A novel iterative solution for time-fractional Boussinesq equation by reproducing kernel method

Abstract

In this study, iterative reproducing kernel method (RKM) will be applied in order to observe the effect of the method on numerical solutions of fractional order Boussinesq equation. Hilbert spaces and their kernel functions, linear operators and base functions which are necessary to obtain the reproducing kernel function are clearly explained. Iterative solution is constituted in a serial form by using reproducing kernel function. Then convergence of RKM solution is shown with lemma and theorem. Two problems, “good” Boussinesq and generalized Boussinesq equations, are examined by using RKM for different fractional values. Results are presented with tables and graphics.

Appendix

In this part, the general definition and inner product of $W_{2}^{n}[0,1]$ will be given. Definition of $W_{2}^{5}$ and $W_{2}^{3}$ spaces and their kernel functions will be presented for obtaining reproducing kernel function of $W_{2}^{(5,3)}(\varOmega )$. Then, it will be explained how to obtain the kernel functions of $W_{2}^{(1,1)}(\varOmega )$ and $W_{2}^{(5,3)}(\varOmega )$ spaces.

$\mathbf {W}_{2}^{n}[0,1]$ space and its kernel function

$W_{2}^{n} \left[ {0,\,1} \right]$ space is described as

$$\begin{aligned} W_{2}^{n}[0,1]=\{w(\eta )|w,w^{'},...,w^{(n-1)} \,\ \hbox {are AC functions}, \,\ w^{(n)}\in L^2[0,1]\}. \end{aligned}$$

Inner product of $W_{2}^{n} \left[ {0,\,1} \right]$ can be written as:

$$\begin{aligned} \langle w\left( \eta \right) ,p\left( \eta \right) \rangle _{W_{2}^{n}} =\sum _{i=0}^{n-1}w^{(i)}(0)p^{(i)}(0)+\int \limits _0^1 {w^{(n)}(\eta )p^{(n)}(\eta )d\eta }, \end{aligned}$$

(73)

and norm of $W_{2}^{n} \left[ {0,\,1} \right]$ is given as follow:

$$\begin{aligned} \left\| \omega \right\| _{W_{2}^{n}}^2 =\langle \omega ,\omega \rangle _{W_{2}^{n}} ,\,\,\,\omega ,\sigma \in W_{2}^{n} \left[ {0,\,1} \right] . \end{aligned}$$

(74)

It will be given some kernel functions for different initial-boundary conditions and special cases.

Case I: $n=1$;

$$\begin{aligned} K_y^{\{1\}} \left( \eta \right) = \left\{ {{\begin{array}{*{20}c} {1 + \eta , \,\,\, \eta \le y,} \\ {1 + y , \,\,\, y >\eta ,} \\ \end{array} }} \right. \end{aligned}$$

(75)

and

$$\begin{aligned} K_z^{\{1\}} \left( \zeta \right) = \left\{ {{\begin{array}{*{20}c} {1 + \zeta , \,\,\, \zeta \le z,} \\ {1 + z , \,\,\, y >\zeta .} \\ \end{array} }} \right. \end{aligned}$$

(76)

Case II: $n=3$ and $w(0)={w}'(0)=0$

$$\begin{aligned} K_z^{\{3\}}(\zeta ) = \left\{ {{\begin{array}{*{20}c} {\sum _{i=1}^{6}c_i(z)\zeta ^{i-1},\,\,\,\zeta \le z,} \\ \\ {\sum _{i=1}^{6}d_i(z)\zeta ^{i-1},\,\,\,\zeta > z,} \\ \end{array} }} \right. \end{aligned}$$

(77)

For $\zeta \le z$, the kernel function $K_z^{\{3\}}(\zeta )$ is given as:

$$\begin{aligned} \frac{1}{4}z^2\zeta ^2+\frac{1}{12}z^2\zeta ^3-\frac{1}{24}z\zeta ^4+\frac{1}{120}\zeta ^5 \end{aligned}$$

(78)

Case III: $n=5$ and $w(0)=w(1)={w}''(0)={w}''(1)=0$

$$\begin{aligned} K_y^{\{5\}}(\eta ) = \left\{ {{\begin{array}{*{20}c} {\sum _{i=1}^{10}c_i(y)\eta ^{i-1},\,\,\,\eta \le y,} \\ \\ {\sum _{i=1}^{10}d_i(y)\eta ^{i-1},\,\,\,\eta > y,} \\ \end{array} }} \right. \end{aligned}$$

(79)

For $\eta \le y$, the kernel function $K_y^{\{5\}}(\eta )$ is given as:

$$\begin{aligned}&((2969/1654073)y-(91/33081460)y^9+(819/33081460)y^8 \\&\quad -(579/8270365)y^7+(231/8270365)y^6+(4347/16540730)y^5\\&\quad +(4347/3308146)y^4-(5544/1654073)y^3)\eta +((127013/19848876)y^3\\&\quad -(91723/3572797680)y^7+(636517/3572797680)y^6\\&\quad -(635131/1190932560)y^5-(635131/238186512)y^4-(11/1190932560)y^9\\&\quad +(11/132325840)y^8-(5544/1654073)y)\eta ^3\\&\quad +((222389/158791008)y^4-(88249/28582381440)y^7\\&\quad +(635131/28582381440)y^6-(319739/4763730240)y^5\\&\quad -(635131/238186512)y^3+(23/6351640320)y^9-(69/2117213440)y^8\\&\quad +(4347/3308146)y)\eta ^4+((222389/793955040)y^4\\&\quad -(88249/142911907200)y^7+(635131/142911907200)y^6\\&\quad -(319739/23818651200)y^5-(635131/1190932560)y^3\\&\quad +(23/31758201600)y^9-(69/10586067200)y^8+(4347/16540730)y)\eta ^5\\&\quad +(-(127013/2381865120)y^3\\&\quad +(91723/428735721600)y^7-(636517/428735721600)y^6\\&\quad +(635131/142911907200)y^5+(635131/28582381440)y^4\\&\quad +(11/142911907200)y^9-(11/15879100800)y^8+(231/8270365)y)x^6\\&\quad +((1/10080)y^2-(105619/3001150051200)y^7\\&\quad +(91723/428735721600)y^6-(88249/142911907200)y^5\\&\quad -(88249/28582381440)y^4-(91723/3572797680)y^3\\&\quad -(193/1000383350400)y^9+(193/111153705600)y^8\\&\quad -(579/8270365)y)\eta ^7+(-(2969/66692223360)y\\&\quad +(13/190549209600)y^9-(13/21172134400)y^8+(193/111153705600)y^7\\&\quad -(11/15879100800)y^6-(69/10586067200)y^5\\&\quad -(69/2117213440)y^4+(11/132325840)y^3)\eta ^8+(-(13/1714942886400)y^9\\&\quad +(13/190549209600)y^8\\&\quad -(193/1000383350400)y^7+(11/142911907200)y^6\\&\quad +(23/31758201600)y^5+(23/6351640320)y^4\\&\quad -(11/1190932560)y^3-(91/33081460)y+1/362880)\eta ^9 \end{aligned}$$

For obtaining of coefficients $c_i(y)$ and $d_i(y)$, please see [20].

Note: The reproducing kernel function of $W_{2}^{(5,3)}(\varOmega )$ is found by multiplying of kernel functions $K_z^{\{3\}}(\zeta )$ and $K_y^{\{5\}}(\eta )$. Similarly, the kernel function of $W_{2}^{(1,1)}(\varOmega )$ is found by multiplying of kernel functions $K_z^{\{1\}}(\zeta )$ and $K_y^{\{1\}}(\eta )$.