CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHODS FOR VOLTERRA INTEGRAL EQUATIONS WITH A WEAKLY SINGULAR KERNEL

Transcription

1 COVERGECE AALYSIS OF THE JACOBI SPECTRAL-COLLOCATIO METHODS FOR VOLTERRA ITEGRAL EQUATIOS WITH A WEAKLY SIGULAR KEREL YAPIG CHE AD TAO TAG Abstract. In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of second kind with a weakly singular kernel. We use some function transformation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [, ], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomials approximation theory for orthogonal polynomials and the operator theory. The spectral rate of convergence for the proposed method is established in the L -norm and weighted L -norm. umerical results are presented to demonstrate the effectiveness of the proposed method.. Introduction In practical applications one frequently encounters the Volterra integral equations of second kind with a weakly singular kernel of the form yt = gt + t t s µ Kt, sysds, t T,. where the unknown function yt is defined in t T <, gt is a given source function and Kt, s is a given kernel. Date: March 8, Mathematics Subject Classification. 35Q99, 35R35, 65M, 65M7. The first author is supported by ational Science Foundation of China, the ational Basic Research Program under the Grant 5CB373, and Scientific Research Fund of Hunan Provincial Education Department. The second author is supported by Hong Kong Research Grant Council, and Ministry of Education of China through a Changjiang Scholar Program.

2 YAPIG CHE AD TAO TAG For any positive integer m, if g and K have continuous derivatives of order m, then from [5] there exists a function Z = Zt, v possessing continuous derivatives of order m, such that the solution of. can be written as yt = Zt, t µ. This implies that near t = the first derivative of the solution yt behaves like y t t µ. Several methods have been proposed to recover high order convergence properties for. using collocation type methods, see, e.g., [3, 4, 9, 5, 6, 7] and using multi-step method, see, e.g., [7]. The singular behavior of the exact solution makes the direct application of the spectral methods difficult. More precisely, for any positive integer m, we have y m t t µ m, which indicates that y Hω m, T, where Hω m is a standard Soblev space associated with a weight ω. To overcome this difficulty, we first apply the transformation to change. to the equation where ỹt = t µ [yt y] = t µ [yt g]. t ỹt = gt + t µ s µ t s µ Kt, sỹsds, t T,.3 gt = t µ [gt g] + t µ g It is easy to see that the solution of.3 is a regular function t t s µ Kt, sds..4 ỹt C m [, T ]..5 To use the theory of orthogonal polynomials, we make the change of variable t = T + x, x = t,.6 T to rewrite problem.3 as follows [ ] µ T T +x µ T ux = fx + + x s µ + x s K where x [, ], and By using a linear transformation: T + x, s ỹsds,.7 ux = ỹ T + x/, fx = g T + x/..8 s = T + τ, τ [, x],.9 the equation.7 becomes [ ] µ T x ux = fx + + x + τ µ x τ µ Kx, τuτdτ,.

3 for x [, ], where JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS 3 Kx, τ = K T + x, T + τ. Recently, in [8], a Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind whose kernel and solutions are sufficiently smooth. Then, in [7], a Chebyshev-collocation method is proposed and analyzed for the special case µ = for.. The main purpose of this work is to use Jacobi collocation methods to numerically solving the Volterra integral equations.. We will provide a rigorous error analysis which theoretically justify the spectral rate of convergence of the proposed method. This paper is organized as follows. In section, we introduce the Jacobi-collocation spectral approaches for the Volterra integral equations.. Some preliminaries and useful lemmas are provided in Section 3. The convergence analysis is given in Section 4. We prove the error estimates in L norm and weighted L norm. The numerical experiments are carried out in Section 5, which will be used to verify the theoretical results obtained in Section 4. The final section contains conclusions and remarks. Throughout the paper C will denote a generic positive constant that is independent of but which will depend on T and on the bounds for the given functions g and K.. Jacobi-collocation methods Let ω α,β x = x α + x β be a weight function in the usual sense, for α, β >. As defined in [6,, 3, 4, 5, 9], the set of Jacobi polynomials {Jn α,β x} n= forms a complete L, -orthogonal system, where L, is a weighted space defined ω α,β ω α,β by equipped with the norm and the inner product L ω α,β, = {v : v is measurable and v ω α,β < }, u, v ω α,β = v ω α,β = vx ω α,β xdx uxvxω α,β xdx, u, v L ω α,β,. For a given positive integer, we denote the collocation points by {x i } i=, which is the set of + Jacobi Gauss, or Jacobi Gauss-Radau, or Jacobi Gauss-Lobatto points,,

4 4 YAPIG CHE AD TAO TAG and by {w i } i= the corresponding weights. Let P denote the space of all polynomials of degree not exceeding. For any v C[, ], see, e.g., [6, 3, 4, 5], we can define the Lagrange interpolating polynomial I α,β v P, satisfying I α,β vx i = vx i, i. It can be written as an expression of the form I α,β vx = vx i F i x, i= where F i x is the Lagrange interpolation basis function associated with the Jacobi collocation points {x i } i=. Firstly, assume that Eq.. holds at the collocation points {x i } i= on [, ], namely, ux i = fx i + [ ] µ T xi + x i + τ µ x i τ µ Kxi, τuτdτ,. for i. In order to obtain high order accuracy of the approximated solution, we use the Gauss-type quadrature formula relative to the Jacobi weight to compute the integral term in.. Based on this idea, we need to transfer the integral interval [, x i ] to the unit interval [, ] xi + τ µ x i τ µ Kxi, τuτdτ = by using the transformation τ = τ i θ = + x i θ µ Kxi, τ i θuτ i θdθ,. θ + x i, θ [, ]..3 ext, using a + -point Gauss quadrature formula relative to the Jacobi weight {w i } i=, the integration term in. can be approximated by θ µ Kxi, τ i θuτ i θdθ k= Kx i, τ i θ k uτ i θ k w k, where the set {θ k } k= coincides with the collocation points {x i} i= on [-,]. We use u i, i, to indicate the approximate value for ux i, i, and use u x = u j F j x.4 j=

5 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS 5 to approximate the function ux, namely, ux i u i, ux u x, uτ i θ k u j F j τ i θ k. j= The Jacobi collocation method is to seek u x such that {u i } i= satisfies the following collocation equations: [ ] µ T u i = fx i + + x i u j Kx i, τ i θ k F j τ i θ k w k,.5 j= k= for i. We denote the error function by It follows from. and.8 that ex := u u x, x [, ]..6 yt = y + [ ] µ T + x ux..7 Consequently, the approximate solution to. is given by [ ] µ T y t = y + + x u x..8 Then the corresponding error functions satisfy [ ] µ T ɛt := y y t = + x ex = t µ ex Some preliminaries and useful lemmas We first introduce some weighted Hilbert spaces. For simplicity, denote x vx = / xvx, etc. For non-negative integer m, define H m ω α,β, := { v : k xv L ω α,β,, k m }, with the semi-norm and the norm as m / v m,ω α,β = x m v ω α,β, v m,ω α,β = v k,ω, α,β respectively. Let ωx = ω, x denote the Chebyshev weight function. In bounding some approximation error of Chebyshev polynomials, only some of the L -norms appearing on the right-hand side of above norm enter into play. Thus, it is convenient to k=

6 6 YAPIG CHE AD TAO TAG introduce the seminorms v H m; ω, = m k=minm,+ xv k L ω, For bounding some approximation error of Jacobi polynomials, we need the following non-uniformly weighted Sobolev spaces: H m ω α,β,, := { v : k xv L ω α+k,β+k,, k m }, equipped with the inner product and the norm as m u, v m,ω α,β, = xu, k xv k ω α+k,β+k, v m,ω α,β, = k=. v, v m,ω α,β,. Furthermore, we introduce the orthogonal projection π,ω α,β : L, P ω α,β, which is a mapping such that for any v L,, ω α,β v π,ω α,βv, φ ω α,β =, φ P. The following result follows from Theorem.8. in [5] and 3.8 in [3], also see []. Lemma 3.. Let α, β >. Then for any function v H m, and any nonnegative integer m, we ω α,β, have k xv π,ω α,βv ω α+k,β+k C k m m x v ω α+m,β+m, k m. 3. In particular, v π,ω α,βv ω α,β C v,ω α+,β+. 3. Applying Theorem.8.4 in [5] and Theorem 4.3, 4.7, 4. in [4], we have the following optimal error estimate for the interpolation polynomials based on the Jacobi Gauss points, Jacobi Gauss-Radau points, and Gauss-Lobatto points. Lemma 3.. For any function v satisfying x v H m,, we have, for k m, ω α,β, k xv I α,β v ω α+k,β+k C k m m x v ω α+m,β+m, 3.3 v I α,β v,ω α,β C + α + β m/ m x v ω α+m,β+m. 3.4 Define a discrete inner product, for any continuous functions u, v on [, ], by u, v = ux j vx j w j. 3.5 j= By Lemmas 3. and 3. we can obtain an estimate for the integration error produced by a Gauss-type quadrature formula relative to the Jacobi weight.

7 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS 7 Lemma 3.3. If v H m ω α,β,, for some m and φ P, then for the Jacobi Gauss and Jacobi Gauss-Radau integration we have v, φ ω α,β v, φ v I α,β v ω α,β φ ω α,β C m m x v ω α+m,β+m φ ω α,β, 3.6 and for the Jacobi Gauss-Lobatto integration, we have v, φ ω α,β v, φ C v π,ω α,βv ω α,β + v I α,β v ω φ α,β ω α,β C m m x v ω α+m,β+m φ ω α,β. 3.7 We have the following result on the Lebesgue constant for the Lagrange interpolation polynomials associated with the zeros of the Jacobi polynomials, see, e.g., [8]. Lemma 3.4. Assume that F j x is the corresponding -th Lagrange interpolation polynomials associated with the Gauss, or Gauss-Radau, or Gauss-Lobatto points of the Jacobi polynomials. Then I α,β := max x, { O log, < α, β F j x =, O γ+, γ = maxα, β, otherwise. j= 3.8 In our analysis, we shall apply the Gronwall s lemma. We call such a function v = vt locally integrable on the interval [, T ] if for each t [, T ], its Lebesgue integral t vsds is finite. Lemma 3.5. Assume that vt is a non-negative, locally integrable function defined on, which satisfies vt bt + B t s α t s β vsds t [, T ], where bt and B. Then, there exists a constant C such that vt bt + C t s α t s β bsds t [, T ]. We now introduce some notations. For r and κ [, ], C r,κ [, T ] will denote the space of functions whose r-th derivatives are Hölder continuous with exponent κ, endowed with the usual norm r,κ. When κ =, C r, [, T ] denotes the space of functions with r continuous derivatives on [, T ], also denote C r [, T ], and with norm r.

8 8 YAPIG CHE AD TAO TAG We will make use of a result of Ragozin [, ] see also [], which states that, for each non-negative integer r and κ [, ], there exists a constant C r,κ > such that for any function v C r,κ [, T ], there exists a polynomial function T v P such that v T v C r,κ r+κ v r,κ, 3.9 where is the norm of the space L [, T ], and when the function v C[, T ] we also denote v = v C[,T ]. Actually, as stated in [, ], T is a linear operator from C r,κ [, T ] to P. For convenience, we define a linear operator with a weakly singular integral kernel: t Mvt = t µ s µ t s µ Kt, svsds, t [, T ]. 3. We will need the fact that M is compact as an operator from C[, T ] to C,κ [, T ] for any < κ < µ. Lemma 3.6. Let κ, µ, and M be defined by 3.. Then, for any function vx C[, T ], there exists a positive constant C such that Proof. We first prove that M is Hölder continuous, i.e., for κ, µ. Let We then have Mv,κ C v. 3. Mvˆt Mvť ˆt ť κ C v ˆt < ť T, 3. kt, s = s µ t s µ Kt, s. 3.3 Mvˆt Mvť ť ˆt κ ˆt ť = ť ˆt ˆt κ µ kˆt, svsds ť µ kť, svsds E + E + E 3, 3.4

9 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS 9 where E = ť ˆt κˆt ˆt µ kˆt, s kť, s vs ds E = ť ˆt κ ť µ ˆt µ ť kť, s vs ds E 3 = ť ˆt κˆt ť µ kť, s vs ds. ˆt We now estimate the three terms one by one. Observe E E + E, 3.5 where E = ť ˆt κˆt ˆt µ s [ µ ˆt s µ ť s µ] Kˆt, s vs ds, E = ť ˆt κˆt ˆt µ s µ ť s µ Kˆt, s Kť, s 3.6 vs ds. Recall the definition of the Beta function This gives that Observe that x a x b dx = Ba, b, a, b >. 3.7 z τ a z τ b dτ = z a+b Ba, b. 3.8 ˆt s µ [ ˆt s µ ť s µ] ds = ˆt µ ˆt s µ ť s µ ds 3.9 for < µ <, and ˆt s µ [ ˆt s µ ť s µ] ds = B µ, µ ˆt µ ť µ + ť ˆt ť ˆt s µ ť s µ ds s µ ť s µ ds 3.

10 YAPIG CHE AD TAO TAG for < µ. The above observation, together with 3.6, yield E C v ť ˆt κ [ˆt µ C v ť ˆt κ [ˆt µ ˆt ˆt s µˆt µ ť s µ ds ť s µ ds [ ] C v ť ˆt κ µ ť ˆt µ + ˆt µ ť µ C v ť ˆt µ κ C v 3. ] ] for < µ < and κ, µ; and E ť C v ť ˆt κ s µˆt µ ť s µ ds ť C v ť ˆt κ ť s µ ds ˆt ˆt C v ť ˆt µ κ C v 3. for < µ and κ, µ. Furthermore, we have E C v ˆt µ ˆt s µ ť s µ Kˆt, s Kť, s ť ˆt κ ds ˆt C v K,κˆt µ s µ ť s µ ds ť C v ˆt µ s µ ť s µ ds C v, 3.3 where we have used the fact ˆt < ť and 3.8. Using the fact that ť ν ˆt ν ť ˆt C, ˆt < ť < T, ν,, ν we have for κ, µ, ť E Cť ˆt κ ť µ ˆt µ v s µ ť s µ ds Cť ˆt κ ť µ ˆt µ ť µ v { Cť ˆt [ť µ κ ť/ˆt ] µ ˆt µ v, µ, /, Cť ˆt µ κ ť ˆt µ v, µ /,. Cť ˆt µ κ v C v. 3.4

11 Finally, we have JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS E 3 C v ť ˆt κˆt ť µ s µ ť s µ ds C v, 3.5 ˆt where we have used the estimate for E, i.e., 3.. The desired result 3. is established by combining 3.4 with the estimates for E, E and E 3 above. To prove the error estimate in weighted L norm, we need the generalized Hardy s inequality with weights see, e.g., [, 6, 4]. Lemma 3.7. For all measurable function f, the following generalized Hardy s inequality b /q b /p Kfx uxdx q C fx p vxdx holds if and only if a b /q x /p sup utdt v p tdt <, p = p a<x<b x a p for the case < p q <. Here, K is an operator of the form Kfx = x a a kx, tftdt with kx, t a given kernel, u, v weight functions, and a < b. Using Theorem in [9], we have the following estimate for the Lagrange interpolation associated with the Jacobi Gaussian collocation points. Lemma 3.8. For every bounded function vx, there exists a constant C independent of v such that sup vx j F j x C v, j= L ω α,β, where F i x is the Lagrange interpolation basis function associated with the Jacobi collocation points {x i } i=.

12 YAPIG CHE AD TAO TAG 4. Convergence analysis The objective of this section is to analyze the approximation scheme.5. Firstly, we derive the error estimate in L norm of the Jacobi collocation method. Theorem 4.. Let u be the exact solution of the Volterra equation.. Assume the approximated solution u of the form.4 is given by the spectral collocation scheme.5 with the Jacobi Gauss, or Jacobi Gauss-Radau, or Jacobi Gauss-Lobbatto collocation points. If the given data gt and Kt, s in. belong to C m [, T ], then for sufficiently large, C m log u H m; ω, + K u, < µ <, u u C u m H m; ω, + log K u, µ =, 4. C u µ m H m; ω, + K u, < µ <, where K = max i m θ Kx i, τ i ω m µ,m µ. 4. Proof. Since the given data gt and Kt, s in. belong to C m [, T ], based on the analysis in Section we have u C m [, ]. Consequently, u Hω m, L,. By using.-. and the definition of the weighted inner product, we first observe that the solution u of. satisfies [ ] µ T ux i = fx i + + x i Kxi, τ i, uτ i 4.3 ω µ, µ for i. By using the definition of the discrete inner product 3.3, we set Kxi, τ i, φτ i = k= Kx i, τ i θ k φτ i θ k w k. Then, the numerical scheme.5 can be written as [ ] µ T u i = fx i + + x i Kxi, τ i, u τ i, 4.4 for i, where u is defined by.4. We now subtract 4.4 from 4.3 to get the error equation ux i u i = t µ i Kxi, τ i, eτ i + t µ ω µ, µ i I i, = t µ i xi + τ µ x i τ µ Kxi, τeτdτ + t µ i I i,, 4.5

13 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS 3 for i, where t i = T + x i, ex = ux u x is the error function and I i, = Kxi, τ i, u τ i Kxi, τ i, u τ i. ω µ, µ In 4.5, the integral transformation. was used. Applying again the transformation.6 and.9, we change 4.5 to ux i u i = t µ i = t µ i ti ti t i s µ Kt i, sɛsds + t µ i I i, s µ t i s µ Kt i, s ɛsds + t µ i I i,, 4.6 where ɛt = t µ ɛt, and ɛt was defined in.9. Multiplying F j x on both sides of the error equation 4.6 and summing up from i = to i = yield I µ, µ u u = M ɛt i F i x + i= t µ i I i,f i x, 4.7 where M was defined in 3.. Consequently, recalling the relation of error function.9 gives that where From 4.8, we have where t ɛt = t µ s µ t s µ Kt, s ɛsds + I + I + I 3, 4.8 I = u I µ, µ u, I = t µ i I i,f i x, i= i= I 3 = I µ, µ M ɛ M ɛ. t ɛt bt + Bt µ s µ t s µ ɛs ds t [, T ], 4.9 bt = I + I + I 3, B = max Kt, s. 4. s<tt Using the generalized Gronwall inequality, i.e., Lemma 3.5, we have t ɛt bt + Bt µ s µ t s µ bs ds t [, T ]. 4. In fact, from.9 we then write Then, it follows from 4. and 3.8 that e = ɛ C b C ex = t µ ɛt = ɛt. 4. I + I + I

14 4 YAPIG CHE AD TAO TAG Firstly, let I c u P denote the interpolant of u at any of the three families of Chebyshev Gauss points. From in [6], the interpolation error estimate in the maximum norm is given by ote that I = u I c u C / m u H m; ω,, µ =. 4.4 I µ, µ px = px, i.e., I µ, µ Ipx =, px P., 4.5 where I denotes the identical operator. By using 4.5, Lemma 3.4, and 4.4, we obtained that I = u I µ, µ u = u I c u + I µ, µ ext, it follows from Lemma 3.3 that so that Iu c I µ, µ u u Iu c + I µ, µ Iu c u + I µ, µ u I c u { C m log u H m; ω,, < µ <, C µ m u H m; ω,, < µ <. 4.6 I i, C m m θ Kx i, τ i ω m µ,m µ u τ i ω µ, µ, 4.7 max I i, C m max i i m Kx θ i, τ i ω m µ,m µ max i u τ i ω µ, µ C m K u C m K e + u, 4.8 where K is defined by 4.. Hence, by using Lemma 3.4 and 4.8, we have I = t µ i I i,f i x C max I i, max F j x i x, i= j= { C m log K e + u, µ <, C µ m K e + u, < µ <, 4.9 for sufficiently large. We now estimate the third term I 3. It follows from 4. that ɛ C[, T ]. Consequently, using 3.9 and Lemma 3.6 that M ɛ T M ɛ C κ M ɛ,κ C κ ɛ = C κ e, 4.

15 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS 5 where κ, µ and T M ɛ P. It follows from 4.5, Lemma 3.4, and the above estimate that I 3 = I µ, µ IM ɛ = I µ, µ IM ɛ T M ɛ + I µ, µ M ɛ T M ɛ { C κ log e, µ <, C κ µ e, < µ <. 4. The desired estimate 4. follows from , 4.6, 4.9, and 4.. Our next goal is to derive the error estimate in weighted L norm. Theorem 4.. Let u and u be the same as those in Theorem 4.. If the given data gt and Kt, s in. belong to C m [, T ], then u u ω µ, µ C m U + κ log U + / log U 3, < µ <, C m U + κ U + µ U 3, µ =, 4. C m U + µ κ U + µ U 3, < µ <, for sufficiently large and for any κ, µ, where U = m x u ω m µ,m µ + K u, and K is defined by 4.. U = u H m; ω,, U 3 = K u H ω,, 4.3 Proof. Using the transformation.6 and.9, we change 4. to µ T x ex bx + B + x + τ µ x τ µ bτ dτ x [, ]. 4.4 where b and B are defined by 4.. It follows from the generalized Hardy s inequality Lemma 3.7 that e ω µ, µ C I ω µ, µ + I ω µ, µ + I 3 ω µ, µ. 4.5 Firstly, by Lemma 3., we see that I ω µ, µ = u I µ, µ u ω µ, µ C m m x u ω m µ,m µ. 4.6 ext, it follows from Lemma 3.8 and 4.8 that I ω µ, µ = t µ i I i,f i x C max I i, C m K u. 4.7 i i= ω µ, µ

16 6 YAPIG CHE AD TAO TAG By the convergence result in Theorem 4., we have u e + u { C / log u H ω, + u, < µ <, C µ u H ω, + u, < µ, 4.8 which, together with 4.7, gives I ω µ, µ { C m K / log u H ω, + u, < µ <, C m K µ u H ω, + u, < µ, 4.9 for sufficiently large. Moreover, it follows from 4.5, Lemma 3.8, and 3.9 that I 3 ω µ, µ = I µ, µ = I µ, µ IM ɛ ω µ, µ IM ɛ T M ɛ ω µ, µ I µ, µ M ɛ T M ɛ ω µ, µ + M ɛ T M ɛ ω µ, µ C M ɛ T M ɛ C κ M ɛ,κ C κ ɛ = C κ e, 4.3 where, in the last step we used Lemma 3.6 for any κ, µ. By the convergence result in Theorem 4., we obtain that C κ m log u H m; ω, + K u, < µ <, I 3 ω µ, µ C u κ m H m; ω, + log K u, µ =, 4.3 C u µ κ m H m; ω, + K u, < µ <, for sufficiently large and for any κ, µ. The desired estimate 4. is obtained by combining , 4.9 and 4.3. Finally, we state the main result of this paper, i.e., the error estimates for the numerical solutions to the Volterra integral equation.. Theorem 4.3. Let y be the exact solution of the Volterra integral equation.. Assume the approximated solution y is given by.8 using the spectral collocation scheme.4-.5 with the Jacobi Gauss, or Gauss-Radau, or Gauss-Lobbatto collocation points. If

17 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS 7 the given data gt and Kt, s in. belong to C m [, T ], then max yti y t i i+ C 3 m log u H m; ω, + K u, < µ <, C 3 u m H m; ω, + log K u, µ =, 4.3 C u µ m H m; ω, + K u, < µ <, where u C m [, ] is defined by. and.8, K is defined by 4.; and y y L ω,t C m U + κ log U + / log U 3, < µ <, C m U + κ U + µ U 3, µ =, 4.33 C m U + µ κ U + µ U 3, < µ <, for any κ, µ, where U, U, U 3 are defined by 4.3, and ωt := t µ Proof. Using.9 we have y y t = t µ u u x. ote that max i t µ i T max + x i µ C. i This, together with 4., leads to 4.3. The estimate 4.33 is obtained by using 4. and the observation T µ ωty y dt = + x µ u u dx T This completes the proof of Theorem 4.3. C x µ u u dx. 5. umerical experiments Let U = [u,, u ] T and F = [fx,, fx ] T. The numerical scheme.5 leads to a system of equation of the form U = F + AU, 5.

18 8 YAPIG CHE AD TAO TAG where the entries of the matrix A is given by [ ] µ T a ij = + x i Kx i, τ i θ k F j τ i θ k w k. k= Here, we simply introduce the computation of Gauss-Jacobi quadrature rule nodes and weights see the detailed algorithm and download related codes in []. The Gauss-Jacobi quadrature formula is used to numerically calculate the integral by using the formula x µ + x µ fxdx, fx [, ], x µ + x µ fxdx w i fx i. With the help of a change in the variables which changes both weights w i and nodes x i, we can get onto the arbitrary interval [a, b]. If you perform a change of variables, you should take into account that the formula of error term is changed along with the nodes and weights you can get a new form by changing the variables in the formula. The Gauss-Jacobi quadrature formula for a given order is completely defined by the set of nodes x i and weights w i see more details in [3, 4]. Example 5.. Consider the linear Volterra integral equations of second kind with a weakly singular kernel: yt = bt t i= t s µ ysds, t T, 5. with bt chosen so that yt = t µ sin t for < µ <. By calculation, bt = sin t t + π Γ µ t / µ sin t µ besselj µ, t, where besseljν, z is the Bessel function defined by k z ν besseljν, z = z 4 k!γν + k +. This problem has the property stated at the beginning of this paper, i.e., y t = t µ cost + t µ sint t µ at t = +. So according to the method in this paper, we need to compute ỹt = t µ yt = sint first. We can see the numerical results and convergence are given in Table, Figure, and Figure. k=

19 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS L Error 9.87e e e e-5.37e-6 L ω Error.44e-.938e-3.878e-4.547e e L Error 4.937e-8.35e-9.87e- 5.59e-3.7e-3 L ω Error.68e e e-.94e-3.345e-3 Table. Example 5.: the L and L ω errors for ỹt. Approximate Solution Exact Solution Figure. Example 5.: umerical and exact solution yt = sint t µ with T = 4π and µ =.6. Example 5.. Consider the following nonlinear Volterra integral equation of second kind with weakly singular kernels: where yt = gt t t s µ tan ys ds, t T, 5.3 gt = arctant 5 µ + t 6 µ B µ, 5 µ.

20 YAPIG CHE AD TAO TAG L Error L ω Error Figure. Example 5.: L and L ω errors versus the number of collocation points with the relationship between t and x is presented in paper: t = T x L Error 7.64e-3.438e e e e-6 L ω Error 9.99e-4.357e e e e L Error.57e-6.53e-7.594e-8.55e-9.458e- L ω Error.435e-7.473e-8.393e-9.76e-.49e- Table. Example 5.: the L and L ω errors for ỹt. It is easy to check that this problem has an unique solution yt = arctant 5 µ. We assume that ỹt := t µ yt and we can get ỹt C 5 I. The numerical results are presented in Table and Figure 3.

21 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS data data Figure 3. Example 5.: L and L ω errors versus the number of collocation points with the relationship between t and x is presented in paper: t = T x Conclusion and future work This work has been concerned with the Jacobi-collocation spectral analysis of the second kind Volterra integral equations which have a weakly singular kernel of the form t s µ, where µ,. The derivative y t of the solution behaves like t µ near the origin and this is expected to cause a loss in the global convergence order of collocation methods. To this end, the original equation was changed into a new Volterra integral equation which possesses better regularity, by applying some function transformation and variable transformations. ext, we directly presented a discretization scheme for the new Volterra integral equation. We proved the convergence of the method and obtained the error estimates in L norm and weighted L norm of the approximated solution. These results were confirmed by some numerical examples. We have also implemented the Jacobi-collocation method based on the Gauss-Jacobi quadrature formula.

22 YAPIG CHE AD TAO TAG In our future work, the stability will be established for these spectral collocation methods and spectral-galerkin methods also be studied for Volterra integral equations of second kind, with a weakly singular kernel. Acknowledgment. The authors are grateful to Mr. Xiang Xu of Fudan University for his assistance in providing us the numerical results in Section 5. References [] S. Bochkanov and V. Bystritsky, Computation of Gauss-Jacobi quadrature rule nodes and weights, [] H. Brunner, onpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. umer. Anal., 983, pp [3] H. Brunner, The numerical solutions of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comp., , pp [4] H. Brunner, Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. umer. Anal., 6 986, pp [5] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations Methods, Cambridge University Press 4. [6] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods Fundamentals in Single Domains, Springer-Verlag 6. [7] Y. Chen and T. Tang, Convergence analysis for the Chebyshev collocation methods to Volterra integral equations with a weakly singular kernel, submitted to SIAM J. umer. Anal.. [8] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93, Springer-Verlag, Heidelberg, nd Edition 998. [9] T. Diogo, S. McKee, and T. Tang, Collocation methods for second-kind Volterra integral equations with weakly singular kernels, Proceedings of The Royal Society of Edinburgh, 4A, 994, pp [] A. Gogatishvill and J. Lang, The generalized hardy operator with kernel and variable integral limits in Banach function spaces, Journal of Inequalities and Applications, 4 999, Issue, pp. -6. [] I. G. Graham and I. H. Sloan, Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in R 3, umerische Mathematik, 9, pp [] B. Guo, J. Shen and L. Wang, Optimal spectral-galerkin methods using generalized Jacobi polynomials, J. Sci. Comput. 7 6, [3] B. Guo and L. Wang, Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math. 4, pp [4] B. Guo and L. Wang, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory, 8 4, pp. -4. [5] Q. Hu, Stieltjes derivatives and polynomial spline collocation for Volterra integro-differential equations with singularities. SIAM J. umer. Anal., , 8-.

23 JACOBI COLLOCATIO METHODS FOR ITEGRAL EQUATIOS 3 [6] A. Kufner and L.E. Persson, Weighted Inequalities of Hardy Type, World Scientific, ew York, 3. [7] Ch. Lubich, Fractional linear multi-step methods for Abel-Volterra integral equations of the second kind, Math. Comp., , pp [8] G. Mastroianni and D. Occorsio, Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey, Journal of Computational and Applied Mathematics, 34, pp [9] P. evai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc., 8 984, MR 85c:49 [] C. K. -Qu and R. Wong, Szegö s Conjecture on Lebesgue Constants for Legendre Series, Pacific J. Math., , pp [] D. L. Ragozin, Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc., 5 97, pp [] D. L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., 6 97, pp [3] H. J.J. te Riele, Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution, IMA J. umer. Anal., 98, pp [4] S. G. Samko and R. P. Cardoso, Sonine integral equations of the first kind in L p, b, Fract. Calc. & Appl. Anal. 3, vol. 6, o 3, [5] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press Beijing 6. [6] T. Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, umer. Math., 6 99, pp [7] T. Tang, A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. umer. Anal., 3 993, pp [8] T. Tang, X. Xu, and J. Cheng, On spectral methods for Volterra type integral equations and the convergence analysis, submitted to J. Comput. Math. [9] Zheng-su Wan, Ben-yu Guo and Zhong-qing Wang, Jacobi pseudospectral method for fourth order problems, J. Comp. Math., 4 6, pp [3] D. Willett, A linear generalization of Gronwall s inequality, Proceedings of the American Mathematical Society, Vol. 6, o. 4. Aug., 965, pp School of Mathematical Sciences, South China ormal University, Guangzhou 563, Hunan, China address: yanpingchen@scnu.edu.cn Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong & Faculty of Science, Beijing University of Aeronautics and Astronautics, Beijing, China address: ttang@math.hkbu.edu.hk