Spectral Collocation Methods for Fractional Integro-Differential Equations with Weakly Singular Kernels

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 L 2 norm. Finally, some numerical examples are considered to verify the obtained theoretical results.


Introduction
Fractional integro-di erential equations (FIDEs) have been frequently utilized in modeling real phenomena in earthquake engineering, statistical mechanics, thermal systems, control theory, astronomy, turbulence, and other application elds and have attracted more and more attention among researchers. Note that, obtaining an analytical solution of FIDEs is very di cult and sometimes even impossible. erefore, e ective numerical methods have been widely used for solving this kind of equations in recent years, such as fractional di erential transform methods [1], Taylor expansion [2], operational Tau methods [3], Adomian decomposition methods [4], spline collocation methods [5], wavelets [6][7][8], piecewise polynomial collocation methods [9], and Laplace decomposition methods [10]. Recently, the kernels methods have also received much attention, for the detail, see [11][12][13]. It is well known, that fractional di erential 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 su ciently 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 in nity norm and weighted L 2 norm. Here, we consider the following initial value problems for FIDEs in the form: where y(t) is the unknown function to be determined, f(t), a(t) and K(t, s) are known smooth functions on their respective domains, y (i) 0 (i � 0, 1, . . . , n − 1) are given real numbers, n: � μ 1 is the smallest integer which is bigger than the real number μ 1 . In this paper, D μ 1 denotes the Caputo fractional derivative of order μ 1 defined as follows: where Γ(·) denotes the Gamma function, and is called the Riemann-Liouville fractional integral of order μ 1 .
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 equation where μ � μ 1 − μ 1 and From [20], we known that the m-th derivative of the solution y(t) of (5) behaves like y (m) (t) ∼ t 1− m− μ as t ⟶ 0, which indicates that y ∉ C m [0, T]. To eliminate this singularity of the solution, we introduce the smoothing transformations (see [21][22][23]) where q is a positive integer number, then equation (5) becomes It is easy to see that the solution of Equation (8) satisfies Y (m) (z) ∼ z q(1− m)− μ as z ⟶ 0. en, by choosing a suitable q, we can obtain the regularity of Y(z) as we like. erefore, spectral methods can be applied for solving the resulting equation. e 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. e 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.

Jacobi Spectral Collocation Method
In this section, we derive the Jacobi spectral collocation method with the numerical implementation. Set ω α,β (x) � (1 − x) α (1 + x) β be a weight function in the usual sense, for α, β > − 1. As described in [24], the set of Jacobi polynomials J α,β 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 N, we denote the collocation points by x i N i�0 which is the set of (N + 1) Jacobi-Gauss points corresponding to the weight functions ω α,β (x). In this paper, we take the special collocation points x i N i�0 which are respect to the case that α � β � − μ. Obviously, (13) holds Using the linear transformation yields Applying the Jacobi-Gauss integration formula, we have the following equation: Where θ k N k�0 and ω k N k�0 denote the Jacobi-Gauss points and the weights with respect to the weight function ω − μ,0 (θ), respectively.
Let P N denote the space of all polynomials of degree not exceeding N. We use u i to indicate the approximate values for u(x i ), 0 ≤ i ≤ N. en, the Jacobi collocation method is to seek an approximate solution u N (x) ∈ P N of the following form: where are the Lagrange interpolation basis functions associated with x i N i�0 , and u i N i�0 are determined by the following discrete collocation equations:

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 m ≥ 1, we introduce a Sobolev space equipped with the norm and seminorm Hereafter, we use C to denotes a positive constant which is independent of N and may have different values in different occurrences.
Lemma 4 (see [27,28]). Suppose L ≥ 0, 0 < μ < 1, and let v(t) be a non-negative and locally integrable function defined on Then, we have the following equation: Lemma 5 (see [29]). For any measurable function f ≥ 0, the following generalized Hardy's inequality holds if and only if for the case 1 < p ≤ q < ∞. Here, T is an operator of the form with a given kernel R(x, t), weight functions � ω, ω, and − ∞ ≤ a < b ≤ ∞.

Convergence Analysis
In this section, we devote to analyzing the convergence of the approximation method (20). e goal is to show that the proposed method possesses spectral accuracy in the infinity norm and weighted L 2 norms.

Theorem 1. Let u(x) be the exact solution of equation (13). Assume that u N (x) is the numerical solution obtained by the proposed spectral method. If
, then for sufficiently large N, Journal of Mathematics Proof. 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: where From Lemma 1, we have the following equation: Subtracting (38) from (15) yields the error equation: where e(x) � u(x) − u N (x). Multiplying F i (x) on both sides of (41) and summing up from i � 0 to N give that For convenience, we define integral operators M as follows: Consequently, where It follows from (44) and Lemma 4 that By applying Lemma 2 and the estimate (40), we obtain that Using Lemma 3 gives the following equation: From [33], we known that M are linear and compact operators from C[− 1, 1] into C 0,κ [− 1, 1]. is implies that for any function v ∈ C[− 1, 1], there exists a positive constant C such that Journal of Mathematics Hence, it follows from (35) and Lemma 2 that Now, we can choose 1/2 − μ < κ < 1 − μ when 0 < μ < (1/2), and 0 < κ < 1 − μ when (1/2) ≤ μ < 1, then a combination of (46)-(48) and (50) yields the desired estimate (36), provided that N is sufficiently large.

Theorem 2.
If the hypotheses given in eorem 1 hold, then for any κ ∈ (0, 1 − μ) and for sufficiently large N, we have the following error estimate Proof. By Lemma 4 and Lemma 5, it follows from (44) that Now, using (40) and Lemma 6 gives the following equation: Using Lemma 3, we obtain that Finally, it follows from (35), (49) and Lemma 6 that Therefore, the estimate (51) is obtained by combining (36), (52)-(55), provided that N is sufficiently large.

Numerical Experiments
In this section, we present some numerical experiments to confirm the efficiency and accuracy of the suggested spectral method.
Equation (56) has nonsmooth solution y(t) � t 2/3 . We introduce the smoothing transformations t � z 3 , σ � w 3 to the equivalent Volterra integral equation and implement the spectral collocation method to solve the transformed equation. e obtained L ∞ and L ω − μ,− μ 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 con rmed by spectral accuracy. is is in accordance with our theoretical results.
Example 2. Consider the following fractional integro-differential equation with weakly kernels: e exact solution of above equation is given by y(t) t 11/6 . It is very clear, y(t) is not smooth at t 0.
Similar to the previous example, we introduce the smoothing transformations t z 6 , σ w 6 to the     Table 2 and Figure 2 for di erent values of N. Again we can see the spectral approximation gives spectral accuracy.

Conclusion
In this work, we have elaborated a Jacobi spectral collocation approximation for fractional integro-di erential equation with weakly kernels and with nonsmooth solutions. e converge analysis in L ∞ norm and L 2 ω − μ,− μ 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 e author declares that the data supporting the ndings of this study are available within the article.