Abstract

In this article, barycentric interpolation collocation method (BICM) is presented to solve the fractional linear Fredholm-Volterra integro-differential equation (FVIDE). Firstly, the fractional order term of equation is transformed into the Riemann integral with Caputo definition, and this integral term is approximated by the Gauss quadrature formula. Secondly, the barycentric interpolation basis function is used to approximate the unknown function, and the matrix equation of BICM is obtained. Finally, several numerical examples are given to solve one-dimensional differential equation.

1. Introduction

The concept of the fractional calculus dates back to 1695. Fractional differential equations, as a generalization of integer differential equations, are suitable for describing materials and processes with genetic and memory properties. Compared with integer order model, fractional order model can simulate dynamic system and natural physical phenomena more accurately. Fractional models are widely used in many fields, such as biological engineering [1–3], mechanics [4, 5], physics [6], electromagnetism [7, 8], viscoelastic system [9, 10], and heat conduction engineering [11]. Moreover, many researchers have proposed some efficient methods to investigate the existence and uniqueness of the solutions of fractional differential equations [12–18].

Lately, many researchers insinuated some standards to classify fractional differential operators. The notion of offering a guideline in a field was satisfactory enough, although the list of items that were suggested presented a limitation along with the critics brought up that were not academically acceptable. As a result of these criticisms, numerous researchers investigated the list along with their outcomes rejected the index law; in [19], their outcomes invalidated that inclusion of index law in the field. In another research work, the authors did overall investigation of the diffusive function of some kernel [20] and the outcomes they presented suggested that only operators with nonindex law properties can have crossover diffusive behaviors. However, Caputo and Fabrizio proved that the suggested index law was not right or it was a restriction to the field, and in their turn, they offered a list of items to be followed [21]. Further, they also proved the necessity of nonsingular differential operators along with their applications to nature applications to nature. In [22], the authors presented an optimal control of diffusion using the Atangana–Baleanu fractional differential operator. They proved that the existence of the solution with Atangana–Baleanu derivatives was obtained when the fractional order , and they also mentioned that the existence of the solution with Riemann-Liouville and Caputo was achieved during .

Furthermore, definitions of two well-known fractional derivatives, namely, Riemann-Liouville and Caputo [23], included a singular kernel. However, Caputo and Fabrizio introduced another definition having a nonsingular kernel and properties can be found in [24]. Another derivatives with nonsingular kernel were suggested in [25] which fundamentally generalized the Caputo and Fabrizio definition [26]. However, Riemann-Liouville fractional derivative be essential in the development of theory of fractional derivatives and integrals. But, this derivative barely able to generate physical interpretation of the initial conditions that are compulsory for the initial value issues containing fractional differential equations and also the boundary value issue both of the issues can be solved with the Caputo definition of fractional derivative for further details, refer [27]. Another difference is that the derivative of a constant is not zero for Riemann-Liouville, but it is equal to zero for Caputo. Additionally, the Riesz fractional derivatives have some shortcomings, such as it relies upon the values of whole interval also not sustaining the Leibniz rule for the product of two functions [28]. Besides, the Caputo fractional definition is easy to calculate and program. So the Caputo derivative is chosen in this manuscript.

In this paper, we mainly solve the FVIDE where is the Volterra part, is the Fredholm part, is the fractional derivative part, and the fractional derivative is defined as the Caputo definition as follows: where is the Gamma function.

The initial condition of one-dimensional differential equation is given as

In recent years, many methods are proposed to solve fractional differential equations. In [29], the Bell polynomials are introduced to solve fractional differential equations based on matrix and collocation points. In [30], the central difference and Crank-Nicolson method are used to obtain the full discrete scheme of spatial fractional convection-diffusion equation; then, the Richardson extrapolation method is used to further improve the calculation accuracy. In [31, 32], the finite element method is presented to solve fractional convection-diffusion equations. In [33–35], the element free Galerkin method is used to solve fractional differential equations. Compared with other algorithms, BICM has the advantages of high precision, easy programming, and simple formula. Therefore, this method has been applied to solve various equations, such as heat conduction equation [36], generalized Poisson equation [37], fractional differential equation [38], and fractional reaction-diffusion equation [39]. At the same time, the BICM is also utilized to solve some engineering problems, such as the plane elasticity problem [40], the bending problem of elliptic plate [41], and the numerical approximation of Darcy flow [42].

In this article, BICM is introduced to solve FVIDE. In Section 2, we provide relevant definitions of barycentric interpolation. In Sections 3–5, barycentric interpolation basis function is applied to approximate the unknown function, and matrix equations of the fractional derivative part, Volterra part, and Fredholm part are given. In Section 6, we obtain the matrix equation of FVIDE, and initial condition is dealt with by replacement method or additive method. In Section 7, some numerical examples are shown to prove feasibility of the algorithm.

2. Barycentric Interpolation

In this section, we will introduce barycentric interpolation for solving one-dimensional differential equation. First, equidistant nodes or Chebyshev’s nodes are chosen as collocation points on the domain, i.e., (), . The barycentric interpolation function is defined as

where and

According to different definitions of weight functions , barycentric interpolation can be divided into barycentric rational interpolation and barycentric Lagrange interpolation. The weight functions of barycentric Lagrange interpolation are defined as the weight functions of barycentric rational interpolation are defined as where , the parameter is integer, and .

3. Matrix Equation of Fractional Derivative Part

Fractional terms are dealt with by integration by parts; then, we get where .

Substituting equation (4) into equation (8), we obtain where

Let , formula (9) can be expressed as where

Let us write the integral term of the formula (11) as the following form: Then, we have

The integral term (12) is calculated using the Gauss quadrature formula with weights ; we get

where and are the Gauss points and Gauss weights and is the number of the Gauss points.

Using the Gauss-Legendre quadrature formula, equation (15) is given as

where and are integral points and integral weights, is the number of the integral points, is transformed coefficient, and .

Then, the formula (16) with the Gauss quadrature formula is obtained as where is the identity matrix and is the Kronecker product.

Briefly, the formula (16) can be written as where The relations between differential matrices and basis functions are defined as follows: where and

Hence, we can get

4. Matrix Equation of the Volterra Part

The Volterra part is expressed as ; equation (22) is shown as follows:

Substituting equation (4) into equation (22), we obtain where is defined as shown in equation (10).

is replaced by of formula (23), and we have where .

Formula (25) is expressed in the following form:

Using the Gauss quadrature formula with weights , we get where and are the Gauss points and Gauss weights and is the number of the Gauss points.

Formula (25) is calculated by the Gauss-Legendre quadrature formula, and we obtain where and are integral points and integral weights, is the number of the integral points, is transformed coefficient, and .

Combining equation (24), equation (25), and equation (26), equation (28) is obtained

Let ; formula (29) is obtained as follows: where

5. Matrix Equation of the Fredholm Part

The Fredholm part is expressed as the following form:

Substituting equation (4) into equation (31), we obtain where the definition of is as shown in equation (10).

Let , ; we have

Equation (34) is written as follows:

Formula (34) is calculated by the Gauss quadrature formula with weights ; we have where and are the Gauss points and Gauss weights and is the number of the Gauss points.

Using the Gauss-Legendre quadrature formula, we obtain where and are integral points and integral weights, is the number of the integral points, is transformed coefficient, and .

Formula (33) is calculated by the Gauss quadrature formula; equation (37) is written as follows:

Let ; formula (38) is obtained where

6. Matrix Equation for FVIDE

Equation (1) is treated by integration by parts; then, we get

Substituting equation (4) into equation (40), equation (41) is obtained

Taking , , we get