x</sup><sup>5</sup> <sup>þ</sup> 5 <sup>624</sup> <sup>y</sup><sup>4</sup>

8 >>>>>><

>>>>>>:

<sup>x</sup><sup>2</sup> � <sup>1</sup> <sup>1872</sup> <sup>y</sup><sup>5</sup> x3

From (33)–(36), the unknown coefficients cið Þy and dið Þy ð Þ i ¼ 1; 2;…; 6 can be obtained. Thus

<sup>78</sup> xy<sup>3</sup> <sup>þ</sup>

<sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>1872</sup> <sup>x</sup><sup>3</sup>

<sup>78</sup> yx<sup>3</sup> <sup>þ</sup>

<sup>x</sup><sup>4</sup> � <sup>1</sup> <sup>1872</sup> <sup>y</sup><sup>3</sup>

v xð Þ∣v xð Þ, v<sup>0</sup>

<sup>v</sup>ð Þ<sup>4</sup> ð Þ<sup>x</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup>

<sup>x</sup><sup>2</sup> <sup>þ</sup>

<sup>y</sup><sup>2</sup> <sup>þ</sup>

3 <sup>13</sup> xy <sup>þ</sup>

5 <sup>1872</sup> <sup>x</sup><sup>4</sup>

3 <sup>13</sup> xy <sup>þ</sup>

5 <sup>1872</sup> <sup>y</sup><sup>4</sup>

, x > y

are absolutely continuous in 0½ � ; 1 ,

∂k

<sup>2</sup>½ � 0; 1 , it follows that

<sup>156</sup> xy<sup>5</sup> � <sup>5</sup>

1 <sup>3744</sup> <sup>x</sup><sup>4</sup>

<sup>624</sup> <sup>x</sup><sup>5</sup>

<sup>156</sup> yx<sup>5</sup> � <sup>5</sup>

<sup>26</sup> <sup>y</sup><sup>2</sup><sup>x</sup> <sup>þ</sup>

1 <sup>3744</sup> <sup>y</sup><sup>4</sup>

<sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>26</sup> <sup>x</sup><sup>2</sup> y þ

<sup>y</sup><sup>4</sup> <sup>þ</sup>

<sup>x</sup><sup>3</sup> � <sup>5</sup>

<sup>y</sup><sup>4</sup> <sup>þ</sup>

W<sup>4</sup> <sup>2</sup>½ �¼ 0; 1

The inner product and the norm in W<sup>4</sup>

<sup>y</sup><sup>5</sup> � <sup>1</sup> <sup>624</sup> <sup>y</sup><sup>5</sup>

<sup>18720</sup> <sup>x</sup><sup>5</sup>y<sup>5</sup> � <sup>1</sup>

<sup>y</sup> ð Þ¼ x δð Þ x � y ,

Ry� ð Þy , k ¼ 0; 1; 2; 3; 4, (34)

Ryð Þ¼ 0 0, Ryð Þ¼ 1 0, (36)

<sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>624</sup> <sup>x</sup><sup>2</sup>

<sup>y</sup><sup>2</sup> � <sup>5</sup> <sup>936</sup> <sup>x</sup><sup>3</sup>

<sup>104</sup> <sup>x</sup><sup>4</sup><sup>y</sup> � <sup>1</sup>

<sup>x</sup><sup>4</sup> � <sup>1</sup> <sup>624</sup> <sup>y</sup><sup>2</sup>

<sup>x</sup><sup>2</sup> � <sup>5</sup> <sup>936</sup> <sup>x</sup><sup>3</sup>

<sup>104</sup> <sup>y</sup><sup>4</sup><sup>x</sup> � <sup>1</sup>

9 >>>>>>=

>>>>>>;

<sup>y</sup><sup>5</sup> <sup>þ</sup> 21 <sup>104</sup> <sup>x</sup><sup>2</sup> y2

<sup>156</sup> <sup>x</sup><sup>5</sup> y þ

> <sup>x</sup><sup>5</sup> <sup>þ</sup> 21 <sup>104</sup> <sup>x</sup><sup>2</sup> y2

<sup>156</sup> <sup>y</sup><sup>5</sup> x þ

<sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>78</sup> <sup>y</sup><sup>3</sup> x

<sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>78</sup> <sup>x</sup><sup>3</sup> y

> 1 <sup>3744</sup> <sup>x</sup><sup>5</sup> y4

Reproducing Kernel Functions

107

http://dx.doi.org/10.5772/intechopen.75206

1 <sup>3744</sup> <sup>y</sup><sup>5</sup> x4

(37)

5 <sup>624</sup> <sup>x</sup><sup>2</sup>

<sup>y</sup><sup>3</sup> � <sup>1</sup>

5 <sup>624</sup> <sup>y</sup><sup>2</sup>

<sup>x</sup><sup>3</sup> � <sup>1</sup>

ð Þx , v00ð Þx , v000ð Þx

½ � 0; 1 , x∈ ½ � 0; 1

<sup>2</sup>½ � 0; 1 are defined, respectively, by

<sup>y</sup><sup>5</sup> <sup>þ</sup> 7 <sup>104</sup> <sup>x</sup><sup>3</sup>

<sup>x</sup><sup>5</sup> <sup>þ</sup> 7 <sup>104</sup> <sup>y</sup><sup>3</sup>

Ry� ð Þ¼� y 1: (35)

Through several integrations by parts for (31), we have

$$\begin{split} \langle u(\mathbf{x}), R\_{\mathcal{Y}}(\mathbf{x}) \rangle\_{\mathcal{W}\_2^{\ell}} &= \sum\_{i=0}^{2} u^{(i)}(0) \left[ R\_{\mathcal{Y}}^{(i)}(0) - (-1)^{(2-i)} R\_{\mathcal{Y}}^{(5-i)}(0) \right] \\ &+ \sum\_{i=0}^{2} (-1)^{(2-i)} u^{(i)}(1) R\_{\mathcal{Y}}^{(5-i)}(1) \\ &- \int\_{0}^{1} u(\mathbf{x}) R\_{\mathcal{Y}}^{(6)}(\mathbf{x}) d\mathbf{x}. \end{split} \tag{32}$$

Note that property of the reproducing kernel

$$\langle \mathfrak{u}(\mathfrak{x}), \mathcal{R}\_{\mathfrak{y}}(\mathfrak{x}) \rangle\_{W^3\_2} = \mathfrak{u}(\mathfrak{y})\_{\mathfrak{y}}$$

If

$$\begin{cases} \mathcal{R}\_y''(0) - \mathcal{R}\_y^{(3)}(0) = 0, \\ \mathcal{R}\_y'(0) + \mathcal{R}\_y^{(4)}(0) = 0, \\ \mathcal{R}\_y^{(3)}(1) = 0, \\ \mathcal{R}\_y^{(4)}(1) = 0, \end{cases} \tag{33}$$

then by (31), we have the following equation:

$$-R\_y^{(6)}(\mathbf{x}) = \delta(\mathbf{x} - \mathbf{y}),$$

$$\text{when } \mathbf{x} \neq \mathbf{y},$$

$$R\_y^{(6)}(\mathbf{x}) = \mathbf{0},$$

therefore,

$$R\_{\mathcal{Y}}(\mathbf{x}) = \begin{cases} \sum\_{i=1}^{6} c\_i(\mathbf{y}) \mathbf{x}^{i-1}, & \mathbf{x} \le \mathbf{y}, \\\\ \sum\_{i=1}^{6} d\_i(\mathbf{y}) \mathbf{x}^{i-1}, & \mathbf{x} > \mathbf{y}. \end{cases}$$

Since

Proof. We have

106 Differential Equations - Theory and Current Research

If

therefore,

u xð Þ;Ryð Þ<sup>x</sup> � �

W<sup>6</sup> 2 <sup>¼</sup> <sup>X</sup> 2

i¼0

þ X 2

i¼0

u xð Þ; Ryð Þ<sup>x</sup> � �

<sup>y</sup>ð Þ� <sup>0</sup> <sup>R</sup>ð Þ<sup>3</sup>

<sup>y</sup>ð Þþ <sup>0</sup> <sup>R</sup>ð Þ<sup>4</sup>

<sup>y</sup> ð Þ¼ x δð Þ x � y ,

cið Þ<sup>y</sup> xi�<sup>1</sup>

dið Þ<sup>y</sup> xi�<sup>1</sup>

, x ≤ y,

, x > y,

when x 6¼ y,

Rð Þ<sup>6</sup> <sup>y</sup> ð Þ¼ x 0,

X 6

8 >>>>>><

>>>>>>:

i¼1

X 6

i¼1

R 00

8 >>>>>>><

>>>>>>>:

R 0

Rð Þ<sup>3</sup> <sup>y</sup> ð Þ¼ 1 0,

Rð Þ<sup>4</sup> <sup>y</sup> ð Þ¼ 1 0,

�Rð Þ<sup>6</sup>

Ryð Þ¼ x

� <sup>Ð</sup> <sup>1</sup>

Through several integrations by parts for (31), we have

u xð Þ; Ryð Þ<sup>x</sup> � �

Note that property of the reproducing kernel

then by (31), we have the following equation:

W<sup>3</sup> 2 <sup>¼</sup> <sup>X</sup> 2

<sup>u</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>R</sup>ð Þ<sup>i</sup>

<sup>0</sup> u xð ÞRð Þ<sup>6</sup>

W<sup>3</sup> 2

i¼0

<sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>u</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>R</sup>ð Þ<sup>i</sup>

<sup>0</sup> <sup>u</sup>ð Þ<sup>3</sup> ð Þ<sup>x</sup> <sup>R</sup>ð Þ<sup>3</sup>

ð Þ �<sup>1</sup> ð Þ <sup>2</sup>�<sup>i</sup> <sup>u</sup>ð Þ<sup>i</sup> ð Þ<sup>1</sup> <sup>R</sup>ð Þ <sup>5</sup>�<sup>i</sup>

<sup>y</sup> ð Þx dx:

¼ u yð Þ,

<sup>y</sup> ð Þ¼ 0 0,

<sup>y</sup> ð Þ¼ 0 0,

<sup>y</sup> ð Þ0

<sup>y</sup> ð Þ� � <sup>0</sup> ð Þ<sup>1</sup> ð Þ <sup>2</sup>�<sup>i</sup> <sup>R</sup>ð Þ <sup>5</sup>�<sup>i</sup>

h i

<sup>y</sup> ð Þ1

<sup>y</sup> ð Þ0

<sup>y</sup> ð Þx dx:

(31)

(32)

(33)

�Rð Þ<sup>6</sup> <sup>y</sup> ð Þ¼ x δð Þ x � y ,

we have

$$
\partial^k R\_{y^+} (y) = \partial^k R\_{y^-} (y), \quad k = 0, 1, 2, 3, 4,\tag{34}
$$

and

$$
\partial^5 R\_{y^+} (y) - \partial^5 R\_{y^-} (y) = -1. \tag{35}
$$

Since Ryð Þ<sup>x</sup> <sup>∈</sup> <sup>W</sup><sup>3</sup> <sup>2</sup>½ � 0; 1 , it follows that

$$R\_y(0) = 0,\\ R\_y(1) = 0,\tag{36}$$

From (33)–(36), the unknown coefficients cið Þy and dið Þy ð Þ i ¼ 1; 2;…; 6 can be obtained. Thus Ryð Þx is given by

Ryð Þ¼ x 5 <sup>516</sup> xy<sup>4</sup> � <sup>1</sup> <sup>156</sup> xy<sup>5</sup> � <sup>5</sup> <sup>26</sup> xy<sup>2</sup> � <sup>5</sup> <sup>78</sup> xy<sup>3</sup> <sup>þ</sup> 3 <sup>13</sup> xy <sup>þ</sup> 5 <sup>624</sup> <sup>x</sup><sup>2</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>624</sup> <sup>x</sup><sup>2</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup> 21 <sup>104</sup> <sup>x</sup><sup>2</sup> y2 � 5 <sup>312</sup> <sup>x</sup><sup>2</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>26</sup> <sup>x</sup><sup>2</sup><sup>y</sup> <sup>þ</sup> 5 <sup>1872</sup> <sup>x</sup><sup>3</sup> <sup>y</sup><sup>4</sup> � <sup>1</sup> <sup>1872</sup> <sup>x</sup><sup>3</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup> 7 <sup>104</sup> <sup>x</sup><sup>3</sup> <sup>y</sup><sup>2</sup> � <sup>5</sup> <sup>936</sup> <sup>x</sup><sup>3</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>78</sup> <sup>x</sup><sup>3</sup> y � <sup>5</sup> <sup>3744</sup> <sup>x</sup><sup>4</sup> <sup>y</sup><sup>4</sup> <sup>þ</sup> 1 <sup>3744</sup> <sup>x</sup><sup>4</sup> <sup>y</sup><sup>5</sup> <sup>þ</sup> 5 <sup>624</sup> <sup>x</sup><sup>4</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> 5 <sup>1872</sup> <sup>x</sup><sup>4</sup> <sup>y</sup><sup>3</sup> � <sup>1</sup> <sup>104</sup> <sup>x</sup><sup>4</sup> <sup>y</sup> � <sup>1</sup> <sup>156</sup> <sup>x</sup><sup>5</sup> y þ 1 <sup>3744</sup> <sup>x</sup><sup>5</sup> y4 � <sup>1</sup> <sup>18720</sup> <sup>x</sup><sup>5</sup>y<sup>5</sup> � <sup>1</sup> <sup>624</sup> <sup>x</sup><sup>5</sup> <sup>y</sup><sup>2</sup> � <sup>1</sup> <sup>1872</sup> <sup>x</sup><sup>5</sup> y3 , x ≤ y 5 <sup>516</sup> yx<sup>4</sup> � <sup>1</sup> <sup>156</sup> yx<sup>5</sup> � <sup>5</sup> <sup>26</sup> yx<sup>2</sup> � <sup>5</sup> <sup>78</sup> yx<sup>3</sup> <sup>þ</sup> 3 <sup>13</sup> xy <sup>þ</sup> 5 <sup>624</sup> <sup>y</sup><sup>2</sup> <sup>x</sup><sup>4</sup> � <sup>1</sup> <sup>624</sup> <sup>y</sup><sup>2</sup> <sup>x</sup><sup>5</sup> <sup>þ</sup> 21 <sup>104</sup> <sup>x</sup><sup>2</sup> y2 � 5 <sup>312</sup> <sup>y</sup><sup>2</sup> <sup>x</sup><sup>3</sup> � <sup>5</sup> <sup>26</sup> <sup>y</sup><sup>2</sup><sup>x</sup> <sup>þ</sup> 5 <sup>1872</sup> <sup>y</sup><sup>3</sup> <sup>x</sup><sup>4</sup> � <sup>1</sup> <sup>1872</sup> <sup>y</sup><sup>3</sup> <sup>x</sup><sup>5</sup> <sup>þ</sup> 7 <sup>104</sup> <sup>y</sup><sup>3</sup> <sup>x</sup><sup>2</sup> � <sup>5</sup> <sup>936</sup> <sup>x</sup><sup>3</sup> <sup>y</sup><sup>3</sup> � <sup>5</sup> <sup>78</sup> <sup>y</sup><sup>3</sup> x � <sup>5</sup> <sup>3744</sup> <sup>x</sup><sup>4</sup> <sup>y</sup><sup>4</sup> <sup>þ</sup> 1 <sup>3744</sup> <sup>y</sup><sup>4</sup> <sup>x</sup><sup>5</sup> <sup>þ</sup> 5 <sup>624</sup> <sup>y</sup><sup>4</sup> <sup>x</sup><sup>2</sup> <sup>þ</sup> 5 <sup>1872</sup> <sup>y</sup><sup>4</sup> <sup>x</sup><sup>3</sup> � <sup>1</sup> <sup>104</sup> <sup>y</sup><sup>4</sup> <sup>x</sup> � <sup>1</sup> <sup>156</sup> <sup>y</sup><sup>5</sup> x þ 1 <sup>3744</sup> <sup>y</sup><sup>5</sup> x4 � <sup>1</sup> <sup>18720</sup> <sup>x</sup><sup>5</sup> <sup>y</sup><sup>5</sup> � <sup>1</sup> <sup>624</sup> <sup>y</sup><sup>5</sup> <sup>x</sup><sup>2</sup> � <sup>1</sup> <sup>1872</sup> <sup>y</sup><sup>5</sup> x3 , x > y 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: W<sup>4</sup> <sup>2</sup>½ �¼ 0; 1 v xð Þ∣v xð Þ, v<sup>0</sup> ð Þx , v00ð Þx , v000ð Þx are absolutely continuous in 0½ � ; 1 , <sup>v</sup>ð Þ<sup>4</sup> ð Þ<sup>x</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup> ½ � 0; 1 , x∈ ½ � 0; 1 8 >>>>>>< >>>>>>: 9 >>>>>>= >>>>>>; (37)

The inner product and the norm in W<sup>4</sup> <sup>2</sup>½ � 0; 1 are defined, respectively, by

$$
\langle \boldsymbol{v}(\mathbf{x}), \boldsymbol{g}(\mathbf{x}) \rangle\_{\mathcal{W}\_2^4} = \sum\_{i=0}^3 \boldsymbol{v}^{(i)}(\mathbf{0}) \boldsymbol{g}^{(i)}(\mathbf{0}) + \int\_0^1 \boldsymbol{v}^{(4)}(\mathbf{x}) \boldsymbol{g}^{(4)}(\mathbf{x}) d\mathbf{x}, \quad \boldsymbol{v}(\mathbf{x}), \boldsymbol{g}(\mathbf{x}) \in \mathcal{W}\_2^4[0, 1]. \tag{38}
$$

$$
\|\boldsymbol{v}\|\_{\mathcal{W}\_2^4} = \sqrt{\langle \boldsymbol{v}, \boldsymbol{v} \rangle\_{\mathcal{W}\_2^4}} \quad \boldsymbol{v} \in \mathcal{W}\_2^4[0, 1].
$$

The space W<sup>4</sup> <sup>2</sup>½ � 0; 1 is a reproducing kernel space, that is, for each fixed. <sup>y</sup><sup>∈</sup> ½ � <sup>0</sup>; <sup>1</sup> and any v xð Þ<sup>∈</sup> <sup>W</sup><sup>4</sup> <sup>2</sup>½ � 0; 1 , there exists a function Ryð Þx such that

$$\left\langle \boldsymbol{v}(\boldsymbol{y}) = \left\langle \boldsymbol{v}(\boldsymbol{x}), \boldsymbol{R}\_{\boldsymbol{y}}(\boldsymbol{x}) \right\rangle\_{\mathcal{W}\_2^{\boldsymbol{\theta}}} \tag{39}$$

where the inner product and the norm in W<sup>2</sup>

<sup>2</sup> <sup>¼</sup> <sup>X</sup> 1

i¼0

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>ɡ</sup>ð Þ<sup>i</sup> ð Þþ <sup>0</sup>

8 ><

>:

<sup>2</sup> <sup>¼</sup> <sup>v</sup>ð Þ<sup>0</sup> <sup>ɡ</sup>ð Þþ <sup>0</sup> <sup>Ð</sup> <sup>T</sup>

∥v∥W<sup>1</sup> <sup>2</sup> ¼ q

∥v∥<sup>W</sup><sup>2</sup> ¼

Qyð Þ¼ x

<sup>2</sup>½ � 0; T is defined by

8 ><

>:

ðT 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>W</sup><sup>2</sup> 2

<sup>1</sup> <sup>þ</sup> xy <sup>þ</sup> <sup>y</sup>

1 þ xy þ

2 <sup>x</sup><sup>2</sup> � <sup>1</sup> 6 x3

x 2 <sup>y</sup><sup>2</sup> � <sup>1</sup> 6 y3

v tð Þ<sup>∈</sup> <sup>L</sup><sup>2</sup>

<sup>0</sup> v<sup>0</sup> ð Þt ɡ<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>W</sup><sup>1</sup> 2

�

v ∂x<sup>3</sup>∂t

∂6 v ∂x<sup>4</sup>∂t

<sup>2</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup>

q

qsðÞ¼ t

v xð Þ ; <sup>t</sup> <sup>∣</sup> <sup>∂</sup><sup>4</sup>

and the inner product and the norm in Wð Þ Ω are defined, respectively, by

8

>>>>>>>>>><

>>>>>>>>>>:

, v ∈ W<sup>2</sup>

<sup>2</sup>½ � 0; 1 is a reproducing kernel space, and its reproducing kernel function Qyð Þx is

v tð Þ∣v tð Þ is absolutely continuous in 0½ � ; T ,

½ � 0; T , t∈½ � 0; T

<sup>2</sup>½ � 0; T are defined, respectively, by

, v ∈ W<sup>1</sup>

<sup>2</sup>½ � 0; T is a reproducing kernel space, and its reproducing kernel function qsð Þt is

, is completely continuous,

ð Þ Ω ,v xð Þ¼ ; 0 0

1 þ t, t ≤ s, 1 þ s, t > s:

inΩ ¼ ½ �� 0; 1 ½ � 0; T ,

h i v tð Þ; ɡð Þt <sup>W</sup><sup>2</sup>

The space W<sup>2</sup>

Similarly, the space W<sup>1</sup>

The space W<sup>1</sup>

given by

W<sup>1</sup>

The inner product and the norm in W<sup>1</sup>

Further, we define the space Wð Þ Ω as

Wð Þ¼ Ω

h i v tð Þ; ɡð Þt <sup>W</sup><sup>1</sup>

<sup>2</sup>½ �¼ 0; T

given by

<sup>2</sup>½ � 0; 1 are defined, respectively, by

<sup>2</sup>½ � 0; 1 :

, x ≤ y,

, x > y:

ð Þ<sup>t</sup> <sup>d</sup>t, v tð Þ, <sup>ɡ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>W</sup><sup>1</sup>

<sup>v</sup>00ð Þ<sup>t</sup> <sup>ɡ</sup>00ð Þ<sup>t</sup> <sup>d</sup>t, v tð Þ, <sup>ɡ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>W</sup><sup>2</sup>

<sup>2</sup>½ � 0; 1 ,

http://dx.doi.org/10.5772/intechopen.75206

Reproducing Kernel Functions

9 >=

>;

<sup>2</sup>½ � 0; T ,

<sup>2</sup>½ � <sup>0</sup>; <sup>T</sup> : (47)

9

>>>>>>>>>>=

>>>>>>>>>>;

(44)

109

(45)

(46)

(48)

(49)

Similarly, we define the space

$$\mathcal{W}\_2^2[0, T] = \left\{ \begin{array}{c} \upsilon(t) \vert \upsilon(t), \upsilon'(t) \\\\ \text{are absolutely continuous in } [0, T]\_\prime \\\\ \upsilon'(t) \in L^2[0, T], t \in [0, T], \upsilon(0) = 0 \end{array} \right\} \tag{40}$$

The inner product and the norm in W<sup>2</sup> <sup>2</sup>½ � 0; T are defined, respectively, by

$$
\langle v(t), g(t) \rangle\_{\mathcal{W}\_2^2} = \sum\_{i=0}^1 v^{(i)}(0) g^{(i)}(0) + \int\_0^T v''(t) g''(t) dt, \quad v(t), g(t) \in \mathcal{W}\_2^2[0, T]. \tag{41}
$$

$$
\|v\|\_{\mathcal{W}\_1} = \sqrt{\langle v, v \rangle\_{\mathcal{W}\_2^2}}, \quad v \in \mathcal{W}\_2^2[0, T].
$$

Thus, the space W<sup>2</sup> <sup>2</sup>½ � 0; T is also a reproducing kernel space, and its reproducing kernel function rsð Þt can be given by

$$r\_s(t) = \begin{cases} st + \frac{s}{2}t^2 - \frac{1}{6}t^3, & t \le s, \\ st + \frac{t}{2}s^2 - \frac{1}{6}s^3, & t > s, \end{cases} \tag{42}$$

and the space

$$W\_2^2[0,1] = \left\{ \begin{array}{c} v(\mathbf{x}) | v(\mathbf{x}), v'(\mathbf{x}) \\\\ \text{are absolutely continuous in } [0,1]\_\prime \\\\ v''(\mathbf{x}) \in L^2[0,1]\_\prime \mathbf{x} \in [0,1] \end{array} \right\} \tag{43}$$

where the inner product and the norm in W<sup>2</sup> <sup>2</sup>½ � 0; 1 are defined, respectively, by

$$
\langle \boldsymbol{v}(t), \boldsymbol{g}(t) \rangle\_{\mathcal{W}\_2^2} = \sum\_{i=0}^{1} \boldsymbol{v}^{(i)}(0) \boldsymbol{g}^{(i)}(0) + \int\_0^T \boldsymbol{v}''(t) \boldsymbol{g}''(t) \mathrm{d}t, \quad \boldsymbol{v}(t), \boldsymbol{g}(t) \in \mathcal{W}\_2^2[0, 1]. \tag{44}
$$

$$
\|\boldsymbol{v}\|\_{\mathcal{W}\_2} = \sqrt{\langle \boldsymbol{v}, \boldsymbol{v} \rangle\_{\mathcal{W}\_2^2}} \quad \boldsymbol{v} \in \mathcal{W}\_2^2[0, 1].
$$

The space W<sup>2</sup> <sup>2</sup>½ � 0; 1 is a reproducing kernel space, and its reproducing kernel function Qyð Þx is given by

$$Q\_y(\mathbf{x}) = \begin{cases} 1 + \mathbf{x}y + \frac{y}{2}\mathbf{x}^2 - \frac{1}{6}\mathbf{x}^3, & \mathbf{x} \le y, \\ 1 + \mathbf{x}y + \frac{\mathbf{x}}{2}y^2 - \frac{1}{6}y^3, & \mathbf{x} > y. \end{cases} \tag{45}$$

Similarly, the space W<sup>1</sup> <sup>2</sup>½ � 0; T is defined by

h i v xð Þ; ɡð Þx <sup>W</sup><sup>4</sup>

108 Differential Equations - Theory and Current Research

<sup>y</sup><sup>∈</sup> ½ � <sup>0</sup>; <sup>1</sup> and any v xð Þ<sup>∈</sup> <sup>W</sup><sup>4</sup>

Similarly, we define the space

W<sup>2</sup>

The inner product and the norm in W<sup>2</sup>

h i v tð Þ; ɡð Þt <sup>W</sup><sup>2</sup>

Thus, the space W<sup>2</sup>

rsð Þt can be given by

and the space

<sup>2</sup>½ �¼ 0; T

<sup>2</sup> <sup>¼</sup> <sup>X</sup> 1

W<sup>2</sup> <sup>2</sup>½ �¼ 0; 1

i¼0

v00 ð Þ<sup>t</sup> <sup>∈</sup>L<sup>2</sup>

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>ɡ</sup>ð Þ<sup>i</sup> ð Þþ <sup>0</sup>

q

∥v∥<sup>W</sup><sup>1</sup> ¼

rsðÞ¼ t

8

>>>>>>>><

>>>>>>>>:

8

>>>>>>>><

>>>>>>>>:

The space W<sup>4</sup>

<sup>2</sup> <sup>¼</sup> <sup>X</sup> 3

i¼0

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>ɡ</sup>ð Þ<sup>i</sup> ð Þþ <sup>0</sup>

∥v∥W<sup>4</sup> <sup>2</sup> ¼ ð1 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>W</sup><sup>4</sup> 2

<sup>2</sup>½ � 0; 1 , there exists a function Ryð Þx such that

v tð Þ∣v tð Þ, v<sup>0</sup>

are absolutely continuous in 0½ � ; T ,

v yð Þ¼ v xð Þ;Ryð Þ<sup>x</sup> � �

ðT 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>W</sup><sup>2</sup> 2

st þ s 2 t <sup>2</sup> � <sup>1</sup> 6 t <sup>3</sup>, t ≤ s,

8 ><

>:

st þ t 2 s <sup>2</sup> � <sup>1</sup> 6 s 3

, v ∈ W<sup>4</sup>

W<sup>4</sup> 2

ð Þt

9

>>>>>>>>=

>>>>>>>>;

9

>>>>>>>>=

>>>>>>>>;

<sup>2</sup>½ � 0; T ,

½ � 0; T , t∈ ½ � 0; T , vð Þ¼ 0 0

<sup>v</sup>00ð Þ<sup>t</sup> <sup>ɡ</sup>00ð Þ<sup>t</sup> <sup>d</sup>t, v tð Þ, <sup>ɡ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>W</sup><sup>2</sup>

<sup>2</sup>½ � 0; T :

<sup>2</sup>½ � 0; T are defined, respectively, by

, v ∈ W<sup>2</sup>

<sup>2</sup>½ � 0; T is also a reproducing kernel space, and its reproducing kernel function

, t > s,

ð Þx

½ � 0; 1 , x∈ ½ � 0; 1

v xð Þ∣v xð Þ, v<sup>0</sup>

are absolutely continuous in 0½ � ; 1 ,

<sup>v</sup>00ð Þ<sup>x</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup>

q

<sup>2</sup>½ � 0; 1 is a reproducing kernel space, that is, for each fixed.

<sup>v</sup>ð Þ<sup>4</sup> ð Þ<sup>x</sup> <sup>ɡ</sup>ð Þ<sup>4</sup> ð Þ<sup>x</sup> <sup>d</sup>x, v xð Þ, <sup>ɡ</sup>ð Þ<sup>x</sup> <sup>∈</sup> <sup>W</sup><sup>4</sup>

<sup>2</sup>½ � 0; 1 :

<sup>2</sup>½ � 0; 1 ,

(38)

(39)

(40)

(41)

(42)

(43)

$$\mathcal{W}^1\_2[0, T] = \left\{ \begin{array}{l} v(t)|v(t) \ \text{is absolutely continuous in } [0, T] \\\\ v(t) \in L^2[0, T], t \in [0, T] \end{array} \right\} \tag{46}$$

The inner product and the norm in W<sup>1</sup> <sup>2</sup>½ � 0; T are defined, respectively, by

$$\begin{aligned} \langle \boldsymbol{v}(t), \boldsymbol{g}(t) \rangle\_{W^1\_2} &= \boldsymbol{v}(0)\boldsymbol{g}(0) + \int\_0^T \boldsymbol{v}'(t)\boldsymbol{g}'(t) \, \mathrm{d}t, \quad \boldsymbol{v}(t), \boldsymbol{g}(t) \in \mathcal{W}^1\_2[0, T]. \\ &\quad \|\boldsymbol{v}\|\_{W^1\_2} = \sqrt{\langle \boldsymbol{v}, \boldsymbol{v} \rangle\_{W^1\_2}} \quad \boldsymbol{v} \in \mathcal{W}^1\_2[0, T]. \end{aligned} \tag{47}$$

The space W<sup>1</sup> <sup>2</sup>½ � 0; T is a reproducing kernel space, and its reproducing kernel function qsð Þt is given by

$$q\_s(t) = \begin{cases} 1+t, & t \le s, \\ 1+s, & t > s. \end{cases} \tag{48}$$

Further, we define the space Wð Þ Ω as

$$W(\Omega) = \left\{ \begin{array}{l} \upsilon(\mathbf{x},t) | \frac{\partial^4 \upsilon}{\partial \mathbf{x}^3 \partial t'}, \text{ is completely continuous}, \\\\ \dot{m}\Omega = [0,1] \times [0,T], \\\\ \frac{\partial^6 \upsilon}{\partial \mathbf{x}^4 \partial t'^2} \in L^2(\Omega), \upsilon(\mathbf{x},0) = 0 \end{array} \right\} \tag{49}$$

and the inner product and the norm in Wð Þ Ω are defined, respectively, by

$$\begin{split} \langle v(\mathbf{x},t), g(\mathbf{x},t) \rangle\_{\mathcal{W}} &= \sum\_{i=0}^{3} \int\_{0}^{T} \left[ \frac{\partial^{2}}{\partial t^{2}} \frac{\partial^{i}}{\partial \mathbf{x}^{i}} v(\mathbf{0},t) \frac{\partial^{2}}{\partial t^{2}} \frac{\partial^{i}}{\partial \mathbf{x}^{i}} g(\mathbf{0},t) \right] \mathbf{d}t \\ &+ \sum\_{j=0}^{1} \left\langle \frac{\partial^{j}}{\partial t^{j}} v(\mathbf{x},0), \frac{\partial^{j}}{\partial t^{j}} g(\mathbf{x},0) \right\rangle\_{\mathcal{W}^{4}\_{2}} \\ &+ \int\_{0}^{T} \int\_{0}^{1} \left[ \frac{\partial^{4}}{\partial \mathbf{x}^{4}} \frac{\partial^{2}}{\partial t^{2}} v(\mathbf{x},t) \frac{\partial^{4}}{\partial \mathbf{x}^{4}} \frac{\partial^{2}}{\partial t^{2}} g(\mathbf{x},t) \right] \mathbf{d}x \mathbf{d}t, \\ \|v\|\_{\mathcal{W}} &= \sqrt{\langle v,v \rangle\_{\mathcal{W}}} \qquad v \in \mathcal{W}(\Omega). \end{split} \tag{50}$$

v xð Þ; Ryð Þ<sup>x</sup> � �

Note that property of the reproducing kernel

then by (54), we obtain the following equation:

If

when x 6¼ y,

therefore,

Since

W<sup>4</sup> 2 <sup>¼</sup> <sup>X</sup> 3

i¼0

þ X 3

i¼0

v xð Þ;Ryð Þ<sup>x</sup> � �

Ryð Þþ <sup>0</sup> <sup>R</sup>ð Þ<sup>7</sup>

<sup>y</sup>ð Þ� <sup>0</sup> <sup>R</sup>ð Þ<sup>6</sup>

<sup>y</sup>ð Þþ <sup>0</sup> <sup>R</sup>ð Þ<sup>5</sup>

<sup>y</sup> ð Þ� <sup>0</sup> <sup>R</sup>ð Þ<sup>4</sup>

Rð Þ<sup>4</sup> <sup>y</sup> ð Þ¼ 1 0,

Rð Þ<sup>5</sup> <sup>y</sup> ð Þ¼ 1 0,

Rð Þ<sup>6</sup> <sup>y</sup> ð Þ¼ 1 0,

Rð Þ<sup>7</sup> <sup>y</sup> ð Þ¼ 1 0,

R0

R<sup>00</sup>

R‴

Rð Þ<sup>8</sup>

Ryð Þ¼ x

Rð Þ<sup>8</sup>

X 8

8 >>>>><

>>>>>:

cið Þ<sup>y</sup> xi�<sup>1</sup>

dið Þ<sup>y</sup> xi�<sup>1</sup>

, x ≤ y,

, x > y:

i¼1

X 8

i¼1

<sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>R</sup>ð Þ<sup>i</sup>

ð Þ �<sup>1</sup> ð Þ <sup>3</sup>�<sup>i</sup>

W<sup>4</sup> 2

<sup>y</sup> ð Þ¼ 0 0,

<sup>y</sup> ð Þ¼ 0 0,

<sup>y</sup> ð Þ¼ 0 0,

<sup>y</sup> ð Þ¼ 0 0,

<sup>0</sup> v xð ÞRð Þ<sup>8</sup>

<sup>y</sup> ð Þ� � <sup>0</sup> ð Þ<sup>1</sup> ð Þ <sup>3</sup>�<sup>i</sup> <sup>R</sup>ð Þ <sup>7</sup>�<sup>i</sup>

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>1</sup> <sup>R</sup>ð Þ <sup>7</sup>�<sup>i</sup>

<sup>y</sup> ð Þx dx:

h i

<sup>y</sup> ð Þ1

<sup>y</sup> ð Þ0

Reproducing Kernel Functions

http://dx.doi.org/10.5772/intechopen.75206

¼ v yð Þ: (55)

<sup>y</sup> ð Þ¼ x δð Þ x � y , (57)

<sup>y</sup> ð Þ¼ x 0; (58)

(54)

111

(56)

(59)

Now, we have the following theorem:

Theorem 1.6. The space W<sup>4</sup> <sup>2</sup>½ � 0; 1 is a complete reproducing kernel space, and its reproducing kernel function Ryð Þx can be denoted by

$$R\_y(\mathbf{x}) = \begin{cases} \sum\_{i=1}^8 c\_i(y)\mathbf{x}^{i-1}, & \mathbf{x} \le y\_\prime \\\\ \sum\_{i=1}^8 d\_i(y)\mathbf{x}^{i-1}, & \mathbf{x} > y\_\prime \end{cases} \tag{51}$$

where

$$\begin{aligned} c\_1(y) &= 1, & c\_2(y) &= y, & c\_3(y) &= \frac{1}{4}y^2, \\ c\_4(y) &= \frac{1}{36}y^3, & c\_5(y) &= \frac{1}{144}y^3, & c\_6(y) &= -\frac{1}{240}y^2, \\ & & c\_7(y) &= \frac{1}{720}y, & c\_8(y) &= -\frac{1}{5040}y^2, \\ & d\_1(y) &= 1 - \frac{1}{5040}y^7, & d\_2(y) &= y + \frac{1}{720}y^6, \\ & d\_3(y) &= \frac{1}{4}y^2 - \frac{1}{240}y^5, & d\_4(y) &= \frac{1}{36}y^3 + \frac{1}{144}y^4, \\ & d\_5(y) &= 0, & d\_6(y) &= 0, & d\_8(y) &= 0. \end{aligned} \tag{52}$$

Proof. Since

$$\left< v(\mathbf{x}), R\_{\mathcal{Y}}(\mathbf{x}) \right>\_{\mathcal{W}^4\_2} = \sum\_{i=0}^3 v^{(i)}(0) R\_{\mathcal{Y}}^{(i)}(0) + \int\_0^1 v^{(4)}(\mathbf{x}) R\_{\mathcal{Y}}^{(4)}(\mathbf{x}) d\mathbf{x},\tag{53}$$

$$\left< v(\mathbf{x}), R\_{\mathcal{Y}}(\mathbf{x}) \in \mathcal{W}^4\_2[0, 1] \right>$$

through iterative integrations by parts for (53), we have

#### Reproducing Kernel Functions http://dx.doi.org/10.5772/intechopen.75206 111

$$\begin{split} \left< v(\mathbf{x}), R\_{\mathbf{y}}(\mathbf{x}) \right>\_{\mathcal{W}^{\mathbf{1}}\_{2}} &= \quad \sum\_{i=0}^{3} v^{(i)}(0) \left[ R\_{\mathbf{y}}^{(i)}(0) - (-1)^{(3-i)} R\_{\mathbf{y}}^{(7-i)}(0) \right] \\ &+ \sum\_{i=0}^{3} (-1)^{(3-i)} v^{(i)}(1) R\_{\mathbf{y}}^{(7-i)}(1) \\ &+ \int\_{0}^{1} v(\mathbf{x}) R\_{\mathbf{y}}^{(8)}(\mathbf{x}) d\mathbf{x}. \end{split} \tag{54}$$

Note that property of the reproducing kernel

$$
\langle \boldsymbol{\upsilon}(\mathbf{x}), \boldsymbol{R}\_{\mathcal{Y}}(\mathbf{x}) \rangle\_{\mathcal{W}\_2^4} = \boldsymbol{\upsilon}(\boldsymbol{y}).\tag{55}
$$

If

h i v xð Þ ; <sup>t</sup> ; <sup>ɡ</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>W</sup> <sup>¼</sup> <sup>X</sup>

Now, we have the following theorem:

110 Differential Equations - Theory and Current Research

c4ð Þ¼ y

1 <sup>36</sup> <sup>y</sup><sup>3</sup>

v xð Þ;Ryð Þ<sup>x</sup> � �

through iterative integrations by parts for (53), we have

Theorem 1.6. The space W<sup>4</sup>

where

Proof. Since

function Ryð Þx can be denoted by

3

ðT 0

> ∂j ∂t

> > ∂4 ∂x<sup>4</sup> ∂2 ∂t <sup>2</sup> v xð Þ ; t

X 8

8 >>>>><

>>>>>:

i¼1

X 8

i¼1

c1ð Þ¼ y 1, c2ð Þ¼ y y, c3ð Þ¼ y

1 <sup>144</sup> <sup>y</sup><sup>3</sup>

1

<sup>y</sup><sup>2</sup> � <sup>1</sup> <sup>240</sup> <sup>y</sup><sup>5</sup>

<sup>v</sup>ð Þ<sup>i</sup> ð Þ<sup>0</sup> <sup>R</sup>ð Þ<sup>i</sup>

<sup>y</sup> ð Þþ 0

<sup>5040</sup> <sup>y</sup><sup>7</sup>

d5ð Þ¼ y 0, d6ð Þ¼ y 0, d7ð Þ¼ y 0, d8ð Þ¼ y 0:

∂2 ∂t 2 ∂i <sup>∂</sup>xi <sup>v</sup>ð Þ <sup>0</sup>; <sup>t</sup>

<sup>j</sup> v xð Þ ; 0 ;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i v; v <sup>W</sup> q

cið Þ<sup>y</sup> xi�<sup>1</sup>

dið Þ<sup>y</sup> xi�<sup>1</sup>

∂j ∂t <sup>j</sup> ɡðx; 0Þ

\* +

∂2 ∂t 2 ∂i <sup>∂</sup>xi <sup>ɡ</sup>ð0; <sup>t</sup><sup>Þ</sup>

> W<sup>4</sup> 2

dt

dxdt,

(50)

(51)

(52)

(53)

" #

∂4 ∂x<sup>4</sup> ∂2 ∂t <sup>2</sup> ɡðx; tÞ

, v ∈ Wð Þ Ω :

<sup>2</sup>½ � 0; 1 is a complete reproducing kernel space, and its reproducing kernel

, x ≤ y,

, x > y,

, c6ð Þ¼� y

, d2ð Þ¼ y y þ

1 <sup>36</sup> <sup>y</sup><sup>3</sup> <sup>þ</sup>

<sup>v</sup>ð Þ<sup>4</sup> ð Þ<sup>x</sup> <sup>R</sup>ð Þ<sup>4</sup>

v xð Þ; Ryð Þ<sup>x</sup> <sup>∈</sup> <sup>W</sup><sup>4</sup> <sup>2</sup>½ � <sup>0</sup>; <sup>1</sup> � �

, d4ð Þ¼ y

ð1 0

<sup>720</sup> y, c8ð Þ¼� <sup>y</sup>

1 4 y2 ,

1 <sup>240</sup> <sup>y</sup><sup>2</sup> ,

1 <sup>5040</sup> ,

1 <sup>720</sup> <sup>y</sup><sup>6</sup> ,

<sup>y</sup> ð Þx dx,

1 <sup>144</sup> <sup>y</sup><sup>4</sup> ,

� �

i¼0

þ<sup>X</sup> 1

þ ðT 0 ð1 0

Ryð Þ¼ x

, c5ð Þ¼ y

d3ð Þ¼ y

W<sup>4</sup> 2 <sup>¼</sup> <sup>X</sup> 3

c7ð Þ¼ y

<sup>d</sup>1ð Þ¼ <sup>y</sup> <sup>1</sup> � <sup>1</sup>

1 4

i¼0

j¼0

∥v∥<sup>W</sup> ¼

$$\begin{aligned} R\_y(0) + R\_y^{(7)}(0) &= 0, \\ R\_y'(0) - R\_y^{(6)}(0) &= 0, \\ R\_y''(0) + R\_y^{(5)}(0) &= 0, \\ R\_y'(0) - R\_y^{(4)}(0) &= 0, \\ R\_y^{(4)}(1) &= 0, \\ R\_y^{(5)}(1) &= 0, \\ R\_y^{(6)}(1) &= 0, \\ R\_y^{(7)}(1) &= 0, \end{aligned} \tag{56}$$

then by (54), we obtain the following equation:

$$R\_y^{(8)}(\mathbf{x}) = \delta(\mathbf{x} - \mathbf{y})\_\prime \tag{57}$$

when x 6¼ y,

$$R\_y^{(8)}(\mathbf{x}) = \mathbf{0};\tag{58}$$

therefore,

$$R\_{\mathbf{y}}(\mathbf{x}) = \begin{cases} \sum\_{i=1}^{8} c\_i(\mathbf{y}) \mathbf{x}^{i-1}, & \mathbf{x} \le \mathbf{y}, \\\\ \sum\_{i=1}^{8} d\_i(\mathbf{y}) \mathbf{x}^{i-1}, & \mathbf{x} > \mathbf{y}. \end{cases} \tag{59}$$

Since

$$R\_y^{(8)}(\mathbf{x}) = \delta(\mathbf{x} - \mathbf{y}),\tag{60}$$

[6] L X, Cui M, Analytic solutions to a class of nonlinear innite-delay-dierential equations.

Reproducing Kernel Functions

113

http://dx.doi.org/10.5772/intechopen.75206

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[9] Zhou S, Cui M. Approximate solution for a variable-coecient semilinear heat equation with nonlocal boundary conditions. International Journal of Computer Mathematics.

[10] Geng F, Cui M. New method based on the HPM and RKHSM for solving forced dung equations with integral boundary conditions. Journal of Computational and Applied

[11] Du J, Cui M. Solving the forced dung equation with integral boundary conditions in the reproducing kernel space. International Journal of Computer Mathematics. 2010;87(9):

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problems. Applied Mathematics and Computation. 2009;215(5):1937-1948

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137

1365

2088-2100

1558

44-47

2009;86(12):2248-2258

Mathematics. 2009;233(2):165-172

Simulation. 2011;16(9):3639-3645

we have

$$
\partial^k R\_{y^+} (y) = \partial^k R\_{y^-} (y), \quad k = 0, 1, 2, 3, 4, 5, 6,\tag{61}
$$

$$
\partial^{\mathsf{T}} \mathsf{R}\_{y^{+}}(y) - \partial^{\mathsf{T}} \mathsf{R}\_{y^{-}}(y) = 1. \tag{62}
$$

From (56)–(62), the unknown coefficients cið Þy ve dið Þy ð Þ i ¼ 1; 2;…; 8 can be obtained. Thus, Ryð Þx is given by

$$R\_y(\mathbf{x}) = \begin{pmatrix} 1 + y\mathbf{x} + \frac{1}{4}y^2\mathbf{x}^2 + \frac{1}{36}y^3\mathbf{x}^3 + \frac{1}{144}y^3\mathbf{x}^4 \\\\ -\frac{1}{240}y^2\mathbf{x}^5 + \frac{1}{720}y\mathbf{x}^6 - \frac{1}{5040}\mathbf{x}^7, & \mathbf{x} \le y, \\\\ 1 + xy + \frac{1}{4}\mathbf{x}^2y^2 + \frac{1}{36}\mathbf{x}^3y^3 + \frac{1}{144}\mathbf{x}^3y^4 \\\\ -\frac{1}{240}\mathbf{x}^2y^5 + \frac{1}{720}\mathbf{x}y^6 - \frac{1}{5040}y^7, & \mathbf{x} > y. \end{pmatrix} \tag{63}$$
