# Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Authors: Haotao Cai
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We develop a generalized Jacobi-Galerkin method for second kind Volterra integral equations with weakly singular kernels. In this method, we first introduce some known singular nonpolynomial functions in the approximation space of the conventional Jacobi-Galerkin method. Secondly, we use the Gauss-Jacobi quadrature rules to approximate the integral term in the resulting equation so as to obtain high-order accuracy for the approximation. Then, we establish that the approximate equation has a unique solution and the approximate solution arrives at an optimal convergence order. One numerical example is presented to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

In this paper we present a generalized Jacobi-Galerkin method for solving Volterra integral equations of second kind with weakly singular kernels. Specifically, for a given functionK∈CI2 with I≔-1,1 and a parameter μ∈0,1, we define a Volterra integral operator K:CI→CI by (1)Kϕx≔∫-1xx-t-μKx,tϕtdt,x∈I, and then consider the Volterra integral equation of the form(2)u+Ku=f,where f∈CI is a given function and u∈CI is the unknown to be determined.In view of the singularity of the kernel function in the operatorK, the solution u of (2) exhibits a singularity at the point -1 in its derivative even if the forcing term f is a smooth function. There are many numerical attempts based on the spline approximation to overcome the difficulty caused by the singularity of the solution of (2) (see [1–8]). Recently, spectral methods using Jacobi polynomial basis have received considerable attention to approximating the solution of integral equations due to their high accuracy and easy implementation (see [9–17]). In particular, Chen and Tang in [11] proposed a Jacobi-collocation spectral method for second kind Volterra integral equations with weakly singular kernels. Some function transformations and variable transformations are employed to change the equation into new Volterra integral equations possessing better regularity so that the orthogonal polynomial theory can be applied accordingly. In [12], they proposed a spectral Jacobi-Galerkin approach for solving (2). A rigorous error estimate was given in both the infinite norm and the weighted square norm. As far as I am concerned, all existing spectral methods always either suppose that the original equation has a sufficiently smooth solution or convert the equation into a new one with a solution of better regularity than that of the original equation (2) so that the spectral method can be applied. It goes without saying that the function transformation makes the resulting equations and approximations more complicated, which leads us to consider the generalized spectral method involved.We organize this paper as follows. In Section2, we develop a generalized Jacobi-Galerkin method for solving (2) and then show the stability and convergence of this algorithm. For the semidiscrete system proposed in previous section, we construct the efficient numerical integration scheme so as to obtain the fully discrete linear system. In Sections 3 and 4, we give a few technical results and analyze the stability and convergence analysis, respectively. In Section 5, one numerical example is presented to illustrate the efficiency and accuracy of this method. In addition, a conclusion is drawn.


## 2. A Generalized Jacobi-Galerkin Method

In this section, we first introduce some index sets:N0≔N∪0 with N≔1,2,… and Zr≔Zr+∪0 with Zr+≔1,2,…,r for r∈N. We let wα,βx≔1-xα1+xβ,  α,β>-1, be a Jacobi weight function and let Lwα,β2I denote the space of measurable functions whose square is Lebesgue integrable in I relative to the Jacobi weight function wα,β. The inner product and norm of this space are given by (3)ϕ,ψwα,β≔∫Iwα,βxϕxψxdx,ϕwα,β≔ϕ,ϕwα,β1/2. For n∈N0, we let Jnα,β be the Jacobi orthonormal polynomial of degree n relative to the weight function wα,β.The following result regarding the regularity of the solution of (2) comes from [6].Theorem 1.
Suppose thatK∈CmI2 with m∈N. Then the original equation (14) has a unique solution u∈CI. Moreover, if the function f is expressed as(4)fx=∑i,j∈i,jmμei,jμ1+xi+j1-μ+hx,x∈I,where h∈CmI, then the solution u can be written in the form(5)ux=∑i,j∈i,jmμci,jμ1+xi+j1-μ+vx,x∈I.Here i,jmμ≔i,j:i,j∈N0,i+j1-μ<m and h,v∈CmI, and the coefficients ci,jμ and fi,j are some constants.Now we define an index setW by (6)W≔λk≔i+j1-μ<m:λk∈i,jmμ,λk∉Nand suppose that L is the cardinality of the set W, and then we define a nonpolynomial function set W by (7)W≔spanwkx:wkx≔1+xλk,λk∈W,x∈I. It follows from the notations above that Theorem 1 is rewritten as follows.Corollary 2.
Suppose that the kernel functionK∈CmI2. If there exist some constants gi,  i∈ZL+, such that(8)fx≔gx+hx,gx≔∑i∈ZL+giwix,x∈I,where h∈CmI, then there exist some constants ci,  i∈ZL+, such that the solution u has the similar decomposition(9)ux=wx+vx,wx≔∑i∈ZL+ciwixx∈I,where v∈CmI.Now we introduce another finite dimensional spaceYn given by (10)Yn≔spanJiα,β:i∈Zn, and then let (11)Xn≔W⊕Yn.The generalized spectral Galerkin method for solving (2) is to seek a vector un≔a1,1,…,aL,1,a0,2,…,an,2T such that (12)unx≔∑i∈ZL+ai,1wix+∑i∈Znai,2Jiα,βx,x∈I,satisfying the equation (13)I+Kun,ϕwα,β=f,ϕwα,β,forϕ∈Xn. If we use Pn which is the orthogonal projection operator from Lwα,β2I to Xn, then the equation mentioned above has the operator form(14)I+PnKun=Pnf.By expression (8), Pnf can be written as(15)Pnf=g+Pnh.The conventional Jacobi-Galerkin method is to chooseYn as the approximation space and the test space, but when the original solution has a singularity, the approximation solution suffers from possessing lower-order accuracy. In order to overcome this difficulty, the same as in [7], we include the set W of the known nonpolynomial functions reflecting the singularity of the original solution in the usual Jacobi-Galerkin approximation space Yn. Hence, we call this method the generalized spectral Galerkin method.Next we are going to analyze this generalized Chebyshev-Galerkin method. We first show the stability of the original operatorI+K:Lwα,β2I→Lwα,β2I.Theorem 3.
Suppose thatK∈CmI2; then there exists a positive constant ρ such that for ϕ∈Lwα,β2I(16)I+Kϕwα,β≥ρϕwα,β.Proof.
First, it follows fromK∈CI2 and Theorems 2.21 and 4.12 in [18] that the integral operator K:Lwα,β2I→Lwα,β2I is compact. On the other hand, by the fact that -1 is not the eigenvalue of the integral operator K we conclude that I+K is injective from Lwα,β2I into itself. Thus, using Theorem  3.4 in [18], the inverse operator I+K-1:Lwα,β2I→Lwα,β2I exists and is bounded. This completes result (16).On the other hand, for a single function, letDi denote the ith generalized usual differential operator, and for any function of several variables, let ∂xi denote the ith partial generalized differential operator on the variable x. We introduce the nonuniformly weighted Sobolev space Hwα,βrI,  r∈N, by (17)Hwα,βrI≔ϕ:Diϕ∈Lwα+i,β+i2I,i∈Zr, with the norm (18)ϕwα,β,r≔∑i∈ZrDiϕwα+i,β+i. If we use Qn which is the orthogonal projection operator from Lwα,β2I to Yn, it is clear that there holds(19)PnQn=Qn.Throughout the remainder of this paper, we use the symbol c to denote a positive constant which may take different values on different occurrences. Moreover, it follows from [19] that, for ψ∈Hwα,βrI, there exists a positive constant c such that, for i∈Zr,(20)Diψ-Qnψwα,β≤cDrψwα+r,β+rni-r,which implies that(21)DrQnψwα,β≤cDrψwα+r,β+r.In particular, if the function ψ has the decomposition ψ≔ψ1+ψ2 with ψ1∈W and ψ2∈Hwα,βrI, then using (20) yields that(22)ψ-Pnψwα,β≤cDrψ2wα+r,β+rn-r.In the following we consider the stability and convergence result regarding approximation equation (14).Theorem 4.
IfK∈CmI2, there exists a positive integer n0 such that n≥n0 and for ϕ∈Xn,(23)I+PnKϕwα,β≥ρ2ϕwα,β,where ρ appears in (16). Moreover, there exists a positive constant c such that(24)u-unwα,β≤cn-mvwα+m,β+m.Proof.
SinceK is a compact operator from Lα,β2I into itself and Pnϕ→ϕ for all ϕ∈Lwα,β2I as n tends to ∞, we conclude that there exists a positive integer n0 such that, for n≥n0 and for v∈Xn, (25)K-PnKϕwα,β≤ρ2ϕwα,β,where it and (16) and the triangle inequality(26)I+PnKϕwα,β≤I+Kϕwα,β-K-PnKϕwα,βyield conclusion (23).
On the other hand, subtracting (2) from (14) obtains(27)un-u+PnKun-Ku=Pnf-f.By applying the operator Pn to both sides of (2), we have (28)Pnu+PnKu=Pnf.Thus,(29)Pnf-f=PnKu-Ku+Pnu-u.A combination of (27) and (29) gives(30)un-u=I+PnK-1Pnu-u,where it and (16) imply that(31)u-unwα,β≤2ρu-Pnuwα,β.Hence, by the solution expansion of (9) and (22) with ψ≔u,  ψ1≔w, and ψ2≔v, we conclude that(32)u-Pnu≤γn-mvwα+m,β+m.A combination of (31) and (32) presents the desired conclusion.In the remainder of the section, we write the matrix form of (14). To this end, for i,j∈N0, by introducing(33)ai,j,1≔wi,wjwα,β,bi,j,1≔wi,Kwjwα,β,ai,j,2≔wi,Jjα,βwα,β,bi,j,2≔wi,KJjα,βwα,β,ai,j,3≔Jiα,β,wjwα,β,bi,j,3≔Jiα,β,Kwjwα,β,ai,j,4≔Jiα,β,Jjα,βwα,β,bi,j,4≔Jiα,β,KJjα,βwα,β,we then define four block matrices by (34)A′≔ai,j,1:i,j∈ZL+,A′′≔ai,j,2:i∈ZL+,j∈Zn,A′′′≔ai,j,3:i∈Zn,j∈Zm-1,A∗≔ai,j,4:i,j∈Zn.It is clear that(35)A∗=In+1,A′′′=A′′T.Likewise, we define the matrices B′, B′′, B′′′, and B∗. Using these notations, we define An and Bn by (36)An≔A′A′′A′′TIn+1,Bn≔B′B′′B′′′B∗. Associated with Pnf, by letting(37)fi,1≔wi,gwα,β+wi,hwα,β,i∈ZL+,fi,2≔Jiα,β,gwα,β+Jiα,β,hwα,β,i∈Zn, we define the vector fn as (38)fn≔f1,1,…,fL,1,f0,2,…,f0,nT. Thus using the matrices and vectors above, the matrix form of (14) is written as(39)An+Bnun=fn.In order to solve system (39) in previous section, the matrix entries of integral form in (39) must be computed. Hence, the main purpose of this section is going to approximate the integral operator and the inner product based on the Gauss-Jacobi quadrature rule. To this end, for n∈N and i∈Zn+, we denote by xi,nα,β and wi,nα,β the set of n Jacobi-Gauss points and the corresponding weights relative to the weight function wα,β. We use the notation Pn to denote the set of all polynomials of degree not more than n. Moreover, the classical Gauss-Jacobi quadrature rule is given by(40)ϕ,1wα,β=∑i∈Zn+wi,nα,βϕxi,nα,β,ϕ∈P2n-1.Thus, upon relation (35) we only need to give the fully discrete form of A′ and A′′. A direct computation using the Gauss-Jacobi quadrature rule (40) yields that (41)ai,j,1=w1,1α,β+λi+λj,i,j∈ZL+;ai,j,2≔∑k∈Zn+wk,nα,β+λiJjα,βxk,nα,β+λi,i∈ZL+,j∈Zn. In order to give the fully discrete form of matrix Bn, we first approximate the integral operator K. For this purpose, for x∈I, we introduce a variable transformation (42)t≔gx,τ=x+12τ+x-12,x,τ∈I, which converts the interval -1,x into I. Thus, the operator K has the following form: (43)Kϕx=2μ-1w0,1-μx∫Iw-μ,0τKx,gx,τϕgx,τdτ,x∈I. In particular, when ϕ≔wi∈W, then (44)Kwix=2μ-1-λiw0,1-μ+λix∫Iw-μ,λiτKx,gx,τdτ,x∈I. Next we define an integral operator Gξ,η by(45)Gξ,ηϕx≔∫Iwξ,ητbx,τ;ϕdτ,x∈I,where (46)bx,τ;ϕ≔Kx,gx,τ,ϕ=wi∈WKx,gx,τϕgx,τ,otherwisex,τ∈I. Consequently, the integral operator K is rewritten as(47)Kϕx=2μ-1-λiw0,1-μ+λixG-μ,λi1x,w=wi∈W,2μ-1w0,1-μxG-μ,0ϕx,otherwisex∈I.In order to discretize the operator K, we first discretize the operator Gξ,η. To this end, for ϕ∈CI, we define the Lagrange interpolation polynomial Lnα,βϕ∈Pn by (48)Lnα,βϕx≔∑i∈Zn+ϕxi,nα,βLi,nα,βx,x∈I, where (49)Li,nα,βx≔πxx-xi,nα,βπ′xi,nα,β,πx≔x-x1,nα,βx-x2,nα,β⋯x-xn,nα,β,x∈I. Thus, for N∈N, let bNx,τ denote the N-degree interpolation polynomial of b about the variable τ relative to the weight wα,β; that is, (50)bNx,τ;ϕ≔∑i∈ZN+bx,xi,Nα,β;ϕLi,Nα,βτ,x,τ∈I, and then replacing b by bN in (45) yields the discrete form of GNξ,η as follows:(51)GNξ,ηϕx≔∫Iwξ,ητbNx,τ;ϕdτ,x∈I.It follows from the Gauss-Jacobi quadrature rule (40) that(52)GNξ,ηϕx=∑i∈ZN+∑k∈ZN+wi,Nξ,ηbx,xk,Nα,β;ϕLk,Nα,βxi,Nξ,η,x∈I.Subsequently, we can obtain the fully discrete form KN of the operator K,(53)KNϕx=2μ-1-λiw0,1-μ+λixGN-μ,λi1x,w=wi∈W,2μ-1w0,1-μxGN-μ,0ϕx,otherwisex∈I.In order to approximate the inner product and easily analyze the stability and convergence of fully discrete equation, we define the operator K~N by(54)K~Nϕx=2μ-1-λiw0,1-μ+λixLNα,β+1-μGN-μ,λi1x,ϕ=wi∈W,2μ-1w0,1-μxLNα,β+1-μGN-μ,0ϕx,otherwisex∈I.On the other hand, let(55)L~nα,βf≔g+Lnα,βh.Using these notations above, we replace K and Pnf by K~N and L~nα,βf, obtaining(56)I+PnK~Nu~n=L~nα,βf,where u~n is given by (57)u~n≔∑i∈ZL+a~i,1wi+∑i∈Zna~i,2Jiα,β. In order to observe that (56) is a fully discrete form, we have to write its matrix. To this end, suppose N≥n+1; replacing the operator K in (33) by the operator K~N given in (54) and then using Gauss-Jacobi quadrature (40) produce that (58)b~i,j,1≔2μ-1-λj∑k∈ZN+∑l∈ZN+wk,Nα,β+1-μ+λi+λjGN-μ,λi1xl,Nα,β+1-μLl,Nα,β+1-μxk,Nα,β+1-μ+λi+λj,b~i,j,2≔2μ-1∑k∈ZN+∑l∈ZN+wk,Nα,β+1-μ+λiGN-μ,λiJjα,βxl,nα,β+1-μLl,Nα,β+1-μxk,Nα,β+1-μ+λi,b~i,j,3≔2μ-1-λj∑k∈ZN+∑l∈ZN+wk,Nα,β+1-μ+λjJiα,βxk,Nα,β+1-μ+λjGN-μ,λj1xl,nα,β+1-μLl,Nα,β+1-μxk,Nα,β+1-μ+λj,b~i,j,4≔2μ-1∑k∈ZN+wk,Nα,β+1-μJiα,βxk,Nα,β+1-μGN-μ,0Jjα,βxk,Nα,β+1-μ. The same as before, we define four matrices B~′, B~′′, B~′′′, and B~∗ and then set (59)B~n≔B~′B~′′B~′′′B~∗. On the other hand, replacing the function h in (37) by the interpolation polynomial LNα,βh and then using Gauss-Jacobi quadrature (40) produce that (60)f~i,1≔∑j∈ZL+gjw1,1α,β+λi+λj+∑j∈ZM+∑k∈ZM+wj,Mα,β+λihxk,Mα,βLk,Mα,βwj,Mα,β+λi,f~i,2≔∑j∈ZL+∑k∈Zn+gjwk,nα,β+λjJiα,βxk,nα,β+λj+∑k∈ZM+wk,Mα,βJiα,βxk,Mα,βhxk,Mα,β.Using the notations above, in a similar manner, we let (61)f~n≔f~1,1,…,f~L,1,f~0,2,…,f~n,2T. Hence, we have the following matrix form of (56):(62)A~n+B~nu~n=f~n,where (63)u~n≔a~1,1,…,a~L,i,a~1,2,…,a~n,2T.


## 3. Some Useful Results

In this section we are going to give some technical results so as to analyze the fully discrete equation (56).Lemma 5.
Suppose the kernel functionK∈CmI2. If ϕ∈CmI, then there exists a positive constant c such that, for x,τ∈I,(64)∂τmbx,τ;ϕ2≤cw0,2mx∑i∈ZmDgiϕgx,τ2,∂xmbx,τ;ϕ2≤c∑i∈Zmw0,2iτDgiϕgx,τ2.Proof.
We only need to show the first inequality in (64) since the other is the same. In fact, a direct application of the high-order derivative formula to b yields that(65)∂τmbx,τ;ϕ=∑i∈ZmCmi∂τivgx,τ∂τm-iKx,gx,τ,x,τ∈I,where Cmi is the binomial coefficient given by Cmi≔mm-1⋯m-i+1. Clearly, (66)∂τivgx,τ=2-iw0,ixDgivgx,τ,∂τm-iKx,gx,τ=2i-mw0,m-ixDgm-iKx,gx,τ. Substituting the above result into the right hand side of (65) yields that(67)∂τmbx,τ;ϕ=2-mw0,mx∑i∈ZmCmiDgivgx,τDgm-iKx,gx,τ,x,τ∈I.Using the Cauchy-Schwartz inequality to the right hand side of (67) yields the desired conclusion with c being given by c≔∑i∈ZmCmi2maxi∈Zm⁡∂tiK∞2.Now we give the difference betweenG-μ,0ϕ and G-μ,0ϕ for ϕ∈CmI. To this end, we introduce the result in [8]:  for ψ∈Hwα,βrI, there exists a positive constant c such that, for i∈Zr,(68)Diψ-Lnα,βψwα,β≤cDrψwα+r,β+rni-r.Lemma 6.
Suppose the kernel functionK∈CmI2. If three parameters α,  β, and  μ satisfy -1<α<1-2μ,  -1<β<1,  and α+β<1-2μ, then there exists a positive constant c such that, for ϕ∈CmI,(69)G-μ,0ϕ-GN-μ,0ϕ∞≤cN-mϕwα,β,m.Similarly, there also exists a positive constant c such that, for i∈ZL+,(70)G-μ,λi1-GN-μ,λi1∞≤cN-m.Proof.
We only prove the first result (69), and the other is the same. We first observe that(71)G-μ,0ϕx-GN-μ,0ϕx=∫Iw-μ,0τbx,τ;ϕ-bNx,τ;ϕdτ,x∈I.Associated with the above equation, we define S1x by the left hand side of (71) and then define S2 and S3 by (72)S2τ≔w-μ-α/2,-β/2τ,S3τ;x≔wα/2,β/2τbx,τ;ϕ-bNx,τ;ϕ. It is clear that(73)S1x=∫IS2τS3τdτ.Applying Cauchy inequality to the right hand side of (73) produces that(74)S12x≤∫IS22τdτ∫IS32τ;xdτ.It follows from the hypothesis that α+2μ<1 and -1<β<1 yield that(75)∫IS22τdτ≤1w-α-2μ,-β<+∞.On the other hand, an application of (68) produces that(76)∫IS32τ;xdτ≤cN-2m∂τmbwα+m,β+m2.It follows from the first estimation in (64) that(77)∂τmbwα+m,β+m2≤cw0,2mx∑i∈ZmDgiϕwα+m,β+m2.For i∈Zm, a direct observation for Dgiϕwα+m,β+m2 using the condition that α+β<1-2μ can obtain that(78)w0,2mxDgiϕwα+m,β+m2≤cw0,2m-2i+1-α-β-2μx∑i∈ZmDiϕwα+i,β+i2.A combination of (73)–(78) yields the desired conclusion (69).Now we introduce the operatorK¯N as (79)K¯Nϕx=2μ-1-λiw0,1-μ+λixLNα,β+2-2μ+2λiG-μ,λi1x,ϕ=wi,i∈ZL,2μ-1w0,1-μxLNα,β+2-2μG-μ,0ϕx,otherwisex∈I, and then we estimate the difference between Kϕ and K¯Nϕ.Lemma 7.
Suppose the kernel functionK∈CmI2. If three parameters α,  β,  and μ satisfy -1<α,β<1-2μ,α+β<1-2μ, then there exists a positive constant c such that, for ϕ∈CmI,(80)Kϕ-K¯Nϕwα,β≤cN-mϕwα,β,m.Similarly, there also exists a positive constant c such that, for i∈ZL+,(81)Kwi-K¯Nwiwα,β≤cN-mwiwα,β,m.Proof.
The same as Lemma6, we only need to show that (80) holds. For ϕ∈CmI, by the definition of the operators K and K¯n,(82)Kϕx-K¯Nϕx=2μ-1w0,1-μxG-μ,0ϕx-LNα,β+2-2μG-μ,0ϕx,x∈I.A direct estimation yields that(83)Kϕ-K¯Nϕwα,β2≤G-μ,0ϕ-LNα,β+2-2μG-μ,0ϕwα,β+2-2μ2,in which combining result (68) with ψ≔G-μ,0ϕ implies that(84)Kϕ-K¯Nϕwα,β2≤cn-2mDmG-μ,0ϕwα+m,β+2-2μ+m2.In the following we estimate ∥DmG-μ,0ϕ∥wα+m,β+2-2μ+m2. If we let (85)Sx≔∫Iw-μ,0τ∂xmbx,τ;ϕdτ,x∈I, then(86)DmG-μ,0ϕwα+m,β+2-2μ+m2=∫Iwα+m,β+2-2μ+mxS2xdx.By the assumption that α+β,β<1-2μ, there exists a positive constant δ such that max⁡-1,α+β,β<δ<1-2μ. Thus we define two functions S1 and S2 by (87)S1τ≔w-μ-α/2,-δ/2τ,S2τ,x;ϕ≔wα/2,δ/2τ∂xmbx,τ;ϕ,τ∈I.It is obvious that(88)Sx=∫IS1τS2τ,x;ϕdτ.Applying Cauchy inequality to the right hand side of (88) yields that(89)Sx2≤∫IS12τdτ∫IS2τ,x;ϕdτ,x∈I.Clearly,(90)∫IS12τdτ≤1w-α-2μ,-δ<+∞.By using the second estimation in (64), we have(91)∫IS22τdτ≤c∑i∈ZmDgiϕα,δ+2i2.Substituting results (89)–(91) into (86) can obtain that(92)DmG-μ,0ϕwα+m,β+2-2μ+m2≤c∑i∈ZmIi,where Ii is defined by (93)Ii≔∫Iwα,δ+2iτdτ∫Iwα+m,β+2-2μ+mxDxiϕgx,τ2dx,i∈Zm.The left thing is to estimate Ii for i∈Zm. To this end, we let (94)ϵi≔δ-β2,i=00,otherwise.Then, for i∈Zm, a direct estimation for Ii yields that (95)Ii≤c∫Iwα,δ-α-β-ϵi-1τdτ∫Iwα+i,β+itDiϕt2dt,which implies that(96)Ii≤cDiϕwα+i,β+i2.A consequence of (83), (92), and (96) produces the desired conclusion.The next result is concerned with the difference betweenKϕ and K~Nϕ. To this end, we introduce the result proposed in [9–11]: for any ψ∈CI, there exists a positive constant c independent of n,(97)Lnα,βψwα,β≤cψ∞.Lemma 8.
Suppose the kernel functionK∈CmI2. If three parameters α,  β, and  μ satisfy -1<α,β<1-2μ,α+β<1-2μ, then there exists a positive constant c such that, for ϕ∈CmI,(98)Kϕ-K~Nϕwα,β≤cN-mϕwα,β,m.Moreover, if ϕ≔wi for i∈ZL+, then there exists a positive constant c such that(99)Kwi-K~Nwiwα,β≤cN-m.Proof.
We only show that (98) holds, since the proof of the result (99) is the same. For ϕ∈CmI, using triangle inequality produces that(100)Kϕ-K~Nϕwα,β≤Kϕ-K¯Nϕwα,β+K¯Nϕ-K~Nϕwα,β.It follows from result (80) in Lemma 7 that we only need to estimate the second term in the right hand side of (100). By definition of K¯Nϕ and K~Nϕ(101)K^Nϕ-K~Nϕwα,β=2μ-1Lnα,β+2-2μG-μ,0ϕ-Lnα,β+2-2μGN-μ,0ϕwα,β+2-2μ,in which combining (97) produces that there exists a positive constant c such that (102)K^Nϕ-K~Nϕwα,β≤cG-μ,0ϕ-GN-μ,0ϕ∞.This and (69) conclude the desired conclusion (98).As a consequence of Lemma8, for v∈Pn, by using the inverse inequality relative to two norms weighted with different Jacobi weight functions in Theorem  3.31 in [19], we can easily obtain the following.Corollary 9.
Suppose the conditions in Lemma8 hold. Then there exists a positive constant c such that, for ϕ∈Pn,(103)Kϕ-K~Nϕwα,β≤cϕwα,βnNm.


## 4. Convergence Analysis

In the section, we are going to analyze the convergence of the approximate solution of the fully discrete generalized Jacobi-Galerkin method. First we give the stability analysis of the operatorI+PnK~Nϕ:Xn→Xn.Theorem 10.
Suppose the kernel functionK∈CmI2. If three parameters α,  β,  and μ satisfy -1<α,β<1-2μ,α+β<1-2μ and if we choose N as (104)N≥Nmin≔nlog1/mn+1,then there exists a positive integer n0 such that n≥n0 and for ϕ∈Xn,(105)I+PnK~Nϕwα,β≥ρ6ϕwα,β,where ρ appears in (16).Proof.
Forϕ∈Xn, there exists functions ηi∈W for i∈ZL+ and a polynomial function ψ∈Yn such that (106)ϕx≔∑i∈ZL+ηix+ψx,ηix≔ciwixx∈I. By using (99) and (104), there exists a positive constant c such that, for i∈ZL+, (107)PnKηi-PnK~Nηiwα,β≤cn-mηiwα,β. Because of the result (108)limn→∞⁡n-m=0, we conclude that there exists a positive integer n0 such that, for n≥n0,(109)PnKηi-PnK~Nηiwα,β≤ρ6Lηiwα,β,where L denotes the cardinality of the set W as in Section 2.
On the other hand, by estimation (103) in Corollary 9 and the choice of N in (104) there exists a positive constant c such that (110)PnKψ-PnK~Nψwα,β≤clog-1nψwα,β. It follows from the fact that log-1⁡n tends to zero as n→∞ that there exists a positive integer constant n1 such that, for n≥n1,(111)PnKψ-PnK~Nψwα,β≤ρ6ψwα,β.Hence, when n≥max⁡n0,n1, a combination of (109) and (111) produces that (112)PnKϕ-PnK~Nϕwα,β≤ρ6∑i∈ZL+ηiwα,β+ψwα,β.This combining (23) and the next inequality(113)I+PnK~nϕwα,β≥I+PnKϕwα,β-PnKϕ-PnK~Nϕwα,β produces (114)I+PnK~nϕwα,β≥ρ2ϕwα,β-ρ6∑i∈ZL+ηiwα,β+ψwα,β≥ρ6ϕwα,β. Hence, we draw the desired conclusion.Theorem10 ensures that, for sufficiently large n, if we select N as in (104), then the fully discrete system possesses (62) which possesses a unique solution u~n. The next result is concerned with the convergence of the approximation solution u~n.Theorem 11.
Suppose the kernel functionK∈CmI2 and f is given by (8). Three parameters α,  β,  and μ satisfy -1<α,β<1-2μ,α+β<1-2μ and N is given in (104). Then there exist a positive constant c and a positive integer n0 such that, for n≥n0, (115)u-u~nwα,β≤cn-m.Proof.
By using the triangle inequality, we have(116)u-u~nwα,β≤u-Pnuwα,β+Pnu-u~nwα,β.Based on expression (9) of the solution u, an application of relation (19) yields that (117)u-Pnuwα,β≤v-Qnvwα,β, in which combining estimation (20) with ϕ≔v produces(118)u-Pnuwα,β≤cDmvwα+m,β+mn-m.Hence, we only need to estimate the second term in the right hand side of (116). In fact, employing Pn to both sides of (2) yields that (119)Pnu+PnKu=Pnf. A combination of the above equation and (55) and (56) confirms that (120)I+PnK~Nu~n-Pnu=PnKu-PnK~NPnu+Pnh-Lnα,βh. By Theorem 10, there exist a positive constant c and a positive integer n0 such that n≥n0,(121)u~n-Pnuwα,β≤cPnKu-PnK~NPnuwα,β+Pnh-LMα,βhwα,β.Clearly, using (68) with ψ≔h and i≔0 produces(122)Pnϕ-L~Mα,βϕwα,β≤cDmhwα+m,β+mn-m.On the other hand, we let(123)I1≔Ku-KPnuwα,β,I2≔KPnu-K~NPnuwα,β.Obviously,(124)PnKu-PnK~NPnuwα,β≤I1+I2.It follows from the compactness of the operator K and estimate (118) that there exists a positive constant c such that (125)I1≤cDmvwα+m,β+mn-m. Again using the solution decomposition of Theorem 1, we observe that there exist positive constants ci such that(126)ux=∑i∈ZL+ηix+vx,ηix≔ciwix,x∈I.Thus, applying estimate (99) leads to the fact that there exists a positive constant c such that, for i∈ZL+,(127)Kηi-K~Nηiwα,β≤cηiwα,βN-m.In a similar manner, using result (98) produces(128)Kv-K~Nvwα,β≤cN-mvwα,β,m.A combination of (127) and (128) with the help of choice (104) yields that(129)I2≤cn-mlog-1⁡n.
As a consequence, a combination of (121), (122), (123), (124), (126), (127), (128), and (129) completes the proof.Theorem11 illustrates that the approximation solution arrives at the optimal convergence order.


## 5. One Numerical Example

In this section, we present a numerical example to demonstrate the approximation accuracy, the order convergence, and the stability of the proposed method. We also compare it with the conventional Jacobi-Galerkin method. Here, we compute the Gauss-Jacobi quadrature rule nodes and weights by Theorems3.4 and 3.6 discussed in [19].In this example, for simplicity we chooseKx,t≔1. For the numerical comparison purpose, we choose the right hand side function f so that (130)ux≔1+x1/2cos⁡1+x,x∈I,is the exact solution of the equation. Note that the first derivative of this solution has a singularity at x=-1. The purpose of these numerical experiments is to compare the numerical performance between the generalized spectral method with the conventional spectral method. For both of the methods we let Lw-1/2,-1/22I be the solution space of the original solution u. It is obvious that the solution u belongs to the space Hw-1/2,-1/21(I) and u∈Hw-1/2,-1/2mI for m≥2. Now we choose the corresponding conventional finite dimensional subspace Yn as (131)Yn≔spancos⁡marccos⁡x,m∈Zn,x∈I. In this numerical example, our generalized spectral method chooses the nonpolynomial function set W given by (132)W≔span1+x1/2:x∈I. The comparison of numerical results between the conventional method and our proposed method is given in Table 1.Table 1
The comparison of numerical results of the two methods.
n
=
4
n
=
6
n
=
8
n
=
10
n
=
12
n
=
14
Conventional method
3.99
e
-
2
2.53
e
-
2
1.60
e
-
2
1.11
e
-
2
8.37
e
-
3
6.58
e
-
3
Our method
1.59
e
-
3
6.42
e
-
4
2.53
e
-
4
1.24
e
-
4
7.01
e
-
5
4.33
e
-
5From these numerical results we observe that our method is superior to the conventional spectral method, which is consistent with the theoretical results. In summary, the conventional Jacobi-Galerkin method for solving Volterra integral equations with nonsmooth solutions may be of low-order accuracy. In order to obtain the high-order accuracy, in this paper we introduce the setW of nonpolynomial functions in the conventional Jacobi-Galerkin approximation space Yn of Jacobi polynomial basis and then develop a generalized Jacobi-Galerkin method. The price we pay to obtain this optimal convergence rate is the dimension of the increase of the approximation subspace. But we observe that the additional cost to achieve the optimal convergence rate is insignificant in comparison with the acceleration convergence that we obtain.



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*Source: 4751357-2017-08-31.xml*