New implementation of reproducing kernel Hilbert space method for solving a fuzzy integro-differential equation of integer and fractional orders

Let $y^F(x)\in R_F^n$ and $r\in [0\text{,}1]$ , and $R_F^n$ denotes the space of $n$ -dimensional fuzzy number. The $r$ -cut off $y^F(x)$ is the crisp set $[y^F(x)]$ that contains all elements with degree in $y^F(x)$ either greater than or equal to $r$ , that is; ${[y^F(x)]}^r=\{x\in R:y_F(x)⩾r\}$ , for fuzzy number $u_F(x)$ , its $r$ -cut is closed and bounded interval in $R$ and are denoted as follows: $${[y^F(x)]}^r=[y_{1\text{,}1r}(x)\text{,}{y}_{1\text{,}2r}(x)]\text{,}$$ where,

For more details, please refer to Buckley and Qu (1991), Arshad and Lupulescu (2011) and Yue et al. (1998).

The Triangular and trapezoidal fuzzy numbers respectively (Yue et al., 1998; Ahmad et al., 2013), are defined as follows:

where $\mathit{TRF}$ represents the triangular fuzzy and $y^{\mathit{TRF}}(x)\in R_F$ , and its $r$ -cut as follows:

where $\mathit{TLF}$ represents the triangular fuzzy and $y^{\mathit{TLF}}(x)\in R_F$ , and its $r$ -cut as follows:

Let HS { $y(x)|y(x)$ is a real value function or complex function, $x\in X$ is abstract set} is a Hilbert space, with inner product $\langle y(x)\text{,}g(x){\rangle }_{\mathit{HS}}$ , $(y(x)\text{,}g(x))\in \mathit{HS}$

if $\exists U_y(x)\in \mathit{HS}$ $\forall \text{fixed}y\in X\text{,}\text{then}{U}_y(x)\in \mathit{HS}$ and any $y(x)\in \mathit{HS}$ which satisfies the following: $$\langle y(x)\text{,}{U}_y(x)\rangle =y(y)\text{,}\text{for all}y\in X$$ then,

1. $U_y(x)$ is a reproducing kernel of $\mathit{HS}$ .

2. $\mathit{HS}$ is a reproducing kernel Hilbert space (RKHS).

Definition 4.2 (Cui and Lin, 2009)

The function space ${\mathit{FS}}_2^m[a\text{,}b]$ is defined as follows:

where $(i)$ denotes the order of derivative.

Definition 4.3 Cui and Lin, 2009

The inner product in the function space ${\mathit{FS}}_2^m[a\text{,}b]$ for any functions $y(x)\text{,}v(x)\in {\mathit{FS}}_2^1[a\text{,}b]$ is given by:

Definition 4.4 Cui and Lin, 2009

The norm in the function space ${\mathit{FS}}_2^m[a\text{,}b]$ for any functions $y(x)\text{,}v(x)\in {\mathit{FS}}_2^1[a\text{,}b]$ is defined as follows:

Definition 4.5 Cui and Lin, 2009

The inner product space of ${\mathit{FS}}_2^2[0\text{,}1]$ is defined as:

Definition 4.6 Cui and Lin, 2009

The norm in the function space ${\mathit{FS}}_2^2[0\text{,}1]$ for any functions $y(x)\text{,}v(x)\in {\mathit{FS}}_2^m[a\text{,}b]$ is defined as:

Theorem 4.1 Cui and Lin, 2009

${\mathit{FS}}_2^1[0\text{,}1]$ is $\mathit{RKHS}$ with reproducing kernel and the first reproducing kernel $(\mathit{FRK})$ given by:

Theorem 4.2 Cui and Lin, 2009

${\mathit{FS}}_2^2[0\text{,}1]$ is $\mathit{RKHS}$ with reproducing kernel and the second reproducing kernel $(\mathit{SRK})$ given by:

Let $\{x_1\text{,}{x}_2\text{,}\dots x_{\infty }\}={\{x_k\}}_{k=1}^{\infty }$ is a dence set in $[0\text{,}1]$ , Suppose that the inverse operator $L_{1\text{,}\mathit{jr}}^{-1}$ for Eq. (25) exists. Then,

$\forall y_{1\text{,}\mathit{jr}}^H(x)\in {\mathit{FS}}_2^2[0\text{,}1]$ , let $\langle y_{1\text{,}\mathit{jr}}^H(x)\text{,}{\mathrm{\Upsilon }}_k^{1\text{,}\mathit{jr}}(x)\rangle =0$ , for $k=1\text{,}2\text{,}\dots$ Then $$\begin{array}{cc}\langle y_{1\text{,}\mathit{jr}}^H(x)\text{,}{\mathrm{\Upsilon }}_{k\text{,}1\text{,}\mathit{jr}}(x){\rangle }_{{\mathit{FS}}_2^2[0\text{,}1]}& =\langle y_{1\text{,}\mathit{jr}}^H(x)\text{,}{L}_{1\text{,}\mathit{jr}}^{\mathit{ad}}{\phi }_{k\text{,}1\text{,}\mathit{jr}}(x){\rangle }_{{\mathit{FS}}_2^2[0\text{,}1]}=\langle L_{1\text{,}\mathit{jr}}{y}_{1\text{,}\mathit{jr}}^H(x)\text{,}{\phi }_{k\text{,}1\text{,}\mathit{jr}}(x){\rangle }_{{\mathit{FS}}_2^1[0\text{,}1]}\\ & =L_{1\text{,}\mathit{jr}}{y}_{1\text{,}\mathit{jr}}^H(x)\\ & =0\text{,}\end{array}$$

where ${\{x_k\}}_{k=1}^{\infty }$ is dense in $[0\text{,}1]$ , then, $L_{1\text{,}\mathit{jr}}{y}_{1\text{,}\mathit{jr}}^H(x)=0$

from the existence of inverse and the continuity of $y_{1\text{,}\mathit{jr}}^H(x)$ . □

Let $\{x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_{\infty }\}={\{x_k\}}_{k=1}^{\infty }$ is a dence set in $[0\text{,}1]$ , where the solution of Eq. (22) is unique on ${\mathit{FS}}_2^2[0\text{,}1]$ , then, the solution of these Eq. (22) is given by:

And

Note that ${\{{\bar{\mathrm{\Upsilon }}}_k^{1\text{,}\mathit{ir}}(x)\}}_{(k=1\text{,}i=1)}^{(\infty \text{,}2)}$ in Eq. (29) is a complete orthonormal basis of ${\mathit{FS}}_2^{m+1}[0\text{,}1]$ . Thus, $y_{1\text{,}\mathit{ir}}^H(x)$ can be expanded in the Fourier series about the orthonormal system aswhere $i=1$ : $$y_{1\text{,}\mathit{ir}}^H(x)={\displaystyle \sum _k^{\infty }}\langle y_{1\text{,}1r}^H(x)\text{,}{\bar{\mathrm{\Upsilon }}}_k^{1\text{,}\mathit{ir}}(x)\rangle {\mathrm{\Upsilon }}_k^{1\text{,}\mathit{jr}}(x)$$ $$\begin{array}{cc}{y}_{1\text{,}\mathit{ir}}^H(x)\hfill & ={\displaystyle \sum _{k=1}^{\infty }}\langle y_{1\text{,}\mathit{ir}}^H(x)\text{,}{\bar{\mathrm{\Upsilon }}}_k^{1\text{,}\mathit{ir}}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}\mathit{ir}}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}\langle y_{1\text{,}\mathit{ir}}^H(x)\text{,}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}\mathit{ir}}{\mathrm{\Upsilon }}_j^{1\text{,}\mathit{ir}}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}\mathit{ir}}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}1r}\langle y_{1\text{,}\mathit{ir}}^H(x)\text{,}{L}_{1\text{,}\mathit{ir}}^{\mathit{ad}}{\phi }_j^{1\text{,}\mathit{ir}}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}\mathit{ir}}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}1r}\langle L_{1\text{,}\mathit{ir}}{y}_{1\text{,}\mathit{ir}}^H(x)\text{,}{\phi }_j^{1\text{,}1r}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}\mathit{ir}}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}1r}〈g(x\text{,}{y}^H(x)+y_{1\text{,}1r}(x_0))+{\int }_{0^{+}}^{x}f(t\text{,}{y}^H(t)+y_{1\text{,}1r}(x_0))\mathit{dt}\text{,}{\phi }_j^{1\text{,}1r}(x)〉{\bar{\mathrm{\Upsilon }}}_k^{1\text{,}1r}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}1r}(g(x_j\text{,}{y}^H(x_j)+y_{1\text{,}1r}(x_0))+{\int }_{0^{+}}^{x_j}f\left(t\text{,}{y}^H(t)+y_{1\text{,}1r}(x_0)\right)\mathit{dt}){\bar{\mathrm{\Upsilon }}}_k^{1\text{,}1r}(x)\hfill \end{array}$$ Clearly, $y_{1\text{,}\mathit{ir}}^H(x)={\sum }_{k=1}^{\infty }{Q}_k^{1\text{,}\mathit{ir}}{\bar{\mathrm{\Upsilon }}}_k^{1\text{,}1r}\text{,}\mathit{fori}=1\text{.}$ where $i=2$ : $$y_{1\text{,}\mathit{ir}}^H(x)={\displaystyle \sum _k^{\infty }}\langle y_{1\text{,}\mathit{ir}}^H(x)\text{,}{\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}(x)\rangle {\mathrm{\Upsilon }}_k^{1\text{,}2r}(x)$$ $$\begin{array}{cc}{y}_{1\text{,}2r}^H(x)\hfill & ={\displaystyle \sum _{k=1}^{\infty }}\langle y_{1\text{,}\mathit{ir}}^H(x)\text{,}{\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}\langle y_{1\text{,}\mathit{ir}}^H(x)\text{,}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}\mathit{ir}}{\mathrm{\Upsilon }}_j^{1\text{,}\mathit{ir}}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}2r}\langle y_{1\text{,}2r}^H(x)\text{,}{L}_{1\text{,}\mathit{ir}}^{\mathit{ad}}{\phi }_j^{1\text{,}2r}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}2r}\langle L_{1\text{,}\mathit{ir}}{y}_{1\text{,}2r}^H(x)\text{,}{\phi }_j^{1\text{,}2r}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}\mathit{ir}}\langle g(x\text{,}{y}^H(x)+y_{1\text{,}2r}(x_0))+{\int }_{0^{+}}^{x}f\left(t\text{,}{y}^H(t)+y_{1\text{,}2r}(x_0)\right)\mathit{dt}\text{,}{\phi }_j^{1\text{,}2r}(x)\rangle {\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}(x)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \sum _{j=1}^k}{\beta }_{\mathit{kj}}^{1\text{,}2r}\left(g(x_j\text{,}{y}^H(x_j)+y_{1\text{,}2r}(x_0))+{\int }_{0^{+}}^{x_j}f(t\text{,}{y}^H(t)+y_{1\text{,}2r}(x_0))\mathit{dt}\right){\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}(x)\hfill \end{array}$$

Clearly, $y_{1\text{,}2r}^H(x)={\sum }_{k=1}^{\infty }{Q}_k^{1\text{,}2r}{\bar{\mathrm{\Upsilon }}}_k^{1\text{,}2r}\text{,}\text{for}i=2\text{.}$

Through the application of the formula in Eqs. (23), (31) and (32) leads to the next forms, respectively.

And

where $i=2$ .

Now, the approximate solution of $y_{1\text{,}\mathit{ir}}^H(t)$ can be obtained by taking finitely $N$ -terms in Eq. (30), which is given as follows:

For the approximate solution of Eq. (22), let discuss the following two cases.

Case 1. If the initial value is a $\mathit{TRF}$ , then:

And

Case 2. If the initial value is a $\mathit{TLF}$ , then:

And

Consider the following fuzzy integro-differential equations with integer order

$D_{c\text{,}{0}^{+}}^{\beta }y(x)=-2y(x)+5{\int }_0^{x}y(t)\mathit{dt}+1$ , when

$\beta =1,$

Solution.

The exact solution for the last example by Laplace transform method give by:

1. $$y_{1\text{,}1r}(x)=-\frac{e^{(-1-\sqrt{6})x}-e^{(-1+\sqrt{6})x}}{2\sqrt{6}}\text{,}$$ where $r=0.0$ .

2. $$y_{1\text{,}2r}(x)=\frac{1}{12}(-24+{12}^{(-1-\sqrt{6})x}+{\sqrt{6}}^{(-1-\sqrt{6})x}+{12}^{(-1+\sqrt{6})x}-{\sqrt{6}}^{(-1+\sqrt{6})x})+2\text{,}$$ where $r=0.0$ .

3. $$y_{1\text{,}1r}(x)=2.5(-0.2+0.0598e^{-3.4498x}+0.1408e^{1.4495x})+0.5\text{,}$$ where $r=0.5$ .

4. $$y_{1\text{,}2r}(x)=-2.0(-0.75-0.426e^{-3.4495x}-0.324e^{1.4495x})+1.5,$$ where $r=0.5$ .

5. $$y_{1\text{,}1r}(x)=y_{1\text{,}2r}(x)=\frac{1}{2}(-2+e^{(-1-\sqrt{6})x}+e^{(-1+\sqrt{6})x})+1\text{,}$$ where $r=1.0$ .

Approximate solution by $\mathit{RKHS}$ at $N=50$ is listed in Tables 1–5. From these tables, one can see the accuracy in comparing the approximate solution with the exact solution. These results show a convergence in results between the approximate solution and exact solution at different values of $r$ .

Consider the following fuzzy integro-differential equations with fractional order $\beta =0.9$