Abstract

In this paper, we propose and analyze a spectral approximation for the numerical solutions of fractional integro-differential equations with weakly kernels. First, the original equations are transformed into an equivalent weakly singular Volterra integral equation, which possesses nonsmooth solutions. To eliminate the singularity of the solution, we introduce some suitable smoothing transformations, and then use Jacobi spectral collocation method to approximate the resulting equation. Later, the spectral accuracy of the proposed method is investigated in the infinity norm and weighted norm. Finally, some numerical examples are considered to verify the obtained theoretical results.

1. Introduction

Fractional integro-differential equations (FIDEs) have been frequently utilized in modeling real phenomena in earthquake engineering, statistical mechanics, thermal systems, control theory, astronomy, turbulence, and other application fields and have attracted more and more attention among researchers. Note that, obtaining an analytical solution of FIDEs is very difficult and sometimes even impossible. Therefore, effective numerical methods have been widely used for solving this kind of equations in recent years, such as fractional differential transform methods [1], Taylor expansion [2], operational Tau methods [3], Adomian decomposition methods [4], spline collocation methods [5], wavelets [68], piecewise polynomial collocation methods [9], and Laplace decomposition methods [10]. Recently, the kernels methods have also received much attention, for the detail, see [1113]. It is well known, that fractional differential operators are nonlocal and have weakly singular kernels, and so global methods; for example, spectral methods, could be better suited for solving numerically FDIEs. In the past decades, some works are devoted to the spectral approximation of FIDEs with smooth kernels. [14] introduced a Chebyshev spectral collocation method for solving a general form of nonlinear FIDEs with linear functional arguments. In [15], a Chebyshev pseudospectral method was developed for FIDEs. Yang et al. proposed and analyzed a Jacobi spectral collocation approximation for FIDEs of Volterra type or Fredholm-Volterra type in [16, 17], and constructed a general spectral and pseudospectral Jacobi-Galerkin method for FIDEs of Volterra type in [18]. All of these works carried out the analyses under the assumption that the underlying solutions are smooth. However, even if the input functions are sufficiently smooth, the solutions of FDIEs are usually not smooth and will exhibit some weak singularity. And, Yang considered the case of nonsmooth solutions of FIDEs with smooth kernels in [19], where some smoothing transformations were introduced to eliminate the singularity of the solutions, then the Jacobi spectral collocation method was used to solve the transformed equation. Inspired by the work [19], in the present one, we intend to apply spectral methods to solve FIDEs with weakly kernels and with nonsmooth solutions. We will justify that the proposed numerical methods can achieve spectral accuracy in the infinity norm and weighted norm. Here, we consider the following initial value problems for FIDEs in the form:where is the unknown function to be determined, and are known smooth functions on their respective domains, are given real numbers, is the smallest integer which is bigger than the real number . In this paper, denotes the Caputo fractional derivative of order defined as follows:where denotes the Gamma function, andis called the Riemann–Liouville fractional integral of order .

Using the same methods as in the proof of Lemma 2 in [9], we can transform the original Equations (1) and (2) into an equivalent weakly singular Volterra integral equationwhere and

From [20], we known that the -th derivative of the solution of (5) behaves like as , which indicates that . To eliminate this singularity of the solution, we introduce the smoothing transformations (see [2123])where is a positive integer number, then equation (5) becomeswhere , , and

It is easy to see that the solution of Equation (8) satisfies as . Then, by choosing a suitable , we can obtain the regularity of as we like. Therefore, spectral methods can be applied for solving the resulting equation.

The structure of this paper is as follows: In Section 2, we construct a Jacobi spectral collocation approximation for Equation (8). Some elementary definitions and lemmas will be presented in Section 3, and convergence analysis of the proposed approximation will be carried out in Section 4. The numerical experiments are carried out in Section 5, which will be used to verify the theoretical results obtained in Section 4. Finally, Section 6 outlines the conclusions.

2. Jacobi Spectral Collocation Method

In this section, we derive the Jacobi spectral collocation method with the numerical implementation. Set be a weight function in the usual sense, for . As described in [24], the set of Jacobi polynomials forms a complete -orthogonal system, where is a weighted space defined by the following equation:which is equipped with the following inner product and norm:

For the sake of implementing the spectral methods naturally, we take the linear transformations.

and so (8) read as follows:

Here,

For a given positive integer , we denote the collocation points by which is the set of Jacobi–Gauss points corresponding to the weight functions . In this paper, we take the special collocation points which are respect to the case that . Obviously, (13) holds at , i.e.,

Using the linear transformation

yields

Applying the Jacobi-Gauss integration formula, we have the following equation:Where and denote the Jacobi–Gauss points and the weights with respect to the weight function , respectively.

Let denote the space of all polynomials of degree not exceeding . We use to indicate the approximate values for . Then, the Jacobi collocation method is to seek an approximate solution of the following form:where are the Lagrange interpolation basis functions associated with , and are determined by the following discrete collocation equations:

3. Some Useful Lemmas

In this section, we introduce some important definitions and lemmas, which will be used to study the properties of the proposed numerical method later.

For an integer , we introduce a Sobolev spaceequipped with the normand seminorm

Hereafter, we use to denotes a positive constant which is independent of and may have different values in different occurrences.

Lemma 1 (see [24]). Assume that Gauss quadrature formula relative to the Jacobi weight is used to integrate the product , where for some and , thenwhere represents the discrete inner product in space and

Lemma 2 (see [25]). Assume that is the set of -th Lagrange interpolation polynomials associated with the Jacobi-Gauss points , then

Lemma 3 (see [26]). Assume that and denote by its interpolation polynomial associated with the Jacobi–Gauss points , namely,Then, the following estimates hold:where .

Lemma 4 (see [27, 28]). Suppose , and let be a non-negative and locally integrable function defined on satisfyingThen, we have the following equation:

Lemma 5 (see [29]). For any measurable function , the following generalized Hardy’s inequalityholds if and only iffor the case . Here, is an operator of the formwith a given kernel , weight functions , and .

Lemma 6 (see [30]). For every bounded function , we have the following equation:For nonnegative integer and , we denote by the set of continuous function whose -th derivatives are H lder continuous with exponent , equipped with norm (see [26]). From [31, 32], we know that there exists a constant such that for any function , there exists a polynomial function such that

4. Convergence Analysis

In this section, we devote to analyzing the convergence of the approximation method (20). The goal is to show that the proposed method possesses spectral accuracy in the infinity norm and weighted norms.

Theorem 1. Let be the exact solution of equation (13). Assume that is the numerical solution obtained by the proposed spectral method. If , then for sufficiently large ,whereProof. By using (16) and the definition of the weighted inner product and discrete inner product (25), we can rewrite the numerical method (20) as follows:whereFrom Lemma 1, we have the following equation:Subtracting (38) from (15) yields the error equation:where . Multiplying on both sides of (41) and summing up from to give thatFor convenience, we define integral operators as follows:Consequently,whereIt follows from (44) and Lemma 4 thatBy applying Lemma 2 and the estimate (40), we obtain thatUsing Lemma 3 gives the following equation:From [33], we known that are linear and compact operators from into . This implies that for any function , there exists a positive constant such thatHence, it follows from (35) and Lemma 2 thatNow, we can choose when , and when , then a combination of (46)–(48) and (50) yields the desired estimate (36), provided that is sufficiently large.

Theorem 2. If the hypotheses given in Theorem 1 hold, then for any and for sufficiently large , we have the following error estimateProof. By Lemma 4 and Lemma 5, it follows from (44) thatNow, using (40) and Lemma 6 gives the following equation:Using Lemma 3, we obtain thatFinally, it follows from (35), (49) and Lemma 6 thatTherefore, the estimate (51) is obtained by combining (36), (52)–(55), provided that is sufficiently large.

5. Numerical Experiments

In this section, we present some numerical experiments to confirm the efficiency and accuracy of the suggested spectral method.

Example 1. Consider the following fractional integro-differential equation with weakly kernels:Equation (56) has nonsmooth solution . We introduce the smoothing transformations to the equivalent Volterra integral equation and implement the spectral collocation method to solve the transformed equation. The obtained and errors are presented in Table 1. We also plotted the numerical errors in Figure 1. As we can see from Table 1 and Figure 1, the proposed spectral method converges rapidly, which is confirmed by spectral accuracy. This is in accordance with our theoretical results.

Example 2. Consider the following fractional integro-differential equation with weakly kernels:The exact solution of above equation is given by . It is very clear, is not smooth at . Similar to the previous example, we introduce the smoothing transformations to the corresponding Volterra integral equation, and apply the spectral collocation method to approximate the transformed equation. The numerical errors are demonstrated in Table 2 and Figure 2 for different values of . Again we can see the spectral approximation gives spectral accuracy.

6. Conclusion

In this work, we have elaborated a Jacobi spectral collocation approximation for fractional integro-differential equation with weakly kernels and with nonsmooth solutions. The converge analysis in norm and norm was established for the approximation method. Two numerical test examples with nonsmooth solutions was presented to illustrate the spectral accuracy of the proposed method.

Data Availability

The author declares that the data supporting the findings of this study are available within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was supported by the Characteristic Innovation Project of Universities in Guangdong, China (Grant no. 2021KTSCX142).