SPECTRAL JACOBI-GALERKIN METHODS AND ITERATED METHODS FOR FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND WITH WEAKLY SINGULAR KERNEL

. We consider spectral and pseudo-spectral Jacobi-Galerkin methods and corresponding iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. The Gauss-Jacobi quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations, which show that the errors of the approximate solution decay exponentially in L ∞ norm and weighted L 2 -norm. The numerical examples are given to illustrate the theoretical results.


1.
Introduction. The integral equations with weakly singular kernels cover many important applications. Integral equations of this kind arise from potential problems, Dirichlet problems, the description of hydrodynamic interaction between elements of a polymer chain in solution, mathematical problems of radiative equilibrium and transport problems.
Let S is a linear operator with weakly singular kernel defined on the Banach space X = L 2 [a, b] or C[a, b] to X by Sy(t) = 686 YIN YANG AND YUNQING HUANG and K(t, τ ) ∈ C 1 ([a, b] × [a, b]), We are interested for Fredholm integral equations of the second kind, g is a given function, and y ∈ X is an unknown function to be determined. The equation (1) can be reformulated as where I be the identity operator defined on X.
The numerical treatment of (1) is not simple, mainly due to the fact that the solutions of (1) usually have a weak singularity at t = a or t = b, even when the inhomogeneous term g(t) is regular, which is discussed in [3]. In the last decade there has been considerable interest in the numerical analysis of solutions of integral equations with weakly singular kernels. Collocation Spectral methods and the corresponding error analysis have been provided recently [18,20] for for Volterra integral equation without the singular kernel in case of the underlying solutions are smooth. In [26], the authors extended the Legendre-collocation methods to nonlinear Volterra integral equations. Chen and Tang [5,6,21,27,28] developed a novel spectral Jacobi-collocation method to solve for Volterra integral equation with singular kernel and provided a rigorous error analysis which theoretically justifies the spectral rate of convergence, see also [23] for Jacobi spectral-collocation method for fractional integro-differential equations. Recently, in [20], the authors provided a Legendre spectral Galerkin method for second-kind Volterra integral equations, [22,19,24,25] provide general spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential equations.
In this paper, we apply spectral Jacobi-Galerkin and iterated Galerkin method for weakly singular Fredholm integral equations of the second kind. We use Jacobi polynomials as basis functions to find the approximate solution and iterated solution in weakly singular Fredholm integral equations of the second kind. The purpose of this paper is to obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations of the second kind using global polynomial bases. This paper is organized as follows. In Section 2, we demonstrate the implementation of the spectral Jacobi-Galerkin method and corresponding iterated method for Fredholm integral equations with weakly singular kernels. In Section 3, we obtain the convergence results in L ∞ norm and weighted L 2 -norm. In Section 4, we demonstrate the implementation of the pseudo-spectral Jacobi-Galerkin method and corresponding iterated method for Fredholm integral equations with weakly singular kernels. In Section 5, we obtain the convergence results in L ∞ norm and weighted L 2 -norm. In Section 6 we present numerical results. Finally, we end with conclusion and future work.
2. Spectral Jacobi-Galerkin methods. Set Λ = [−1, 1] and ω α,β (x) = (1 − x) α (1 + x) β be a weight function in the usual sense, for α, β > −1. The set of Jacobi polynomials {J α,β n (x)} ∞ n=0 forms a complete L 2 ω α,β (Λ) -orthogonal system, where L 2 ω α,β (Λ) is a weighted space defined by L 2 ω α,β (Λ) = {v : v is measurable and v ω α,β < ∞}, 1 2 , and the inner product Further, define and seminorm For the sake of applying the theory of orthogonal polynomials, we change the interval [a, b] to the standard interval Λ = [−1, 1], we use the variable transformation and let the Fredholm integral equations of the second kind with weakly singular kernel in one dimension (2) is of the form f is a given function, u ∈ X is an unknown to be determined and I denote the identity operator from X into itself, the integral operator G defined on the Banach and G is a compact linear operator on X into X. Then, the problem (3) reads: find u = u(x) such that

YIN YANG AND YUNQING HUANG
Let us demonstrate the numerical implementation of the spectral Jacobi-Galerkin approach first. Denote by N the set of all nonnegative integers. For any N ∈ N, P N denotes the set of all algebraic polynomials of degree at most N in Λ, J α,β j (x) is the j -th Jacobi polynomial corresponding to the weight function ω α,β (x). As a result, span{J α,β 1 (x), J α,β 2 (x), · · · , J α,β N (x)} as an orthonormal bases for the space P N , and P N is the subspaces of L 2 ω α,β . Our spectral Jacobi-Galerkin approximation of (5) is now defined as: Find u N ∈ P N such that where is the continuous inner product. Let Π α,β N : L 2 ω α,β → P N be the orthogonal projection defined by According to (6) and the definition of the projection operator Π α,β N , the spectral Jacobi-Galerkin solution u N satisfies Set u N (x) = N j=0 ξ j J α,β j (x). Substituting it into (6) and which leads to an equation of the matrix form where It is well-known that the iterated solution may improve order of convergence for the original solution of the equations. The iterated spectral Galerkin solution u it N corresponding to spectral Galerkin solution u N given by (8), is defined as follows: 3. Convergence analysis for spectral Jacobi-Galerkin method. In this section, we discuss the convergence rates for spectral Jacobi-Galerkin approximated solution and iterated spectral Jacobi-Galerkin solution for the Fredholm integral equations of the second kind with weakly singular kernel (4) and kernels of this type is common in integral equations resulting from solving the boundary value problems of partial differential equations. We will provide some elementary lemmas, which are important for the derivation of the main results in the subsequent section.
Proposition 1. ( [4,10]) Let Π α,β N : X → X denote the orthogonal projection defined by (7). Then the projection Π α,β N satisfies the following conditions: There exists a constant C > 0 such that for any N ∈ N and u ∈ X, Definition 3.1. Let X be a Banach space and, T and T N are bounded linear operators from X into X.
be the orthogonal projection defined by (7), then for any u ∈ H m ω α,β (Λ) and m ≥ 1, ) Let X be a Banach space and S ⊂ X is a relatively compact set. Assume that T and T N are bounded linear operators from S ⊂ X into S ⊂ X satisfying T N ≤ C for all N ∈ N, and for each x ∈ S, Lemma 3.4. (see [16,17]) For a nonnegative integer r and κ ∈ (0, 1), there exists a constant C r,κ > 0 such that for any where · r,κ is the standard norm in C r,κ ([−1, 1]), K N is a linear operator from C r,κ ([−1, 1]) into P N , as stated in [16,17].
Lemma 3.5. (see [7]) Let κ ∈ (0, 1) and let G be defined by Then, for any function v ∈ C([−1, 1]), there exists a positive constant C such that Here and below, C denotes a positive constant which is independent of N , and whose particular meaning will become clear by the context in which it arises.
Theorem 3.6. Let the kernel k(x, s) satisfies the conditions Let Π α,β N satisfies (C1) and (C2). Then GΠ α,β N is υ-convergent to G and there is a positive integer N such that for all n ≥ N , the inverse (I −GΠ α,β N ) −1 exists as linear operator defined on C[−1, 1] and there exists positive constants M independent of By the condition (A1), it follows that, using the estimate (14) lim which proves (i).
To prove the (ii), let B be a closed unit ball in Now to prove (iii), let the set S be defined by S = {GΠ α,β N u : N ≥ 1, u ∈ B}, where B denotes the closed unit ball in C[−1, 1]. To show that S is relatively compact set, by Arzela-Ascoli Theorem, we need to prove S is bounded and equicontinuous subset of C[−1, 1]. Using the estimate (16), S is bounded. To prove the equicontinuity of , let δ > 0, and The right hand side is independent of x, x , N , u, hence the equicontinuity of S will be proved if the right hand side can be shown to converge to zero as δ → 0. The first factor goes to zero from the condition (A2) and second factor is bounded from the estimate (13). This proves that S is equicontinuous. Hence, S is a relatively compact set in C[−1, 1]. It follows that This completes the proof that GΠ α,β N is υ-convergent to G. Since GΠ α,β N is υ-convergent to G, according to Anselone [1], the operator (I − GΠ α,β N ) −1 exists and are uniformly bounded for all N sufficiently large, i.e., there exists a positive real number M and a positive integer n such that for all N ≥ n, we have This completes the proof. Now, we obtain the convergence rates for the iterated solution in spectral Jacobi-Galerkin method for Fredholm integral equations of the second kind in both weighted L 2 norm and infinity norm.  (3) and (11), and assume that the hypothesis of last theorem holds and (I − G) −1 exists, then we have the following error estimate where C is independent of N .

YIN YANG AND YUNQING HUANG
The compactness of G and the pointwise convergence of {Π α,β exists and is uniformly bounded on N is the orthogonal projection from the space X into P N , then we have ν, (I − Π α,β N )u = 0, ∀ν ∈ P N . Now using Lemma 3.2 and for any ν ∈ P N , we have Hence using Lemma 3.4 and Lemma 3.5, we obtain We obtain the error bounds This completes the proof.
Theorem 3.8. Let u and u N be the solutions of (3) and (8), respectively and the hypotheses of Theorem 3.7 hold, then there exists a positive constant C independent of N , Using the fact that Π α,β N ∞ ≤ C log N (cf. [9]), we obtain  4. Pseudo-spectral Jacobi-Galerkin methods. Now we turn to describe the pseudo-spectral Jacobi-Galerkin method. Set it is clear that Using (N + 1)-point Gauss quadrature formula to approximate yields where {θ k } N k=0 and {θ k } N k=0 are the (N + 1)-degree Jacobi-Gauss points corresponding to the weights {ω −γ,0 k } N k=0 and {ω 0,−γ k } N k=0 corresponding. On the other hand, instead of the continuous inner product, the discrete inner product will be implemented in (6) and (9), i.e.
with {x k } N k=0 are N + 1 Jacobi-Gauss points corresponding to the weight function ω α,β (x). As a result, Substitute (21) and (22) into (6), the pseudo-spectral Jacobi-Galerkin method is to findū the matrix formĀ The iterated solutionū it N corresponding to pseudo-spectral Jacobi-Galerkin solutionū N given by (23), is defined as follows: 5. Convergence analysis for pseudo-spectral Jacobi-Galerkin method. In this section, we discuss the convergence rates for pseudo-spectral Jacobi-Galerkin approximated solution for the Fredholm integral equations of the second kind with weakly singular kernel. Now we investigate the estimate of pseudo-spectral Galerkin solution. Let is the Lagrange interpolation basis function with F j (x) is the Lagrange interpolation basis function associated with {x i } N i=0 which is the set of (N + 1) Jabobi-Gauss points corresponding to the weight function ω α,β (x), i. e, In terms of (23), the pseudo-spectral Galerkin solutionū N satisfies Let Combing (27) and (28), yields which gives rise toū We first consider an auxiliary problem, i.e., we want to findû N ∈ P N , such that In terms of the definition of I α,β N , (31) can be written as which is equivalent toû We quote the following lemma, which helps us to prove Theorems 5.7.
is the Lagrange interpolation basis function associated with (N + 1)-degree Jabobi-Gauss points corresponding to the weight function ω α,β (x).
By virtue of Lemma 3.4, Lamme 3.5 and Lemma 5.4, we have Combing (41), (42) and (43) we obtain, when N is large enough, Now we investigate the · ω α,β -error estimate. It follows from (40) and the Gronwall inequality that By Lemma 5.2, It follows from Lemma 5.3, Lemma 3.4 and Lemma 3.5 that Combing (44), (45) and (46) we obtain This completes the proof of the lemma.
6. Numerical experiments. In this section, we present two numerical examples to confirm the theoretical analysis obtained in the previous sections. For different kernels, and for different values of N , we compute u N and u it N in the spectral Jacobi-Galerkin method,ū N andū it N in the pseudo-spectral Jacobi-Galerkin method, and compare the result with exact solution u. To examine the accuracy of the results, · ∞ and · ω α,β errors are employed to assess the efficiency of the method. All the calculations are supported by the software Matlab.    Figure 1. Example 6.1 Errors of spectral Legendre-Galerkin method (left) and spectral Chebyshev-Galerkin method (right) versus N First we implement the spectral Legendre-Galerkin and Chebyshev-Galerkin methods to solve this example. Fig.1 (left) illustrates the · ∞ and · ω α,β errors of the spectral Legendre-Galerkin method(α = 0, β = 0) . Next the · ∞ and · ω α,β errors of the spectral Chebyshev-Galerkin method(α = − 1 2 , β = − 1 2 ) are demonstrated in Fig.1 (right). Clearly the desired spectral accuracy is obtained in these approaches. where f (t) = 1 − π 2 − 2t 1/2 − 2(1 − t) 1/2 − t log(1 + (1 − t) 1/2 ) − (1 − t) log(1 + t 1/2 ) + 1 2 t log(t) + 1 2 (1 − t) log(1 − t) and exact solution u(t) = 1 + t 1/2 + (1 − t) 1/2 . This problem has the property stated at the beginning of this paper, i.e., which is singular at t = 0 + and t = 1 − . In Fig.2 the errors are given for different values of N . It can be seen that the errors decay algebraically as the exact solution for this example is not sufficiently smooth.
7. Concluding remarks. This work has been concerned with the spectral and pseudo-spectral Jacobi-Galerkin methods and corresponding iterated methods for the Fredholm integral equations with weakly singular kernel. The most important contribution of this work is that we are able to demonstrate rigorously that the errors of spectral approximations decay exponentially in both infinity and weighted norms, which is a desired feature for a spectral method. Although in this work our convergence theory does not cover the nonlinear case, the methods described above remain applicable, it will be possible to extend the results of this paper to nonlinear case which will be the subject of our future work.