Solving a class of variable order nonlinear fractional integral differential equations by using reproducing kernel function

1.   Introduction

Fractional order models ^[1,2,3,4,5] have important applications in materials sciences, information science, and so on ^{[6,7,8,9,10,11,12,13,14]}. Reproducing kernel functions ^[15,16] in reproducing kernel Hilbert spaces ^{[17,18,19,20,21]} and related theory have important application in stochastic processes, signal analysis, machine learning and pattern recognition ^{[22,23,24,25,26,27,28,29,30,31,32]}. the reproducing kernel method ^{[6,7,8,9,10,11,12,13,14]} can not only obtain the exact solution in the form of series but also obtain the approximate solution with higher accuracy, the method has been widely used in linear and nonlinear problems, integral and differential equations, fractional partial differential equation and so on ^{[22,23,24,25,26,27,28,29,30,31,32]}. In ^[33], we use reproducing kernel interpolation collocation method to solve the linear integro differential equations of fractional order.

In this paper, reproducing kernel method with reproducing kernel function in the form of Jacobi polynomials is applied to solving the following variable fractional order nonlinear integral differential equations:

where $0 < \alpha(x)\leqslant 1$, $0 < \beta(x)\leqslant 1$, $f_n(x, u_1(x), u_2(x)), n = 1, 2$ and $k_{ij}(x, t)$, $i, j = 1, 2$ are given functions. ${D}^{\alpha(x)}u_1(x)$ indicates the $\alpha(x)$ is the Caputo fractional derivative defined of $u_1(x)$, ${D}^{\beta(x)}u_2(x)$ indicates the $\beta(x)$ is the Caputo fractional derivative defined of $u_2(x)$.

Definition 1.1. The Caputo fractional derivative operator of variable order $0 < \alpha(x) \leq1$ is defined as

2.   Reproducing kernel space

The well-known Jacobi polynomials are defined on the interval $[-1, 1]$ and can be generated with the aid of the following recurrence formula:

and

The weight function of Jacobi polynomials is $\omega(t) = (1-t)^\mu(1+t)^\nu, t\in [-1, 1]$. If $\mu = \nu = 0$, Jacobi polynomials are Legendre polynomials, if $\mu = \nu = -\frac{1}{2}$, Jacobi polynomials are the first chebyshev polynomials, if $\mu = \nu = \frac{1}{2}$, Jacobi polynomials are the second kind of Chebyshev polynomials. Some polynomials are shown in Figures 1–3.

In order to use these polynomials on the interval $x\in[0, 1]$. Let $t = 2x-1, t\in[-1, 1]$, the shifted Jacobi polynomials are denoted by $J_{i}^{\mu, \nu}(x)$. Then $J_{i}^{\mu, \nu}(x)$ can be generated from:

and

The analytic form of the shifted Jacobi polynomials $J_{i}^{\mu, \nu}(x)$ of degree $i$ is given by

and

Definition 2.1. Let

The inner product and norm are defined as:

where $\omega(x) = x^\nu(1-x)^\mu$ is a weight function. So, the shifted Jacobi polynomials have the following properties:

where

Definition 2.2. Let

Its the norm as same as the norm of $H_n[0, 1]$. From ^[33], we can prove that $H_n[0, 1]$ and $\bar H_n[0, 1]$ are two reproducing kernel Hilbert spaces. The reproducing kernel of $H_n[0, 1]$ is

The reproducing kernel of $\bar H_n[0, 1]$ is

Definition 2.3. The inner product space is defined as:

its inner product and norm are defined by

It is easy to verify that $\bar H_n[0, 1]\bigoplus\bar H_n[0, 1]$ is a Hilbert space with the definition of inner product (2.12).

Some reproducing kernels are shown in Table 1 and Figures 4–7.

3.   The reproducing kernel method

To solve Eq (1), let

So, Eq (1) can be turn into Eq (3.2).

where

The operator L: $\bar H_n[0, 1]\bigoplus \bar H_n[0, 1]\rightarrow H_1[0, 1]\bigoplus H_1[0, 1]$ is a bounded linear operator.

Assuming that $\{x_i\}_{i = 1}^\infty$ is dense on the interval $[0, 1]$, put $\phi_{ijk} = l_{ij}^*k_{x_k}(x),$ where $l_{ij}^*$ is the adjoint operator of $l_{ij}$. From ^[33], we have

Putting

Theorem 3.1. For each fixed $n$, $\{\Psi_{ij}\}_{(1, 1)}^{(n, 2)}$ is linearly independent in $\bar H_n[0, 1]\bigoplus\bar H_n[0, 1]$.

Theorem 3.2. $\{\Psi_{ij}\}_{(1, 1)}^{(\infty, 2)}$is complete in space in $\bar H_n[0, 1]\bigoplus\bar H_n[0, 1]$.

Proof. See of Theorems 3.1 and 3.2 in ^[33].

Using Theorems 3.1 and 3.2, the exact solution of Eq (1) can be expressed as

and truncate the infinite series of the analytic solution, we obtain the approximate solution of Eq (1).

If we can obtain the coefficients of each $\Psi_{ij}(x)$, the approximate solution $U_m(x)$ can be obtained also. Use $\Psi_{ij}(x)$ to do the inner products with both sides of Eq (3.5) and let $u_{1, 0}(x) = u_{2, 0}(x) = 0,$ we have

Letting

It is obvious that the inverse of $A_{2m}$ exists by Theorem 3.1. So, we have

Theorem 3.3. Let $U \in \bar H_n[0, 1]\bigoplus\bar H_n[0, 1]$ be the exact solution of Eq (1), $U_{m, n}$ be the solution of (3.6). If

and

$0 < c < 1$, then $U_{m, n}$ converges uniformly to $U$ ^[17].

4.   Numerical experiment

Example 1. We consider the following nonlinear integro-differential equations of fractional order ^[34].

(1) Where

The exact solution $u_1(x) = x$, $u_2(x) = -x$, The numerical results which of Example 1 for $m = 10, \mu = \nu = 0, n = 2$ are given in Table 2, and the absolute errors of this example for $m = 10, n = 2$ are given in Figures 8 and 9.

(2) Where

The exact solution $u_1(x) = x$, $u_2(x) = -x$. The absolute errors of Example 1 for $m = 10, \mu = \nu = 0, n = 2$ are given in Figures 10 and 11.

(3) Where

The exact solution $u_1(x) = x$, $u_2(x) = -x$. The absolute errors of Example 1 for $m = 10, \mu = \nu = 0, n = 2$ are given in Figures 12 and 13.

Example 2. We consider the following linear integro-differential equations of fractional order ^[34].

Where $\alpha = \beta = \frac{1}{2}$,

The exact solution $u_1(x) = x^2$, $u_2(x) = -x^2$. The absolute errors of Example 1 with $m = 10, n = 5$ are given in Tables 3 and 4 and Figures 14–19.

5.   Conclusions and remarks

In this paper, fractional nonlinear integro-differential equations of variable order have been solved by reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials. By comparing the obtained numerical solutions to the exact solutions of the fractional nonlinear integro-differential equations of variable order, it is indicated that our approach is powerful for variable order time fractional nonlinear integro-differential equations. All computations are performed by the Mathematica 7.0.

Acknowledgments

The work is supported by the Natural Science Foundation of Inner Mongolia [2021MS01009].

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this article.