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

Authors: Xiulian Shi
Journal: Journal of Mathematics
 (2022)

---

## 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 weightedL2 norm. Finally, some numerical examples are considered to verify the obtained theoretical results.


---

## Body

## 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 [6–8], piecewise polynomial collocation methods [9], and Laplace decomposition methods [10]. Recently, the kernels methods have also received much attention, for the detail, see [11–13]. 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 L2 norm. Here, we consider the following initial value problems for FIDEs in the form:(1)Dμ1yt=ft+atyt+∫0tt−s−μ2Kt,sysds,μ1>0,0<μ2<1,t∈0,T,(2)yi0=y0i,i=0,1,…,n−1,where yt is the unknown function to be determined, ft,at and Kt,s are known smooth functions on their respective domains, y0ii=0,1,…,n−1 are given real numbers, n:=⌈μ1⌉ is the smallest integer which is bigger than the real number μ1. In this paper, Dμ1 denotes the Caputo fractional derivative of order μ1 defined as follows:(3)Dμ1yt=Jn−μ1ynt=1Γn−μ1∫0tt−sn−μ1−1ynsds,t>0,where Γ⋅ denotes the Gamma function, and(4)Jμ1yt=1Γμ1∫0tt−sμ1−1ysds,is called the Riemann–Liouville fractional integral of order μ1.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(5)yt=f¯t+∫0tt−σ−μK¯t,σyσdσ,where μ=⌈μ1⌉−μ1 and(6)f¯t=∑k=0n−1yk0tkk!+1Γμ1∫0tt−σμ1−1fσdσ,K¯t,σ=1Γμ1t−σ⌈μ1⌉−1aσ+t−σ1−μ2∫011−τμ1−1τ−μ2Kτt−σ+σ,σdτ.From [20], we known that the m-th derivative of the solution yt of (5) behaves like ymt∼t1−m−μ as t⟶0, which indicates that y∉Cm0,T. To eliminate this singularity of the solution, we introduce the smoothing transformations (see [21–23])(7)t=γz=zq,σ=γw=wq,where q is a positive integer number, then equation (5) becomes(8)Yz=f˜z+∫0tz−w−μK˜z,wYwdw,where z∈0,Tq, Yz=yγz, f˜z=f¯γz and(9)K˜z,w=γz−γwz−w−μγ′wK¯γz,γw,z≠w,γ′z−μγ′wK¯γz,γw,z=w.It is easy to see that the solution of Equation (8) satisfies Ymz∼zq1−m−μ as z⟶0. Then, by choosing a suitable q, we can obtain the regularity of Yz as we like. Therefore, spectral methods can be applied for solving the resulting equation.The structure of this paper is as follows: In Section2, 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. Setωα,βx=1−xα1+xβ be a weight function in the usual sense, for α,β>−1. As described in [24], the set of Jacobi polynomials Jnα,βxn=0∞ forms a complete Lωα,β2−1,1-orthogonal system, where Lωα,β2−1,1 is a weighted space defined by the following equation:(10)Lωα,β2−1,1=v:vis measurable andvωα,β<∞,which is equipped with the following inner product and norm:(11)u,vωα,β=∫−11uxvxωα,βxdx,uωα,β=u,uωα,β1/2.For the sake of implementing the spectral methods naturally, we take the linear transformations.(12)z=Tq21+x,−1≤x≤1,w=Tq21+ξ,−1≤ξ≤x,and so (8) read as follows:(13)ux=f^x+∫−1xx−ξ−μK^x,ξuξdξ.Here,(14)ux=YTq21+x,f^x=f˜Tq21+x,K^x,ξ=Tq21−μK˜Tq21+x,Tq21+ξ.For a given positive integerN, we denote the collocation points by xii=0N which is the set of N+1 Jacobi–Gauss points corresponding to the weight functions ωα,βx. In this paper, we take the special collocation points xii=0N which are respect to the case that α=β=−μ. Obviously, (13) holds at xi,i=0,…,N, i.e.,(15)uxi=f^xi+∫−1xixi−ξ−μK^xi,ξuξdξ.Using the linear transformation(16)ξ=ξiθ=1+xi2θ+xi−12,−1≤θ≤1.yields(17)uxi=f^xi+1+xi21−μ∫−111−θ−μK^xi,ξiθuξiθdθ.Applying the Jacobi-Gauss integration formula, we have the following equation:(18)uxi=f^xi+1+xi21−μ∑k=0NK^xi,ξiθkuξiθkωk.Where θkk=0N and ωkk=0N denote the Jacobi–Gauss points and the weights with respect to the weight function ω−μ,0θ, respectively.LetℙN denote the space of all polynomials of degree not exceeding N. We use ui to indicate the approximate values for uxi,0≤i≤N. Then, the Jacobi collocation method is to seek an approximate solution uNx∈ℙN of the following form:(19)ux≈uNx=∑j=0NujFjx,where Fixi=0N are the Lagrange interpolation basis functions associated with xii=0N, and uii=0N are determined by the following discrete collocation equations:(20)ui=f^xi+1+xi21−μ∑j=0Nuj∑k=0NK^xi,ξiθkFjξiθkωk.


## 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 integerm≥1, we introduce a Sobolev space(21)Hωα,βm−1,1:=vx:vpx∈Lωα,β2−1,1,0≤p≤m,equipped with the norm(22)vm,ωα,β=∑p=0mvpxωα,β21/2,and seminorm(23)vωα,βm;N=∑p=minm,N+1mvpxωα,β21/2.Hereafter, we useC to denotes a positive constant which is independent of N 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 productvϕ, where v∈Hωα,βm−1,1 for some m≥1 and ϕ∈ℙN, then(24)v,ϕωα,β−v,ϕN≤CN−mvωα,βm;Nϕωα,β,where ⋅,⋅N represents the discrete inner product in Lωα,β2−1,1 space and(25)v,ϕN=∑k=0Nvxkϕxkωk.Lemma 2 (see [25]).
Assume thatFjxj=0N is the set of N-th Lagrange interpolation polynomials associated with the Jacobi-Gauss points xii=0N, then(26)/Nα,β∞:=maxx∈−1,1∑j=0NFjx=OlogN,−1<α,β≤−12,ONλ+1/2,λ=maxα,β,otherwise.Lemma 3 (see [26]).
Assume thatv∈Hωα,βm−1,1 and denote by ℐNα,βvx its interpolation polynomial associated with the Jacobi–Gauss points xii=0N, namely,(27)/Nα,βvx=∑i=0NvxiFix.
Then, the following estimates hold:(28)v−/Nα,βvωα,β≤CN−mvωα,βm;N,v−/Nα,βv∞≤CN1/2−mlogNvωcm;N,−1<α,β≤−12,CN1+λ−mvωcm;N,λ=maxα,β,otherwise,where ωc=ω−1/2,−1/2 .Lemma 4 (see [27, 28]).
SupposeL≥0,0<μ<1, and let vt be a non-negative and locally integrable function defined on 0,T satisfying(29)vt≤wt+L∫0tt−s−μvsds.
Then, we have the following equation:(30)vt≤wt+C∫0tt−s−μwsds.Lemma 5 (see [29]).
For any measurable functionf≥0, the following generalized Hardy’s inequality(31)∫abTfxqωˇxdx1/q≤C∫abfxpω^xdx1/p,holds if and only if(32)supa<x<b∫xbωˇtdt1/q∫axω^tv1−p′tdt1/p′<∞,p′=pp−1,for the case 1<p≤q<∞. Here, T is an operator of the form(33)Tfx=∫axRx,tftdt,with a given kernel Rx,t, weight functions ωˇ,ω^, and −∞≤a<b≤∞.Lemma 6 (see [30]).
For every bounded functionvx , we have the following equation:(34)supN/Nα,βvωα,β≤Cvx∞.
For nonnegative integerr and κ∈0,1, we denote by Cr,κ−1,1 the set of continuous function v:−1,1⟶ℝ whose r-th derivatives are H o.. lder continuous with exponent κ, equipped with norm ⋅r,κ (see [26]). From [31, 32], we know that there exists a constant Cr,κ>0 such that for any function v∈Cr,κ−1,1, there exists a polynomial function ℱNv∈ℙN such that(35)v−ℱNv∞≤Cr,κN−r+κvr,κ.


## 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 L2 norms.Theorem 1.
Letux be the exact solution of equation (13). Assume that uNx is the numerical solution obtained by the proposed spectral method. If u∈Hω−μ,−μm−1,1∩Hωcm−1,1m≥1, then for sufficiently large N,(36)u−uN∞≤CN1/2−μ−mK∗u∞+N1/2uωcm;N,0<μ<12,CN−mlogNK∗u∞+N1/2uωcm;N,12≤μ<1,where(37)K∗=max0≤i≤NK^xi,ξi⋅ω−μ,0m;N.
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:(38)ui=f^xi+∫−1xixi−ξ−μK^xi,ξuNξdξ−J1i,where(39)J1i=1+xi21−μK^xi,ξi⋅,uNξi⋅ω−μ,0−1+xi21−μK^xi,ξi⋅,uNξi⋅N.
From Lemma1, we have the following equation:(40)J1i≤CN−mK^xi,ξi⋅ω−μ,0m;NuNξi⋅ω−μ,0.
Subtracting (38) from (15) yields the error equation:(41)uxi−ui=∫−1xixi−ξ−μK^xi,ξeξdξ+J1i,where ex=ux−uNx. Multiplying Fix on both sides of (41) and summing up from i=0 to N give that(42)IN−μ,−μux−uNx=IN−μ,−μ∫−1xx−ξ−μK^x,ξeξdξ+∑i=0NJ1iFix.
For convenience, we define integral operatorsℳ as follows:(43)ℳe=∫−1xx−ξ−μK^x,ξeξdξ.
Consequently,(44)ex=∫−1xx−ξ−μK^x,ξeξdξ+∑l=13Jlx,where(45)J1x=∑i=0NJ1iFix,J2x=ux−IN−μ,−μux,J3x=IN−μ,−μℳe−ℳe.
It follows from (44) and Lemma 4 that(46)e∞≤C∑l=13Jl∞.
By applying Lemma2 and the estimate (40), we obtain that(47)J1∞=∑i=0NJ1iFix∞≤Cmax1≤i≤NJ1imaxx∈−1,1∑j=0NFjx≤CN1/2−μ−mmax1≤i≤NK^xi,ξi⋅ω−μ,0m;Ne∞+u∞,0<μ<12,CN−mlogNmax1≤i≤NK^xi,ξi⋅ω−μ,0m;Ne∞+u∞,12≤μ<1.
Using Lemma3 gives the following equation:(48)J2∞=ux−IN−μ,−μux∞≤CN1−μ−muωcm;N,0<μ<12,CN1/2−mlogNuωcm;N,12≤μ<1.
From [33], we known that ℳ are linear and compact operators from C−1,1 into C0,κ−1,1. This implies that for any function v∈C−1,1, there exists a positive constant C such that(49)ℳv0,κ=Cv∞,0<κ<1−μ.
Hence, it follows from (35) and Lemma 2 that(50)J3∞=IN−μ,−μℳe−ℳe∞=IN−μ,−μ−Iℳe−ℱNℳe∞≤1+IN−μ,−μ∞ℳe−ℱNℳe∞≤1+IN−μ,−μ∞CN−κℳe0,κ≤CN12−μ−κe‖∞,0<μ<12,CN−κlogNe∞,12≤μ<1.
Now, we can choose1/2−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 Theorem1 hold, then for any κ∈0,1−μ and for sufficiently large N, we have the following error estimate(51)u−uNω−μ,−μ≤CN−mK∗u∞+uω−μ,−μm;N+N1−μ−κuωcm;N,0<μ<12,12−μ<κ<1−μ,CN−mK∗u∞+uω−μ,−μm;N+N12−κlogNuωcm;N,12≤μ<1,0<κ<1−μ.
Proof. By Lemma4 and Lemma 5, it follows from (44) that(52)eω−μ,−μ≤C∑l=13Jlxω−μ,−μ.
Now, using (40) and Lemma 6 gives the following equation:(53)J1ω−μ,−μ=∑i=0NJ1iFixω−μ,−μ≤Cmax1≤i≤NJ1i≤CN−mmax1≤i≤NK^xi,ξi⋅ω−μ,0m;Ne∞+u∞.
Using Lemma3, we obtain that(54)J2ω−μ,−μ=ux−IN−μ,−μuω−μ,−μ≤CN−muxω−μ,−μm;N.
Finally, it follows from (35), (49) and Lemma 6 that(55)J3ω−μ,−μ=IN−μ,−μℳe−ℳeω−μ,−μ=IN−μ,−μ−Iℳe−ℱNℳeω−μ,−μ≤IN−μ,−μℳe−ℱNℳeω−μ,−μ+ℳe−ℱNℳeω−μ,−μ≤Cℳe−ℱNℳe∞≤CN−κℳe0,κ≤CN−κe∞.
Therefore, the estimate (51) is obtained by combining (36), (52)–(55), provided that N 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:(56)D2/3yt=Γ53+t11/3−Γ5/3Γ199/200Γ1597/600t997/600−t3yt+∫0tt−s−1/200ysds,t∈0,10,y0=0.
Equation (56) has nonsmooth solution yt=t2/3. We introduce the smoothing transformations t=z3,σ=w3 to the equivalent Volterra integral equation and implement the spectral collocation method to solve the transformed equation. The obtained L∞ and Lω−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 confirmed by spectral accuracy. This is in accordance with our theoretical results.Table 1
TheL∞ and Lω−μ,−μ errors for Example 1.
N481216‖u−uN‖∞1.3305e-037.0244e-062.1050e-068.8371e-07‖u−uN‖ω−μ,−μ1.1124e-034.6223e-061.2920e-065.2854e-07N20242832‖u−uN‖∞4.4501e-072.5190e-071.5485e-071.0124e-07‖u−uN‖ω−μ,−μ2.6308e-071.4807e-079.0772e-085.9271e-08Figure 1
TheL∞ and Lω−μ,−μ2 errors for Example 1.Example 2.
Consider the following fractional integro-differential equation with weakly kernels:(57)D11/6yt=Γ176+t23/6−Γ29/30Γ17/6Γ19/5t14/5−t2yt+∫0tt−s−1/30ysds,t∈0,1,y0=0,y′0=0.
The exact solution of above equation is given byyt=t11/6. It is very clear, yt is not smooth at t=0. Similar to the previous example, we introduce the smoothing transformations t=z6,σ=w6 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 N. Again we can see the spectral approximation gives spectral accuracy.Table 2
TheL∞ and Lω−μ,−μ errors for Example 2.
N4681012‖u−uN‖∞1.2413e-011.1311e-023.1362e-042.9163e-065.9416e-07‖u−uN‖ω−μ,−μ3.8887e-034.6024e-041.0465e-054.6884e-071.4909e-07N1418222630‖u−uN‖∞2.5022e-076.2603e-082.0919-088.4167e-093.8586e-09‖u−uN‖ω−μ,−μ6.2708e-081.5689e-085.2426e-092.1094e-099.6704e-10Figure 2
TheL∞ and Lω−μ,−μ2 errors for Example 2.


## 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 inL∞ norm and Lω−μ,−μ2 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.



---

*Source: 3767559-2022-10-25.xml*