Superconvergence results of Legendre spectral projection methods for weakly singular Fredholm–Hammerstein integral equations

Accepted Manuscript Superconvergence results of Legendre spectral projection methods for weakly singular Fredholm-Hammerstein integral equations Moumita Mandal, Gnaneshwar Nelakanti

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S0377-0427(18)30583-1 https://doi.org/10.1016/j.cam.2018.09.032 CAM 11927

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Journal of Computational and Applied Mathematics

Received date : 22 March 2018 Revised date : 6 September 2018 Please cite this article as: M. Mandal, G. Nelakanti, Superconvergence results of Legendre spectral projection methods for weakly singular Fredholm-Hammerstein integral equations, Journal of Computational and Applied Mathematics (2018), https://doi.org/10.1016/j.cam.2018.09.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Superconvergence results of Legendre spectral projection methods for weakly singular Fredholm-Hammerstein integral equations Moumita Mandala,∗, Gnaneshwar Nelakantia,∗∗ a

Department of Mathematics, Indian Institute of Technology, Kharagpur, Kharagpur 721 302, India

Abstract In this paper, we consider the Galerkin method to approximate the solution of Fredholm-Hammerstein integral equations of second kind with weakly singular kernels, using Legendre polynomial bases. We prove that for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence 1 O(n−r ), whereas the iterated Legendre Galerkin method converges with the order O(n−r−α+ 2 ) for the 1 algebraic kernel, and order O(log n n−r− 2 ) for logarithmic kernel in both L2 -norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the solution. We also propose the Legendre multi-Galerkin and iterated Legendre multiGalerkin methods. We prove that iterated Legendre multi-Galerkin method has order of convergence 1 3 O((1 + c log n)n−r−2α+ 2 ) for the algebraic kernel, and order of convergence O((log n)2 (1 + c log n)n−r− 2 ) for logarithmic kernel in both L2 -norm and infinity norm. Numerical examples are given to illustrate the theoretical results. Keywords: Fredholm-Hammerstein integral equations, Weakly singular kernels, Galerkin method, Multi-Galerkin method, Legendre polynomial, Superconvergence rates 2010 MSC: 45B05, 45G10, 65R20 1. Introduction Consider the following Fredholm-Hammerstein integral equation of the second kind  1 x(t) − k(t, s)ψ(s, x(s)) ds = f(t), − 1 ≤ t ≤ 1,

(1.1)

−1

where the kernel k(t, s) is of the form ⎧ ⎨m(t, s) |t − s|α−1 , if 1 < α < 1, 2 k(t, s) = ⎩m(t, s) log(|t − s|), if α = 1,

and m(t, s) ∈ C 1 ([−1, 1] × [−1, 1]), f and ψ are known functions, x is the unknown function to be determined. ∗ ∗∗

Corresponding author. [email protected] [email protected]

Preprint submitted to Elsevier

September 6, 2018

Integral equations with weakly singular kernels of the algebraic and logarithmic types cover many important applications and these kind of integral equations arise from potential problems, Dirichlet problems, the description of the hydrodynamic interaction between elements of a polymer chain in solution, mathematical problems of radiative equilibrium and transport problems. Many authors have studied for numerical methods to solve Fredholm-Hammerstein integral equations with smooth kernels as well as with weakly singular kernels (see [4], [8], [10], [11], [12], [13], [14], [16], [17]). The Galerkin, collocation, Petrov-Galerkin, degenerate kernel and Nystr¨om methods are commonly used approximation methods for finding the numerical solution of the equation (1.1). In [16] and [17], a new type of collocation method was discussed in certain piecewise-polynomial space to establish superconvergence results for Fredholm-Hammerstein integral equations of second kind with smooth kernels. In [9], Kaneko et al. showed that the solutions of weakly singular Fredholm-Hammerstein integral equations satisfy certain regularity properties. In ([10], [11], [14]), Kaneko et al. discussed the Galerkin and collocation methods to solve the weakly singular Fredholm-Hammerstein integral equations of type (1.1). They proved that in a certain piecewise polynomials space of degree ≤ r−1 for both the algebraic and logarithmic kernels, Galerkin and collocation methods have order of convergence O(hr ), whereas the iterated Galerkin and iterated collocation methods converge with the order O(hr+α ) for the algebraic kernel, and of order O(log h hr+1 ) for logarithmic kernel, where h is the norm of the partition. However, to get better accuracy in piecewise polynomial based projection methods, the number of partition points should be increased. Hence in such cases, one has to solve a large system of non-linear equations, which is computationally expensive. To minimize computational complexity, one can use the approximating subspace Xn as a subspace of global polynomials. In particular one can use Legendre polynomials as basis functions for the subspace Xn . So, in last some years, different spectral methods have been developed rapidly and it has been applied to linear integral equations with weakly singular kernels ([20]) and nonlinear integral equations with smooth kernels ([6], [7]). In [20], Panigrahi et al. solved the linear Fredholm integral equations with weakly singular kernel and showed that for algebraic type kernel, iterated 1 Galerkin solution converges with the order O(n−r−α+ 2 ), and for logarithmic type kernel, it converges with 1 the order O(log n n−r− 2 ) in both L2 - norm and infinity norm. In this paper, we apply the Galerkin method and its iterated version to solve weakly FredholmHammerstein integral equations of type (1.1) for both the algebraic and logarithmic type kernels, using Legendre polynomial bases. We show that for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence O(n−r ) in both L2 -norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the 1 solution. Also we show that the iterated Legendre Galerkin method converges with the order O(n−r−α+ 2 ) 1 for the algebraic kernel and with the order O(log n n−r− 2 ) for logarithmic kernel in both L2 -norm and infinity norm. In ([5], [18]), Legendre multi-projection (Legendre multi-Galerkin and Legendre multi-collocation) methods were discussed to solve Urysohn and Fredholm-Hammerstein integral equations of second kind with smooth kernels and obtained superconvergence results. Here we also discuss the Legendre multiGalerkin and iterated Legendre multi-Galerkin methods to solve weakly Fredholm-Hammerstein integral equations of type (1.1) for both the algebraic and logarithmic type kernels to improve the con2

vergence rates. We prove that for the algebraic kernel, iterated Legendre multi-Galerkin method has 1 order of convergence O((1 + c log n)n−r−2α+ 2 ), and for logarithmic kernel, it has order of convergence 3 O((log n)2 (1 + c log n) n−r− 2 ) in both L2 -norm and infinity norm. Thus under the same assumptions as in the Legendre Galerkin method, the proposed iterated Legendre multi-Galerkin method converges faster than Legendre Galerkin and iterated Legendre Galerkin methods. Moreover, the size of the system of equations that must be solved, remains the same as in Legendre Galerkin method. We illustrate our theoretical results by numerical examples. We organize this paper as follows. In section 2, we apply the Legendre Galerkin and iterated Legendre Galerkin methods to solve the equation (1.1) and discuss the convergence results. In Section 3, we propose the Legendre multi-Galerkin method and its iterated version to obtain superconvergence results. In Section 4, we give the implementation details with numerical results to illustrate the theoretical results. Throughout this paper, we assume that c is a generic constant. 2. Legendre Galerkin method: Fredholm-Hammerstein integral equations with weakly singular kernels Let X = C[−1, 1]. Consider the following Fredholm-Hammerstein integral equation of the second kind  1 x(t) − k(t, s)ψ(s, x(s)) ds = f(t), − 1 ≤ t ≤ 1, (2.1) −1

where the kernel k(t, s) is of the form

⎧ ⎨m(t, s) |t − s|α−1 , if 1 < α < 1, 2 k(t, s) = kt (s) = ⎩m(t, s) log(|t − s|), if α = 1,

(2.2)

and m(t, s) ∈ C 1 ([−1, 1]×[−1, 1]), f and ψ are known functions, x is the unknown function to be determined. Let C r [−1, 1] denotes the space of r-times continuously differentiable functions and for any u ∈ C r [−1, 1], denote ur,∞ = max{u(j) ∞ : 0 ≤ j ≤ r}, where u(j) denotes the j-th derivative of u. Throughout the paper, the following assumptions are made on f, k(., .) and ψ(., x(.)): (i) f ∈ C r [−1, 1], r ≥ 1. (ii) m(., .) ∈ C 1 ([−1, 1] × [−1, 1]). (iii) The nonlinear function ψ(s, x(s)) and the partial derivative ψ (0,1) (s, x(s)) of ψ w.r.t the second variables exists and Lipschitz continuous i.e., for any x1 , x2 ∈ X, s ∈ [−1, 1], ∃ c1 , c2 > 0 such that |ψ(s, x1 (s)) − ψ(s, x2 (s))| ≤ c1 |x1 (s) − x2 (s)|,

|ψ (0,1) (s, x1 (s)) − ψ (0,1) (s, x2 (s))| ≤ c2 |x1 (s) − x2 (s)|. 3

(iv) d = sup |ψ (0,1) (s, x(s))| < ∞. s∈[−1,1]

Let Ky(t) =



1 −1

k(t, s)y(s) ds, t ∈ [−1, 1], y ∈ X.

Note that the kernel k(t, s), for 12 < α ≤ 1 satisfies the following condition  1 |k(t, s)|2 ds = M < ∞. A1. sup t∈[−1,1] −1

A2. lim



1

s→a −1

|k(t, s) − k(a, t)|2 ds = 0,

− 1 ≤ a ≤ 1.

Then K is a compact linear operator on C[−1, 1] (see [15] and [19]). Note that, using assumption A1 and Holder’s inequality, for any y ∈ X, we have   1   k(t, s)y(s) ds Ky∞ = sup |Ky(t)| = sup  t∈[−1,1]

−1

t∈[−1,1]

≤

sup

t∈[−1,1]

√

≤

√

≤



1

−1

2

|k(t, s)| ds

 21 

1

−1

2

|y(s)| ds

 12

MyL2

(2.3)

2My∞ .

(2.4)

which implies that KyL2 ≤

√

2Ky∞ ≤

√

2MyL2 .

(2.5)

This implies KL2 ≤

√

2M.

(2.6)

We will use Kumar and Sloan [17] technique for solving the approximate solution of the equation (2.1). The Galerkin method will now be applied to an equivalent equation for the function z defined by z(t) = ψ(t, x(t)) t ∈ [−1, 1].

(2.7)

Note that ψ(., .) ∈ C r ([−1, 1] × R), and x ∈ C r [−1, 1] (see [9]), using the chain rule for higher derivative it is easy to obtain that z ∈ C r [−1, 1]. The desired exact solution x of (2.1) is obtained by the equation x(t) = f(t) +



1 −1

k(t, s)z(s) ds, − 1 ≤ t ≤ 1.

(2.8)

Then the equation (2.8) will take the form x = f + Kz. 4

(2.9)

For our convenience, we consider a nonlinear operator Ψ : X → X defined by Ψ(x)(t) = ψ(t, x(t)).

(2.10)

z = Ψ(f + Kz).

(2.11)

Then the equation (2.7) becomes Let T (v) = Ψ(f + Kv), v ∈ X, then the equation (2.11) can be written as z = T z.

(2.12)

√ We assume that the constants M and c1 satisfy the condition that 2Mc1 < 1. Then according to the analysis of [9] and from estimate (2.4), it follows that the equation (2.12), has a unique solution, say z0 ∈ X. Note that for any x1 , x2 ∈ X, s ∈ [−1, 1], we have |Ψ(x1 )(s) − Ψ(x2 )(s)| = |ψ(s, x1 (s)) − ψ(s, x2 (s))| ≤ c1 |x1 (s) − x2 (s)|,

|Ψ (x1 )(s) − Ψ (x2 )(s)| = |ψ (0,1) (s, x1 (s)) − ψ (0,1) (s, x2 (s))| ≤ c2 |x1 (s) − x2 (s)|.

Next, we describe the Legendre Galerkin and iterated Legendre Galerkin methods. To do this, let Xn = span{φ0 , φ1 , φ2 , · · · φn }, be the sequence of Legendre polynomial subspaces of X of degree ≤ n, where {φ0 , φ1 , φ2 , · · · φn }, forms an orthonormal basis for Xn . Here φi ’s are given by 2i + 1 φi (s) = Li (s), i = 0, 1, · · · , n, 2 where Li ’s are the Legendre polynomials of degree ≤ i. These Legendre polynomials can be generated by the following three-term recurrence relation L0 (s) = 1, L1 (s) = s,

s ∈ [−1, 1],

and (i + 1)Li+1 (s) = (2i + 1)sLi (s) − iLi−1 (s), i = 1, 2, 3 · · · , n − 1. Orthogonal projection operator: Let Pn : X → Xn be the orthogonal projection operator defined by

Pn u, v = u, v , v ∈ Xn , u ∈ X,

(2.13)

1 where u, v = −1 u(t)v(t)dt. According to the analysis of ([3, pp. 283-289], [21]), Pn satisfies the following Lemmas. Lemma 1. Let Pn : X → Xn is the orthogonal projection operator defined by (2.13). Then i) Pn uL2 ≤ puL2 ,

(2.14)

Pn uL2 ≤ p1 u∞ ,

(2.15)

and where p and p1 are constants independent of n. 5

ii) There exists a constant C > 0 such that for any n ∈ N and u ∈ X, Pn u − uL2 ≤ C inf u − φL2 → 0, n → ∞, φ∈Xn

(2.16)

where C is a constant independent of n. Lemma 2. Let Pn : X → Xn is the orthogonal projection operator defined by (2.13). Then for any u ∈ C r [−1, 1], u − Pn uL2 ≤ cn−r u(r) ∞ ,

u − Pn u∞ ≤ cn u − Pn u∞ ≤ cn

3 −r 4 1 −r 2

u(r) ∞ , V (u(r) ),

(2.17) (2.18) (2.19)

where c is a constant independent of n and V (u(r) ) denotes the total variation of u(r) . From Lemmas 1 and 2, for any u ∈ C[−1, 1], we have u − Pn uL2 → 0 as n → ∞,

(2.20)

u − Pn u∞  0 as n → ∞.

(2.21)

The Galerkin method for equation (2.11) is seeking an approximate solution zn ∈ Xn such that

Let Tn be the operator defined by

zn = Pn Ψ(f + Kzn ).

(2.22)

Tn (u) := Pn Ψ(f + Ku).

(2.23)

Then the equation (2.22) can be written as zn = Tn (zn ).

(2.24)

Corresponding approximate solution xn of x is given by xn = f + K(zn ).

(2.25)

In order to obtain more accurate approximation solution for (2.11), we define the iterated solution as z˜n = Ψ(f + Kzn ).

(2.26)

Applying Pn on both sides of the equation (2.26), we obtain Pn z˜n = Pn Ψ(f + Kzn ).

(2.27)

From equations (2.22) and (2.27), it follows that Pn z˜n = zn . Using this, we see that the iterated solution z˜n satisfies the following equation z˜n = Ψ(f + KPn z˜n ). (2.28) 6

Letting Tn (u) := Ψ(f + KPn u), u ∈ X, the equation (2.28) can be written as z˜n = Tn z˜n . Corresponding approximate solution x˜n of x is given by zn + f. (2.29) x˜n = K˜ Now we discuss the existence and uniqueness of the approximate and iterated approximate solutions. To do this, let BL(X) denote the space of all bounded linear operators on X and we recall the definition of ν-convergence [1], a theorem from [2] and a lemma [20], which are useful in proving existence and convergence of approximated solutions. Definition 1. (ν-convergence) Let T ∈ BL(X) and {Tn } be a sequence in BL(X), then {Tn } is said to be ν convergent to T if Tn  ≤ C, (Tn − T )T  → 0 and (Tn − T )Tn  → 0, as n → ∞. Theorem 1. Let X be a Banach space and T , Tn ∈ BL(X). If Tn is norm convergent or ν-convergent to T and (I − T )−1 exists and is bounded on X, then for sufficiently large n, (I − Tn )−1 exists and is uniformly bounded on X. Lemma 3. [20] Let the kernel k(t, s) is of the form given by (2.2), then for each t ∈ [−1, 1], there exists a polynomial vt of degree less than or equal to n such that 1 O(n−α ), 2 < α < 1,

1  kt (.) − vt L1 = O n log n , α = 1.

We quote the following theorem from [22], which gives us the condition under which the solvability of one equation leads to the solvability of the other equation. Theorem 2. ([22]) Let F and F be continuous operators over an open set Ω in a Banach space X. Let  has an isolated solution u˜0 ∈ Ω and let the following conditions be satisfied. the equation u = Fu (a) The operator F is Frechet differentiable in some neighborhood of the point u˜0 , while the linear operator  (˜ uo ) is continuously invertible. I −F (b) Suppose that for some δ > 0 and 0 < q < 1, the following inequalities are valid (the number δ is assumed to be so small that the sphere u − u˜0  ≤ δ is contained within Ω) sup u−˜ u0 ≤δ

 (˜  (u) − F  (˜ (I − F uo ))−1 (F uo ) ≤ q,

 (˜  uo ) − F(˜  uo ) ≤ δ(1 − q). uo ))−1 (F(˜ α = (I − F

(2.30) (2.31)

 has a unique solution uˆ0 in the sphere u − u˜0  ≤ δ. Moreover, the inequality Then the equation u = Fu α α ≤ ˆ u0 − u˜0  ≤ (2.32) 1+q 1−q is valid.

Lemma 4. Let z0 ∈ C[−1, 1] be the isolated solution of the equation (2.11) and Pn : X → Xn is an orthogonal projection operator defined by (2.13). Let the kernel k(t, s) is of the form given by (2.2). Then there holds √ √ K(I − Pn )z0 L2 ≤ 2K(I − Pn )z0 ∞ ≤ 2 M(I − P)z0 L2 → 0, as n → ∞.

In particular, if z0 ∈ C r [−1, 1], we have



1

O(n−r−α+ 2 ), K(I − Pn )z0 ∞ = 1 O(n−r− 2 log n), 7

1 2

< α < 1, α = 1.

Proof. Using Cauchy-Schwarz inequality and the estimates (2.6) and (2.20), we have   1    sup  k(t, s)(I − Pn )z0 (s) ds K(I − Pn )z0 ∞ = sup |K(I − Pn )z0 (t)| = t∈[−1,1]

t,s∈[−1,1]

=

sup

t,s∈[−1,1]

−1

| < k(t, s), (I − Pn )z0 (s) > |

≤ kL2 (I − Pn )z0 L2 √ 2M(I − Pn )z0 L2 → 0, as n → ∞. ≤ This implies K(I − Pn )z0 L2 ≤

√

√ 2K(I − Pn )z0 ∞ ≤ 2 M(I − Pn )z0 L2 → 0, as n → ∞.

Since Pn is the orthogonal projection from the space X into Xn , then for each t ∈ [−1, 1], there exists a polynomial vt ∈ Xn , we have < vt , (I − Pn )z0 >= 0. Using this with Lemma 3 and estimate (2.19), we have K(I − Pn )z0 ∞ =

sup t,s∈[−1,1]

k(t, s) − vt , (I − Pn )z0 (s) >| ≤ kt (.) − vt L1 (I − Pn )z0 ∞ ≤ =





1

(r)

kt (.) − vt L1 cn−r+ 2 V (z0 ),

kt (.) − vt L1 cn

−r+ 12

1

O(n−r−α+ 2 ), 1 O(n−r− 2 log n),

1 2

(r) V (z0 ),

<α<1

α=1

1 2

< α < 1, α = 1.

This completes the proof.



Lemma 5. Let Pn : X → Xn is an orthogonal projection operator defined by (2.13). Let the kernel k(t, s) is of the form given by (2.2). Then there holds 1 O ((1 + c log n)n−α ) , 2 < α < 1, 

1 K(I − Pn )∞ = O (1 + c log n) n log n , α = 1.

Proof. Using the fact that Pn ∞ ≤ c log n (cf., Page-147, [2]), from Lemma 3, for z0 ∈ C[−1, 1], we have   1   k(t, s)(I − Pn )z0 (s) ds = sup |< k(t, s) − vt , (I − Pn )z0 >| K(I − Pn )z0 ∞ = sup  t∈[−1,1]

This implies

−1

t,s∈[−1,1]

≤ kt (.) − vt L1 (I − Pn )z0 ∞

≤ kt (.) − vt L1 (1 + Pn ∞ )z0 ∞ c(1 + c log n)n−α z0 ∞ ,

 ≤ c(1 + c log n) n1 log n z0 ∞ ,



O ((1 + c log n)n−α ) ,

 K(I − Pn )∞ = O (1 + c log n) n1 log n ,

This completes the proof.

8

1 2

< α < 1, α = 1.

1 2

< α < 1, α = 1. 

Lemma 6. Let T  (z0 ) and Tn (z0 ) be the Fr´echet derivatives of T (z) and Tn (z), respectively at z0 . Then (I − Pn )T  (z0 )L2 → 0, n → ∞, (I − Pn )Tn (z0 )L2 → 0, n → ∞.

Proof. Since Tn (z0 ) = Ψ (f + KPn z0 )KPn , using the Lipschitz continuity of ψ (0,1) (., x(.)), Lemma 4 and boundedness of Ψ (f + Kz0 ), we have Ψ (f + KPn z0 )L2 ≤ Ψ (f + KPn z0 ) − Ψ (f + Kz0 )L2 + Ψ (f + Kz0 )L2 ≤ c2 K(Pn − I)z0 L2 + Ψ (f + Kz0 )L2 ≤ B < ∞,

(2.33)

where B is a constant independent of n. Next, let U := {x ∈ X : xL2 ≤ 1} be the closed unit ball in X. We have Tn (z0 ) = Ψ (f + KPn z0 )KPn . Since KPn is a sequence of compact operators for 12 < α ≤ 1 and Ψ (f + KPn z0 ) is bounded in L2 - norm, Tn (z0 ) are compact operators. Then it can be shown that S = {Tn (z0 )x : x ∈ U, n ∈ N} is relatively compact set. Using the estimate (2.16), we can conclude (I − Pn )Tn (z0 )L2 = sup {(I − Pn )Tn (z0 )xL2 : x ∈ U }

= sup {(I − Pn )yL2 : y ∈ S} → 0, n → ∞.

(2.34)

Similarly, since Ψ (f + Kz0 ) is bounded and K is compact for 12 < α ≤ 1, T  (z0 ) = Ψ (f + Kz0 )K is also compact and we have (I − Pn )T  (z0 )L2 → 0, n → ∞. This complete the proof.



Theorem 3. Let z0 ∈ C r [−1, 1] be an isolated solution of the equation (2.11). Let the kernel k(t, s) is of the form given by (2.2). Assume that 1 is not an eigenvalue of the linear operator T  (z0 ). Let Pn : X → Xn be the orthogonal projection operator defined by (2.13). Then the equation (2.22) has a unique solution zn ∈ B(z0 , δ) = {z : z − z0 L2 < δ} for some δ > 0 and for sufficiently large n. Moreover, there exists a constant 0 < q < 1, independent of n such that αn αn ≤ zn − z0 L2 ≤ , 1+q 1−q where αn = (I − Tn (z0 ))−1 (Tn (z0 ) − T (z0 ))L2 . Further, we obtain zn − z0 L2 ≤ cA1 (I − Pn )z0 )L2 = O(n−r ), where (I − Tn (z0 ))−1 L2 ≤ A1 < ∞ and c is a constant independent of n. Proof. Using Lemma 6, we have Tn (z0 ) − T  (z0 )L2 = Pn Ψ (f + Kz0 )K − Ψ (f + Kz0 )KL2 = (I − Pn )Ψ (f + Kz0 )KL2

= (I − Pn )T  (z0 )L2 → 0 as n → ∞.

Since 1 is not an eigenvalue of T  (z0 ), (I − T  (z0 )) is invertible. Hence by applying Theorem 1, we have (I − Tn (z0 ))−1 exists and is uniformly bounded on X, for some sufficiently large n, i.e., there exists some 9

A1 > 0 such that (I − Tn (z0 ))−1 L2 ≤ A1 < ∞. Now from estimates (2.3) and (2.15), for any z ∈ B(z0 , δ) and v ∈ X, we have [Tn (z0 ) − Tn (z)]vL2 = [Pn Ψ (f + Kz0 )K − Pn Ψ (f + Kz)K]vL2 ≤ p1 [Ψ (f + Kz0 ) − Ψ (f + Kz)]Kv∞

≤ p1 Ψ (f + Kz0 ) − Ψ (f + Kz)∞ Kv∞ √ ≤ p1 MΨ (f + Kz0 ) − Ψ (f + Kz)∞ vL2 . Taking use of the Lipschtiz continuity of ψ (0,1) (., x(.)) and estimate (2.3), we have √ √ Ψ (f + Kz0 ) − Ψ (f + Kz)∞ ≤ c2 K(z0 − z)∞ ≤ c2 Mz0 − zL2 ≤ Mc2 δ.

(2.35)

(2.36)

Using this in the estimate (2.35), we obtain [Tn (z0 ) − Tn (z)]vL2 ≤ p1 Mc2 δvL2 . Thus we have sup z−z0 L2 ≤δ

(I − Tn (z0 ))−1 (Tn (z0 ) − Tn (z))L2 ≤ A1 p1 Mc2 δ ≤ q (say).

Here we choose δ in such a way that, 0 < q < 1. This proves the equation (2.30) of Theorem 2. Taking use of (2.20), we have αn = (I − Tn (z0 ))−1 (Tn (z0 ) − T (z0 ))L2 ≤ A1 (Tn (z0 ) − T (z0 ))L2

≤ A1 Pn Ψ(f + Kz0 ) − Ψ(f + Kz0 )L2

≤ A1 (I − Pn )Ψ(f + Kz0 )L2 ≤ A1 (I − Pn )z0 )L2

→ 0, n → ∞.

(2.37)

By choosing n large enough such that αn ≤ δ(1 − q), the equation (2.31) of Theorem 2 is satisfied. Hence by applying Theorem 2, we obtain αn αn ≤ zn − z0 L2 ≤ , 1+q 1−q and zn − z0 L2 ≤ Hence from estimate (2.17), we have

αn ≤ cA1 (I − Pn )z0 )L2 . 1−q

zn − z0 L2 = O(n−r ). This complete the proof.



In the following theorem we give the error bounds for the Galerkin approximate solution xn , defined by (2.25) and the exact solution x0 .

10

Theorem 4. Let x0 ∈ C r [−1, 1] be an isolated solution of the equation (2.1) and xn be the Legendre Galerkin approximation of x0 . Let the kernel k(t, s) is of the form given by (2.2). Then there hold xn − x0 ∞ = O(n−r ), and xn − x0 L2 ≤

√

2xn − x0 ∞ = O(n−r ).

Proof. From the estimates (2.3), (2.9), (2.25) and Theorem 3, we obtain √ xn − x0 ∞ = K(z0 − zn )∞ ≤ Mz0 − zn L2 = O(n−r ), and xn − x0 L2 ≤

√

2xn − x0 ∞ = O(n−r ).

This completes the proof.



Next we discuss the existance and convergence of the iterated approximate solutions z˜n to z0 . Theorem 5. Tn (z0 ) is ν-convergent to T  (z0 ) in infinity-norm. Proof. Consider

|Tn (z0 )z(t)| = |Ψ (f + KPn z0 )KPn z(t)|

≤ |Ψ (f + KPn z0 ) − Ψ (f + Kz0 )||KPn z(t)| + |Ψ (f + Kz0 )||KPn z(t)|.

(2.38)

Now using the Lipschtiz continuity of ψ (0,1) (., x(.)) and Lemma 4, we have Ψ (f + KPn z0 ) − Ψ (f + Kz0 )∞ ≤ c2 K(I − Pn )z0 ∞ → 0 as n → ∞.

(2.39)

Using estimates (2.3) and (2.15), we have KPn z∞ ≤

√

MPn zL2 ≤

√

Mp1 z∞ .

(2.40)

Now combining the estimates (2.39) and (2.40) with (2.38), we obtain √ Tn (z0 )∞ ≤ Mp1 (c2 K(I − Pn )z0 ∞ + Ψ (f + Kz0 )∞ ) < ∞.

(2.41)

This shows that Tn (z0 )∞ is uniformly bounded. Next consider

|(Tn (z0 ) − T  (z0 ))Tn (z0 )z(t)| = |{Ψ (f + KPn z0 )KPn − Ψ (f + Kz0 )K}Tn (z0 )z(t)| ≤ |Ψ (f + KPn z0 )(KPn − K)T  (z0 )z(t)| n

+ |{Ψ (f + KPn z0 ) − Ψ (f + Kz0 )}KTn (z0 )z(t)|. 



(2.42)

Now for the first term of the above estimate (2.42), using the Lipschitz continuity of ψ (0,1) (., x(.)), Lemma 4 and boundedness of Ψ (f + Kz0 ), we have Ψ (f + KPn z0 )∞ ≤ Ψ (f + KPn z0 ) − Ψ (f + Kz0 )∞ + Ψ (f + Kz0 )∞ ≤ c2 K(Pn − I)z0 ∞ + Ψ (f + Kz0 )∞ ≤ B1 < ∞, 11

(2.43)

and (K(I

− Pn )Tn (z0 )z∞

=

sup t,s∈[−1,1]

≤

≤

≤

√

√

√

   

1

k(t, s)(I −1

 

− Pn )Tn (z0 )z(s) ds

M(I − Pn )Tn (z0 )zL2 M(I − Pn )T  (z0 )L2 zL2 n

2M(I − Pn )Tn (z0 )L2 z∞ .

(2.44)

For the second term of the estimate (2.42), using the Lipschitz continuity of ψ (0,1) (., x(.)) and estimates (2.3) and (2.4), we have {Ψ (f + KPn z0 ) − Ψ (f + Kz0 )}KTn (z0 )z∞ ≤ Ψ (f + KPn z0 ) − Ψ (f + Kz0 )∞ KTn (z0 )z∞ ≤ c2 K(I − Pn )z0 ∞ KTn (z0 )z∞ √ ≤ c2 2M(I − Pn )z0 L2 Tn (z0 )∞ z∞ . (2.45)

Now combining the estimates (2.43), (2.44) and (2.45) with (2.42), we see that √ √ (Tn (z0 ) − T  (z0 ))Tn (z0 )z∞ ≤ {B1 2M(I − Pn )Tn (z0 )L2 + c2 2M(I − Pn )z0 L2 Tn (z0 )∞ }z∞ .

Hence using estimate (2.20), Lemma 6 and the uniform boundedness of Tn (z0 )∞ , we obtain (Tn (z0 ) − T  (z0 ))Tn (z0 )∞ → 0, as n → ∞.

This shows that Tn (z0 ) is ν-convergent to T  (z0 ) in infinity norm. This completes the proof.



Hence by applying the Theorem 1 and Theorem 5 we obtain the following theorem.

Theorem 6. Let z0 ∈ C[−1, 1] be an isolated solution of the equation (2.11). Let the kernel k(t, s) is of the form given by (2.2). Assume that 1 is not an eigenvalue of the linear operator T  (z0 ). Then for sufficiently large n, the operator (I − Tn (z0 )) is invertible on X and there exists a constant L > 0 independent of n such that (I − Tn (z0 ))−1 ∞ ≤ L < ∞.

Theorem 7. Let z0 ∈ C r [−1, 1] be an isolated solution of the equation (2.11). Let the kernel k(t, s) is of the form given by (2.2). Assume that 1 is not an eigenvalue of the linear operator Ψ (f + Kz0 )K. Let Pn : X → Xn be the orthogonal projection operator defined by (2.13). Then the equation (2.26) has a unique solution z˜n ∈ B(z0 , δ) = {z : z − z0 ∞ < δ} for some δ > 0 and for sufficiently large n. Moreover, there exists a constant 0 < q < 1, independent of n such that βn βn ≤ ˜ zn − z0 ∞ ≤ , 1+q 1−q where βn = (I − Tn (z0 ))−1 (Tn (z0 ) − T (z0 ))∞ . Further, we obtain 1 1 < α < 1, O(n−r−α+ 2 ), 2 ˜ z n − z 0 ∞ = −r− 12 log n), α = 1. O(n

12

Proof. Using Theorem 6, we have (I − Tn (z0 ))−1 ∞ ≤ L < ∞. Consider for any z ∈ B(z0 , δ) and v ∈ X,

[Tn (z) − Tn (z0 )]v∞ = [Ψ (f + KPn z) − Ψ (f + KPn z0 )]KPn v∞

≤ Ψ (f + KPn z) − Ψ (f + KPn z0 )∞ KPn v∞ .

Using the Lipschitz continuity of ψ (0,1) (., x(.)) and estimate (2.40), we have √ √ Ψ (f + KPn z) − Ψ (f + KPn z0 )∞ ≤ c2 KPn (z − z0 )∞ ≤ Mp1 c2 z − z0 ∞ ≤ Mp1 c2 δ.

(2.46)

(2.47)

Combining the estimates (2.40), (2.46) and (2.47), we obtain [Tn (z) − Tn (z0 )]vL2 ≤ p21 Mc2 δv∞ . This implies sup z−z0 ∞ ≤δ

(I − Tn (z0 ))−1 (Tn (z) − Tn (z0 ))∞ ≤ Lp21 Mc2 δ ≤ q (say).

Here we choose δ in such a way that, 0 < q < 1. This proves the equation (2.30) of Theorem 2. Now applying the Lipschtiz continuity of ψ(., x(.)) and Lemma 4, we have βn = (I − Tn (z0 ))−1 (Tn (z0 ) − T (z0 ))∞ ≤ LΨ(f + KPn z0 ) − Ψ(f + Kz0 )∞ ≤ Lc1 K(I − Pn )z0 ∞

→ 0, as n → ∞.

By choosing n large enough such that βn ≤ δ(1 − q), the equation (2.31) of Theorem 2 is satisfied. Hence by applying Theorem 2, we obtain βn βn ≤ ˜ z n − z0  ∞ ≤ , 1+q 1−q where Hence from Lemma 4, we have

βn = (I − Tn (z0 ))−1 (Tn (z0 ) − T (z0 ))∞ .

˜ z n − z0  ∞ ≤

βn ≤ c(I − Tn (z0 ))−1 (Tn (z0 ) − T (z0 ))∞ 1−q ≤ c L c1 K(I − Pn )z0 ∞ 1 1 O(n−r−α+ 2 ), 2 < α < 1, = 1 O(n−r− 2 log n), α = 1.

This completes the proof.



In the following theorem we give the error bounds for the iterated Galerkin approximate solution x˜n defined by (2.29) and the exact solution x0 . Theorem 8. Let x0 ∈ C r [−1, 1] be an isolated solution of the equation (2.1) and x˜n be the iterated Legendre Galerkin approximation of x0 . Let the kernel k(t, s) is of the form given by (2.2). Then there holds 1 1 √ O(n−r−α+ 2 ), 2 < α < 1, x n − x 0 ∞ = ˜ xn − x0 L2 ≤ 2˜ 1 O(n−r− 2 log n), α = 1. 13

Proof. From the estimates (2.4) (2.9), (2.29) and Theorem 7, we obtain 1 √ O(n−r−α+ 2 ), ˜ xn − x0 ∞ = K(z0 − z˜n )∞ ≤ 2Mz0 − z˜n ∞ = 1 O(n−r− 2 log n), and ˜ x n − x 0 L 2 ≤

√



1

O(n−r−α+ 2 ), 2˜ x n − x 0 ∞ = 1 O(n−r− 2 log n),

1 2

< α < 1, α = 1,

1 2

< α < 1, α = 1.

This completes the proof.



In the following section we propose the Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods to improve the above convergence rates. 3. Superconvergence results by iterated Legendre multi-Galerkin method In this section, we propose the Legendre multi-Galerkin (Legendre M-Galerkin) and iterated Legendre multi-Galerkin (iterated Legendre M-Galerkin) methods for solving the equation (2.1) and we discuss the superconvergence results. The Legendre M-Galerkin method for the equation (2.11) is seeking an approximate solution znM ∈ X such that znM = Pn Ψ(K(znM ) + f) + Ψ(K(Pn znM ) + f) − Pn Ψ(K(Pn znM ) + f).

(3.1)

Corresponding approximate solution xM n of x is given by M xM n = f + K(zn ),

(3.2)

and iterated Legendre M-Galerkin solution is defined by z˜nM = Ψ(KznM + f).

(3.3)

Corresponding approximate solution x˜M n of x is given by x˜M znM ) + f. n = K(˜

(3.4)

Let TnM (u) = Pn Ψ(K(u) + f) + Ψ(K(Pn u) + f) − Pn Ψ(K(Pn u) + f), u ∈ X, then the equation (3.1) can be written as znM = TnM (znM ). (3.5) The Fr´echet derivative of Tn M (x) at z0 is a linear operator and is given by 

Tn M (z0 ) = Pn Ψ (K(z0 ) + f)K + Ψ (K(Pn z0 ) + f)KPn − Pn Ψ (K(Pn z0 ) + f)KPn = Pn Ψ (K(z0 ) + f)K + (I − Pn )Ψ (K(Pn z0 ) + f)KPn .

Now we discuss the existence and convergence rates of the approximate solution znM to z0 . To do this, we first prove the following lemma. 14

Lemma 7. For any x, y, z ∈ C[−1, 1], the following holds √   [Tn M (x) − Tn M (y)]zL2 ≤ Mc2 [p1 + ( 2 + p1 )p2 ]x − yL2 zL2 , where c2 and p and p1 are constants independent of n. Proof. Using the Lipschtiz continuity of ψ  (., x(.)) and estimates (2.3), (2.14), (2.15), for any x, y, z ∈ C[−1, 1], we have 



[Tn M (x) − Tn M (y)]zL2

= [Pn Ψ (Kx + f)K + (I − Pn )Ψ (KPn x + f)KPn − Pn Ψ (Ky + f)K − (I − Pn )Ψ (KPn y + f)KPn ]zL2

= Pn [Ψ (Kx + f) − Ψ (Ky + f)]Kz + (I − Pn )[Ψ (KPn x + f) − Ψ (KPn y + f)]KPn zL2 √ ≤ p1 [Ψ (Kx + f) − Ψ (Ky + f)]Kz∞ + ( 2 + p1 )[Ψ (KPn x + f) − Ψ (KPn y + f)]KPn z∞ √ ≤ p1 c2 K(x − y)∞ Kz∞ + ( 2 + p1 )c2 KPn (x − y)∞ KPn z∞ √ ≤ p1 Mc2 x − yL2 zL2 + ( 2 + p1 )Mc2 Pn (x − y)L2 Pn zL2 √ ≤ p1 Mc2 x − yL2 zL2 + ( 2 + p1 )Mc2 p2 x − yL2 zL2 √ ≤ Mc2 [p1 + ( 2 + p1 )p2 ]x − yL2 zL2 . This completes the proof.



Theorem 9. Let z0 ∈ C[−1, 1] be an isolated solution of the equation (2.11). Let the kernel k(t, s) is of the form given by (2.2). Let Pn : X → Xn be the orthogonal projection operator defined by (2.13). Assume that 1 is not an eigenvalue of the linear operator T  (z0 ). Then for sufficiently large n, the operator  (I − Tn M (z0 )) is invertible on C[−1, 1] and there exists a constant L1 > 0 independent of n such that  (I − Tn M (z0 ))−1 L2 ≤ L1 < ∞. Proof. Consider 

Tn M (z0 ) − T  (z0 )L2 = Pn Ψ (K(z0 ) + f)K + (I − Pn )Ψ (K(Pn z0 ) + f)KPn − Ψ (K(z0 ) + f)KL2 = (Pn − I)Ψ (K(z0 ) + f)K + (I − Pn )Ψ (K(Pn z0 ) + f)KPn L2

≤ (I − Pn )Ψ (K(z0 ) + f)KL2 + (I − Pn )Ψ (K(Pn z0 ) + f)KPn L2 .

(3.6)

Note that Tn (z0 ) = Ψ (K(Pn z0 ) + f)KPn , hence using Lemma 6, we have and

(I − Pn )Ψ (K(Pn z0 ) + f)KPn L2 = (I − Pn )Tn (z0 )L2 → 0, as n → ∞, (I − Pn )Ψ (K(z0 ) + f)KL2 = (I − Pn )T  (z0 )L2 → 0, as n → ∞.

This implies that 

Tn M (z0 ) − T  (z0 )L2 → 0, as n → ∞.

(3.7)

We assume that 1 is not an eigenvalue of T  (z0 ), i.e., (I −T  (z0 )) is invertible on C[−1, 1]. Then by applying  Theorem 1, (I − Tn M (z0 ))−1 exists and is uniformly bounded on C[−1, 1], for some sufficiently large n,  i.e., there exists a constant L1 > 0 such that (I − Tn M (z0 ))−1 L2 ≤ L1 < ∞. This completes the proof.  15

Theorem 10. Let z0 ∈ C r [−1, 1], r ≥ 1, be an isolated solution of the equation (2.11). Let the kernel k(t, s) is of the form given by (2.2). Assume that 1 is not an eigenvalue of the linear operator Ψ (f + Kz0 )K. Let Pn : X → Xn be the orthogonal projection operator defined by (2.13). Then the equation (3.1) has a unique solution znM ∈ B(z0 , δ) = {z : z − z0 L2 < δ} for some δ > 0 and for sufficiently large n. Moreover, there exists a constant 0 < q < 1, independent of n such that αn αn ≤ znM − z0 L2 ≤ , 1+q 1−q 

where αn = (I − Tn M (z0 ))−1 (Tn M (z0 ) − T (z0 ))L2 . Further, 1 1 O(n−r−α+ 2 ), M 2 < α < 1, zn − z0 L2 = 1 −r− 2 log n), α = 1. O(n 

Proof. From Theorem 9, we have (I − Tn M (z0 ))−1 L2 ≤ L1 < ∞. Now using Lemma 7, for any z ∈ B(z0 , δ) and y ∈ X, we have √   [Tn M (z0 ) − Tn M (z)]yL2 ≤ Mc2 [p1 + ( 2 + p1 )p2 ]z0 − zL2 yL2 . This implies

√   Tn M (z0 ) − Tn M (z)L2 ≤ Mc2 [p1 + ( 2 + p1 )p2 ]z0 − zL2 .

Thus we have sup z−z0 L2 ≤δ

√    (I − Tn M (z0 ))−1 (Tn M (z0 ) − Tn M (x))L2 ≤ L1 Mc2 [p1 + ( 2 + p1 )p2 ]δ ≤ q (say).

Here we choose δ in such a way that 0 < q < 1. This proves the equation (2.30) of Theorem 2. Hence using the Lipschtiz continuity of ψ(., x(.)) and estimates (2.3), (2.15), (2.20), we have 

αn = (I − Tn M (z0 ))−1 (Tn M (z0 ) − T (z0 ))L2 ≤ L1 Tn M (z0 ) − T (z0 ))L2

≤ L1 Pn Ψ(K(z0 ) + f) + Ψ(K(Pn z0 ) + f) − Pn Ψ(K(Pn z0 ) + f) − Ψ(K(z0 ) + f)L2

≤ L1 (I − Pn )[Ψ(K(z0 ) + f) − Ψ(K(Pn z0 ) + f)]L2 √ ≤ L1 ( 2 + p1 )Ψ(K(z0 ) + f) − Ψ(K(Pn z0 ) + f)∞ √ ≤ L1 ( 2 + p1 )c1 K(I − Pn )z0 ∞ √ √ ≤ L1 ( 2 + p1 ) Mc1 (I − Pn )z0 L2

(3.8)

→ 0 as n → ∞.

By choosing n large enough such that αn ≤ δ(1 − q), the equation (2.31) of Theorem 2 is satisfied, i.e., αn αn ≤ znM − z0 L2 ≤ . 1+q 1−q

Hence from Lemma 4 and estimate (3.8), it follows that αn  znM − z0 L2 ≤ ≤ c(I − Tn M (z0 ))−1 (Tn M (z0 ) − T (z0 ))L2 1−q √ ≤ L1 c ( 2 + p1 )c1 K(I − Pn )z0 ∞ 1 1 O(n−r−α+ 2 ), 2 < α < 1, = 1 −r− 2 log n), α = 1. O(n This completes the proof.

(3.9) 

16

Theorem 11. Let x0 ∈ C r [−1, 1] be an isolated solution of the equation (2.1) and xM n is the Legendre multi-Galerkin approximation of x0 , defined by (3.2). Let the kernel k(t, s) is of the form given by (2.2). Then 1 1 √ O(n−r−α+ 2 ), M M 2 < α < 1, xn − x0 L2 ≤ 2xn − x0 ∞ = 1 −r− 2 log n), α = 1. O(n Proof. From the estimates (2.3), (2.9), (3.2) and Theorem 10, we obtain 1 √ O(n−r−α+ 2 ), M M M xn − x0 ∞ = K(z0 − zn )∞ ≤ Mz0 − zn L2 = 1 O(n−r− 2 log n), and xM n

− x 0 L 2 ≤

√

2xM n



1

O(n−r−α+ 2 ), − x0 ∞ = 1 O(n−r− 2 log n),

1 2

< α < 1, α = 1,

1 2

< α < 1, α = 1.

This completes the proof.



Remark 1. Note that from Theorem 8 and Theorem 11, it follows that the iterated Legendre Galerkin −r−α+ 12 ) for solution x˜n and Legendre multi-Galerkin solution xM n have the same order of convergence O(n −r− 12 2 log n) for logarithm kernel in both L - norm and infinity norm. However in algebric kernel, and O(n Theorem 13, below we prove that the iterated Legendre multi-Galerkin solution improves over both iterated Legendre Galerkin and Legendre multi-Galerkin solutions. Next we discuss the superconvergence results for the iterated approximate solution x˜M n of Legendre multiGalerkin method, defined by the equation (3.4). To do this, we first proof the following lemma. Lemma 8. Let z0 ∈ C[−1, 1] be an isolated solution of the equation (2.11) and Pn : X → Xn is an orthogonal projection operator defined by (2.13). Let z˜nM defined by the iterated scheme (3.3) and the kernel k(t, s) is of the form given by (2.2). Then there holds √ 2 ˜ znM − z0 L2 ≤ c1 M1 M2 znM − z0 L2 + c1 ( 2 + M1 p1 d)K(I − Pn )∞ Ψ(KPn z0 + f) − Ψ(Kz0 + f)∞ , where c1 , d and p1 are constants independent of n. Proof. From Theorem 9 and estimate (2.5), for any y ∈ X, it follows that √ √ √    K(I − TnM (z0 ))−1 yL2 ≤ 2M(I − TnM (z0 ))−1 yL2 ≤ 2M(I − TnM (z0 ))−1 L2 yL2 ≤ 2ML1 yL2 . This implies 

K(I − TnM (z0 ))−1 L2 ≤

√

2ML1 ≤ M1 < ∞.

(3.10)

Now using the Lipschtiz continuity of ψ(., x(.)), we have ˜ znM − z0 L2 = Ψ(KznM + f) − Ψ(Kz0 + f)L2 ≤ c1 K(znM − z0 )L2 .

(3.11)

We consider 



znM − z0 = TnM (znM ) − T (z0 ) = TnM (znM ) − TnM (z0 ) − TnM (z0 )(znM − z0 ) + TnM (z0 )(znM − z0 ) + TnM (z0 ) − T (z0 ). 17

This implies 



(I − TnM (z0 ))(znM − z0 ) = TnM (znM ) − TnM (z0 ) − TnM (z0 )(znM − z0 ) + TnM (z0 ) − T (z0 ). Using Mean-value theorem, we have     (znM − z0 ) = (I − TnM (z0 ))−1 TnM (znM ) − TnM (z0 ) − TnM (z0 )(znM − z0 ) + TnM (z0 ) − T (z0 )     = (I − TnM (z0 ))−1 TnM (znM ) − TnM (z0 ) − TnM (z0 )(znM − z0 )    + (I − TnM (z0 ))−1 TnM (z0 ) − T (z0 )   M −1 M M M = (I − Tn (z0 )) Tn (z0 + θ1 (zn − z0 )) − Tn (z0 ) (znM − z0 )    (3.12) + (I − TnM (z0 ))−1 TnM (z0 ) − T (z0 ) .

where 0 < θ1 < 1. Operating K on both side and using the estimate (3.10), we obtain 





K(znM − z0 )L2 ≤ K(I − TnM (z0 ))−1 L2 [TnM (z0 + θ1 (znM − z0 )) − TnM (z0 )](znM − z0 )L2 

+ K(I − TnM (z0 ))−1 [TnM (z0 ) − T (z0 )]L2 



≤ M1 TnM (z0 + θ1 (znM − z0 )) − TnM (z0 )L2 znM − z0 L2 

+ K(I − TnM (z0 ))−1 [TnM (z0 ) − T (z0 )]L2 . 



(3.13)



Using the identity (I − TnM (z0 ))−1 = I + (I − TnM (z0 ))−1 TnM (z0 ), and the estimate (3.10), the second term of the estimate (3.13) becomes 





K(I − TnM (z0 ))−1 [TnM (z0 ) − T (z0 )]L2 = K{I + (I − TnM (z0 ))−1 TnM (z0 )}[TnM (z0 ) − T (z0 )]L2 ≤ K[TnM (z0 ) − T (z0 )]L2 



+ K(I − TnM (z0 ))−1 TnM (z0 )[TnM (z0 ) − T (z0 )]L2 

≤ K[TnM (z0 ) − T (z0 )]L2 + M1 TnM (z0 )[TnM (z0 ) − T (z0 )]L2 . (3.14)

Note that TnM (z0 ) − T (z0 ) = (I − Pn )[Ψ(KPn z0 + f) − Ψ(Kz0 + f)], and



TnM (z0 )[TnM (z0 ) − T (z0 )] = Pn Ψ (Kz0 + f)K(I − Pn )[Ψ(KPn z0 + f) − Ψ(Kz0 + f)]. Hence from the estimates (2.15), (3.13), (3.14) and Lemma 7, we get 



K(znM − z0 )L2 ≤ M1 TnM (z0 + θ1 (znM − z0 )) − TnM (z0 )L2 znM − z0 L2 

+ K[TnM (z0 ) − T (z0 )]L2 + M1 TnM (z0 )[TnM (z0 ) − T (z0 )]L2 . √ 2 ≤ M1 Mc2 [p1 + ( 2 + p1 )p2 ]znM − z0 L2 + K(I − Pn )[Ψ(KPn z0 + f) − Ψ(Kz0 + f)]L2 + M1 Pn Ψ (Kz0 + f)K(I − Pn )[Ψ(KPn z0 + f) − Ψ(Kz0 + f)]L2 2

≤ M1 M2 znM − z0 L2 + K(I − Pn )[Ψ(KPn z0 + f) − Ψ(Kz0 + f)]L2

+ M1 p1 Ψ (Kz0 + f)∞ K(I − Pn )[Ψ(KPn z0 + f) − Ψ(Kz0 + f)]∞ , 18

(3.15)

√ where M2 = Mc2 [p1 + ( 2 + p1 )p2 ] < ∞. Hence using the above estimate, from estimate (3.11), we obtain √ 2 ˜ znM − z0 L2 ≤ c1 M1 M2 znM − z0 L2 + c1 ( 2 + M1 p1 d)K(I − Pn )[Ψ(KPn z0 + f) − Ψ(Kz0 + f)]∞ √ 2 ≤ c1 M1 M2 znM − z0 L2 + c1 ( 2 + M1 p1 d)K(I − Pn )∞ Ψ(KPn z0 + f) − Ψ(Kz0 + f)∞ . This completes the proof.



Theorem 12. Let z0 ∈ C r [−1, 1], r ≥ 1, be an isolated solution of the equation (2.11). Let z˜nM defined by the iterated scheme (3.3) and the kernel k(t, s) is of the form given by (2.2). Then the following holds 1 1 O((1 + c log n)n−r−2α+ 2 ), M 2 < α < 1, ˜ z n − z 0 L 2 = −r− 32 2 (log n) (1 + c log n)), α = 1, O(n where c is a constant independent of n. Proof. From Lemma 8, we obtain √ 2 ˜ znM − z0 L2 ≤ c1 M1 M2 znM − z0 L2 + c1 ( 2 + M1 p1 d)K(I − Pn )∞ Ψ(KPn z0 + f) − Ψ(Kz0 + f)∞ .

(3.16)

Using the Lipschtiz continuity of ψ(., x(.)) and Lemma 4, we have Ψ(KPn z0 + f) − Ψ(Kz0 + f)∞ ≤ c1 K(Pn − I)z0 ∞ 1 1 O(n−r−α+ 2 ), 2 < α < 1, = −r− 12 log n), α = 1. O(n

(3.17)

Hence combining the estimates (3.16), (3.17), Lemma 5 and Theorem 10, we obtain 1 1 O((1 + c log n)n−r−2α+ 2 ), M 2 < α < 1, ˜ zn − z0 L2 = 3 −r− 2 2 ), α = 1. O((log n) (1 + c log n)n This completes the proof.



Theorem 13. Let x0 ∈ C r [−1, 1] be an isolated solution of the equation (2.1) and x˜M n is the iterated Legendre multi-Galerkin approximation of x0 , defined by (3.4). Let the kernel k(t, s) is of the form given by (2.2). Then there hold 1 1 √ O((1 + c log n)n−r−2α+ 2 ), M M 2 < α < 1, x n − x 0 ∞ = ˜ xn − x0 L2 ≤ 2˜ 3 −r− 2 2 ), α = 1, O((log n) (1 + c log n)n where c is a constant independent of n. Proof. From the estimates (2.3), (2.9), (3.4) and Theorem 12, we obtain 1 √ O((1 + c log n)n−r−2α+ 2 ), M M M zn − z0 )∞ ≤ M˜ z n − z0  L 2 = ˜ xn − x0 ∞ = K(˜ 3 O((log n)2 (1 + c log n)n−r− 2 ), and ˜ xM n

− x0 L2 ≤

This completes the proof.

√

2˜ xM n



1

O((1 + c log n)n−r−2α+ 2 ), − x0 ∞ = 3 O((log n)2 (1 + c log n)n−r− 2 ),

1 2

< α < 1, α = 1,

1 2

< α < 1, α = 1. 

19

Remark 2. From Theorems 4 and 8, for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence O(n−r ), whereas the iterated Legendre Galerkin solutions converge with the 1 1 order O(n−r−α+ 2 ) for the algebraic kernel and order O(log n n−r− 2 ) for logarithmic kernel in both L2 -norm and infinity norm. Also from Theorem 13, the iterated multi-Galerkin method has order of convergence 1 3 O((1 + c log n)n−r−2α+ 2 ) for algebric kernel, and order of convergence O((log n)2 (1 + c log n)n−r− 2 ) for logarithmic kernel in both L2 -norm and infinity norm. This shows that the iterated Legendre multi-Galerkin method improves over both Legendre Galerkin and iterated Legendre Galerkin methods. 4. Implementation For multi-Galerkin method, note that znM ∈ X. To solve the equation (3.1), applying Pn and (I − Pn ) to the equation, we have Pn znM = Pn Ψ(K(znM ) + f).

(4.1)

(I − Pn )znM = (I − Pn )Ψ(K(Pn znM ) + f).

(4.2)

⇒ znM = Pn znM + (I − Pn )Ψ(K(Pn znM ) + f).

(4.3)

Substituting (4.3) into (4.1),we get Pn znM = Pn Ψ(K(Pn znM + (I − Pn )Ψ(K(Pn znM ) + f)) + f).

(4.4)

Let WnM = Pn znM , then we can seek WnM ∈ Xn from the equation WnM = Pn Ψ(K(WnM + (I − Pn )Ψ(K(WnM ) + f)) + f),

(4.5)

and obtain znM = WnM + (I − Pn )Ψ(K(WnM ) + f).

(4.6)

 Let WnM = nj=1 βj ej ∈ Xn , where ej are the bases for the space Xn , and taking the inner product with respect to ei , i = 1, 2, · · · , n, equation(4.5) is equivalent to a nonlinear system of equations ⎛ ⎛ ⎛ ⎛ ⎞ ⎞⎞ ⎞ n n n    βj < ej , ei >=< Pn Ψ ⎝K ⎝ βj ej + (I − Pn )Ψ ⎝K ⎝ βj ej ⎠ + f ⎠⎠ + f ⎠ , ei > . j=1

j=1

j=1

(4.7)

Let β = [βj , j = 1, 2, · · · , n], for i = 1, 2, · · · , n, we define [Fi (β)]n×1 =

n  j=1

= 0.

⎛ ⎛ ⎛ ⎛ ⎞ ⎞⎞ ⎞ n n   βj < ej , ei > − < Pn Ψ ⎝K ⎝ βj ej + (I − Pn )Ψ ⎝K ⎝ βj ej ⎠ + f ⎠⎠ + f ⎠ , ei > j=1

j=1

(4.8)

20

We solve the above equation (4.8) iteratively by applying the Newton-Kantorovich method (k+1)

βj

(k)

= βj − [J]−1 Fi (β (k) ),

(4.9)

where Jacobian matrix J of Fi (β) is an n × n matrix.  (k+1) M(k+1) (k+1) Now using β (k+1) = [βj , j = 1, 2, · · · , n], we calculate WnM ≈ Wn = nj=1 βj ej and M(k+1)

from estimate (4.6), we can calculate the solution zn . Thus we can find the Legendre-multi-Galerkin M(k+1) M(k+1) approximate solution xM = K(zn ) + f. n ≈ xn The extra cost in the multi-Galerkin method is the construction of the matrix  Legendre   proposedhere   n n corresponding to < Ψ K + f , ei >, whereas in Legenj=1 βj ej + (I − Pn )Ψ K j=1 βj ej + f     n β e + f , ei >. dre Galerkin method we need to find only the matrix corresponding to < Ψ K j j j=1 This additional cost is compensated by the improvement in the order of the convergence. However note that the size of the system of equations that must be solved, in implementing the Legendre multi-Galerkin method remains the same as the Legendre Galerkin method. Numerical results In this section, we present the numerical results. To do this, we take approximating subspaces Xn to be the Legendre polynomial subspaces of degree ≤ n as defined in section 2. We present the errors of the approximate and iterated approximate solutions by Legendre Galerkin and Legendre M-Galerkin methods in both L2 and infinity norm. We denote the Legendre Galerkin, iterated Legendre Galerkin, Legendre M-Galerkin and iterated Legendre M-Galerkin solutions by xn , x˜n , xM ˜M n and x n , respectively. Denote x − xn  = O(n−α1 ),

x − x˜n  = O(n−α2 ),

−β1 x − xM ), n  = O(n

−β2 x − x˜M ), n  = O(n

where . denotes either L2 norm or infinity norm. For the algebraic and logarithmic kernels, the convergence rates α1 , α2 , β1 , β2 are calculated in the following Tables [1-4] of Example 1 and Example 2, where n denotes the highest degree of Legendre polynomial employed in the computation. For computation we use Newton-Kantorovich method. The numerical tests were performed on a PC Intel(R) Core (TM) i5-3470 CPU @ 3.20GHz Processor, 4.00GB RAM and 32-bit Operating System on Matlab (R2012b). We present the numerical results for Example 1 and Example 2. For algebraic kernel, in Table 1, we present the errors in both Legendre Galerkin, iterated Legendre Galerkin methods and in Table 2, we give the errors for both Legendre M-Galerkin and iterated Legendre M-Galerkin methods. Similarly, for logarithmic kernel, in Table 3, we present the errors in both Legendre Galerkin, iterated Legendre Galerkin methods and in Table 4, we give the errors for both Legendre M-Galerkin and iterated Legendre M-Galerkin methods. Example 1. Consider the following weakly singular Fredholm-Hammerstein integral equations of second kind  1 x(t) − k(t, s)ψ(s, x(s)) ds = f(t), t ∈ [0, 1], 0

21

1

with the algebraic kernel function k(t, s) = m(t, s)|t − s|− 3 , m(t, s) =

2t(1−t) 2

2

3{(1−t) 3 +t 3 }

∈ C 1 ([0, 1] × [0, 1]),

nonlinear function ψ(s, x(s)) = [x(s)]2 and the function f(t) = 1 − t(1 − t). The exact solution is given by x(t) = 1. The transformed integral equation is  1 x(t) − k(t, s)ψ(s, x(s)) ds = f(t), t ∈ [−1, 1], −1

1

s+1 t+1 s+1 − 3 with the algebraic kernel function k(t, s) = 21 m( t+1 , nonlinear function ψ(s, x(s)) = 2 , 2 )| 2 − 2 | s+1 2 t+1 t+1 [x( 2 )] and the function f(t) = 1− 2 (1− 2 ), and the exact solution is given by x(t) = 1, t, s ∈ [−1, 1].

Note that according to the theory we have α = 32 , m(t, s) ∈ C 1 ([−1, 1] × [−1, 1]), f(t) ∈ C r [−1, 1], r ≥ 1, c1 = 2, c2 = 2 and d = 2.

Table 1: Legendre Galerkin and iterated Legendre Galerkin methods

n 3 4 5 6 7 8 9 10

x − xn L2 1.17385256×10−1 2.53828873×10−2 1.57445510×10−3 7.24828644×10−5 8.01786646×10−6 1.40597268×10−6 3.37505732×10−8 7.07945781×10−9

α1 1.95 2.65 4.01 5.32 6.03 6.48 7.83 8.15

x − x˜n L2 5.09332512×10−2 4.12897672×10−3 1.40823546×10−4 2.22154708×10−5 1.41880125×10−6 3.63888551×10−7 4.67157036×10−9 2.81838312×10−10

α2 2.71 3.95 5.51 5.98 6.92 7.13 8.73 9.55

x − xn ∞ 1.19993018×10−1 2.60964974×10−2 1.65233987×10−3 8.67062638×10−5 8.33605899×10−6 1.32094332×10−6 4.02365329×10−8 7.41310243×10−9

α1 1.93 2.63 3.98 5.22 6.01 6.51 7.75 8.13

x − x˜n ∞ 5.20647544×10−2 3.74712546×10−3 1.17974616×10−4 2.10528543×10−5 1.56378291×10−6 2.35136200×10−7 1.43260536×10−9 3.89045134×10−10

α2 2.69 4.03 5.62 6.01 6.82 7.34 8.22 9.41

Table 2: Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods

n 3 4 5 6 7 8 9 10

x − xM n L2 2.36056803×10−2 2.50669120×10−3 8.55056031×10−4 4.35048709×10−6 4.16399115×10−7 2.10734243×10−8 9.39433145×10−10 2.69153652×10−11

β1 3.41 4.32 5.82 6.89 7.55 8.50 9.46 10.5

x − x˜M n L 2 6.17914790×10−3 2.84358785×10−4 4.86805048×10−5 2.54151401×10−6 2.27787246×10−7 6.85593231×10−9 3.81615907×10−10 1.25893342×10−12

β2 4.63 5.89 6.17 7.19 7.86 9.04 9.87 11.9

x − xM n ∞ 2.41300902×10−2 2.09330754×10−3 1.19888706×10−4 4.50921465×10−6 5.46793726×10−7 3.26124937×10−8 1.04852317×10−9 2.51189031×10−11

β1 3.39 4.45 5.61 6.84 7.42 8.29 9.41 10.6

x − x˜M n ∞ 6.31642022×10−3 5.30632257×10−4 5.53697641×10−5 2.32372616×10−6 2.32263189×10−7 8.61799854×10−9 4.45065296×10−10 1.58490976×10−12

Example 2. Consider the following weakly singular Fredholm-Hammerstein integral equations of second kind  1 x(t) − k(t, s)ψ(s, x(s)) ds = f(t), t ∈ [0, 1], 0

22

β2 4.61 5.44 6.09 7.24 7.85 8.93 9.80 11.8

with the logarithmic kernel function k(t, s) = log|t − s|, nonlinear function ψ(s, x(s)) = sin(πx(s)) and the function f(t) = 1. The exact solution is given by x(t) = 1. Note that α = 1. The transformed integral equation is  1 k(t, s)ψ(s, x(s)) ds = f(t), t ∈ [−1, 1], x(t) − −1

s+1 1+s with the logarithmic kernel function k(t, s) = 12 log| t+1 2 − 2 |, and ψ(s, x(s)) = sin(πx( 2 )) and the function f(t) = 1 and its exact solution is given by x(t) = 1, t, s ∈ [−1, 1]. Note that according to the theory we have α = 1, f(t) ∈ C r [−1, 1], r ≥ 1, c1 = π, c2 = π 2 and d = π.

Table 3: Legendre Galerkin and iterated Legendre Galerkin methods

n 3 4 5 6 7 8 9 10

x − xn L2 6.65829933×10−1 2.91183396×10−1 1.90572791×10−1 9.56361009×10−2 4.80448738×10−2 3.21715241×10−3 2.11917264×10−4 1.34896288×10−5

α1 0.37 0.89 1.03 1.31 1.56 2.76 3.85 4.87

x − x˜n L2 4.50190552 ×10−1 8.02141185×10−2 3.93613777×10−2 1.74332426×10−2 1.41912158×10−3 4.82202949×10−5 4.33667720×10−6 1.51356124×10−7

α2 0.72 1.82 2.01 2.26 3.37 4.78 5.62 6.82

x − xn ∞ 7.11245589×10−1 3.07786103×10−1 1.52126577×10−1 1.16471186×10−1 7.66426631×10−2 4.30431685×10−3 3.83541017×10−4 2.23872113×10−5

α1 0.31 0.85 1.17 1.20 1.32 2.62 3.58 4.65

x − x˜n ∞ 4.81149064×10−1 6.98304461×10−2 4.33519354×10−2 1.68195800×10−2 1.19113443×10−3 5.13242440×10−5 4.15023077×10−6 1.14815362×10−7

α2 0.83 1.92 1.95 2.28 3.46 4.75 5.64 6.94

Table 4: Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods

n 3 4 5 6 7 8 9 10

x − xM n L2 3.01458962×10−1 6.42571141×10−2 2.06767506×10−2 7.11709149×10−3 1.63666871×10−4 8.58351505×10−6 3.31660202×10−7 1.44543977×10−8

β1 1.09 1.98 2.41 2.76 4.48 5.61 6.79 7.84

x − x˜M n L 2 1.48536996×10−1 6.16395440×10−2 1.62418983×10−2 4.23290660×10−3 5.83524129×10−5 1.29375870×10−6 3.22995373×10−8 1.09647824×10−9

β2 1.73 2.00 2.56 3.05 5.01 6.52 7.85 8.96

x − xM n ∞ 3.75622636×10−1 4.86977862×10−2 1.59825874×10−2 8.06813462×10−3 1.54385973×10−4 4.43717198×10−5 2.19453200×10−6 3.46736857×10−8

β1 0.89 2.18 2.57 2.69 4.51 4.82 5.93 7.46

x − x˜M n ∞ 2.55693215×10−1 3.95548935×10−2 1.93875749×10−2 4.54741653×10−3 4.53102099×10−5 4.59979182×10−6 1.87322656×10−7 1.44543984×10−9

β2 1.24 2.33 2.45 3.01 5.14 5.91 7.05 8.84

For comparison of numerical results, we have performed 20 Newton iterations for solving the nonlinear systems for the given examples. Table 1 of Example 1 and Table 3 of Example 2, for both the algebraic and logarithm kernels, we see that the iterated Legendre Galerkin solution gives better approximation than the Legendre Galerkin method. From Table 2 of Example 1 and Table 4 of Example 2, we also see that the iterated Legendre multi-Galerkin method gives better convergence rates than the Legendre multi-Galerkin 23

method for both the algebraic and logarithm kernels. Hence the iterated Legendre multi-Galerkin solution improves over the Legendre Galerkin, iterated Legendre Galerkin and Legendre multi-Galerkin solutions. Note that the size of the system of equations that must be solved, remains the same as in Legendre Galerkin method. References [1] M. Ahues, A. Largillier, B. Limaye, Spectral computations for bounded operators, CRC press, 2001. [2] K. Atkinson, The numerical solution of integral equations of the second Kind, vol. 4, Cambridge university press, 1997. [3] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods, Fundamentals in single domains, Springer, 2006. [4] Z. Chen, J. Li, Y. Zhang, A fast multiscale solver for modified Hammerstein equations, Applied mathematics and computation 218 (2011) 3057–3067. [5] P. Das, G. Nelakanti, Error analysis of polynomial-based multi-projection methods for a class of nonlinear Fredholm integral equations, Journal of Applied Mathematics and Computing (2016) 1–24. [6] P. Das, M. M. Sahani, G. Nelakanti, Legendre spectral projection methods for Urysohn integral equations, Journal of Computational and Applied Mathematics 263 (2014) 88–102. [7] P. Das, M. M. Sahani, G. Nelakanti, G. Long, Legendre spectral projection methods for FredholmHammerstein integral equations, Journal of scientific computing (2015) 1–18. [8] L. Grammont, H. Kaboul, An improvement of the product integration method for a weakly singular Hammerstein equation, arXiv preprint arXiv:1604.00881, 2016. [9] H. Kaneko, R. Noren, Y. Xu, Regularity of the solution of Hammerstein equations with weakly singular kernel, Integral equations and operator theory 13 (5) (1990) 660–670. [10] H. Kaneko, R. D. Noren, P. A. Padilla, Superconvergence of the iterated collocation methods for Hammerstein equations, Journal of computational and applied mathematics 80 (1997) 335–349. [11] H. Kaneko, R. D. Noren, Y. Xu, Numerical solutions for weakly singular Hammerstein equations and their superconvergence, Journal of integral equations and applications 4 (1992) 391–407. [12] H. Kaneko, P. Padilla, Y. Xu, Superconvergence of the iterated degenerate kernel method, Applicable analysis 80 (2001) 331–351. [13] H. Kaneko, Y. Xu, Degenerate kernel method for Hammerstein equations, Mathematics of computation 56 (1991) 141–148.

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