J. Appl. Math. Comput. DOI 10.1007/s12190-014-0852-8 ORIGINAL RESEARCH

Convergence analysis of Legendre spectral projection methods for Hammerstein integral equations of mixed type Payel Das · Mitali Madhumita Sahani · Gnaneshwar Nelakanti

Received: 11 August 2014 © Korean Society for Computational and Applied Mathematics 2014

Abstract In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Hammerstein integral equations of mixed type. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the 1 same orders, O(n −r ) in L 2 -norm and O(n 2 −r ) in infinity norm, and the iterated Legendre Galerkin solution converges with the order O(n −2r ) in both L 2 -norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n −r ) in both L 2 -norm and infinity norm, n being the highest degree of Legendre polynomial employed in the approximation and r being the smoothness of the kernels. Keywords Hammerstein integral equations · Mixed type · Smooth kernels · Galerkin method · Collocation method · Legendre polynomials · Superconvergence rates Mathematics Subject Classification

45B05 · 45G10 · 65R20

P. Das (B) · G. Nelakanti Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, India e-mail: [email protected] G. Nelakanti e-mail: [email protected] M. M. Sahani Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar 751024, India e-mail: [email protected]

123

P. Das et al.

1 Introduction In this section, we consider the following Hammerstein integral equation of mixed type m  1  ki (t, s)ψi (s, x(s))ds = f (t), −1 ≤ t ≤ 1, (1.1) x(t) − i=1

−1

where f, ki and ψi (1 ≤ i ≤ m) are known functions and x is the unknown function to be determined in a Banach space X. Various spectral methods are available in literature to solve non-linear integral equations (see [2,3,5,7,8,10–12,14,15]). The Galerkin, collocation, Petrov–Galerkin, Nyström methods and their discretized versions are most commonly used projection methods for finding numerical solutions of nonlinear integral equations. Projection methods for solving the equation of type (1.1) using piecewise polynomial basis functions were studied by Ganesh and Joshi [7] and a discrete version was discussed in Ganesh and Joshi [6]. However to obtain more accurate solutions, in spline based projection methods, one has to increase the number of partitioning points. Therefore a large system of nonlinear equations has to be solved, which is very much expensive computationally. Use of global polynomials imply smaller nonlinear systems, something which is very much useful in case of practical computations. Hence we choose to use global polynomials rather than piecewise polynomial basis functions in this paper. We consider the Galerkin and collocation methods and their iterated versions to approximate the solutions of Hammerstein integral equation of mixed type (1.1) with a smooth kernels, using global polynomial basis functions. In particular, we use Legendre polynomials, which can be generated recursively with ease and possess nice property of orthogonality. We obtain almost similar convergence rates using Legendre polynomial bases as in the case of piecewise polynomial bases discussed in Ganesh and Joshi [7]. We organize this paper as follows. In Sect. 2, we discuss the Legendre spectral Galerkin and Legendre spectral collocation methods to obtain superconvergence results. In Sect. 3, numerical results are given to illustrate the theoretical results. Throughout this paper, we assume that c is a generic constant. 2 Legendre spectral Galerkin and collocation methods: Hammerstein integral equations of mixed type with smooth kernel In this section, we describe the Galerkin and collocation methods for solving Hammerstein integral equation of mixed type using Legendre polynomial basis functions. Let X = L 2 [−1, 1] or C[−1, 1] and consider the following Hammerstein integral equation of mixed type m  1  ki (t, s)ψi (s, x(s))ds = f (t), −1 ≤ t ≤ 1, (2.1) x(t) − i=1

−1

where ki , f and ψi are known functions and x is the unknown function to be determined. Let

123

Convergence analysis of Legendre spectral projection methods

 (Ki ψi ) (x)(t) =

1

−1

ki (t, s)ψi (s, x(s))ds, x ∈ X.

Then the Eq. (2.1) can be written as x−

m 

Ki ψi (x) = f.

(2.2)

i=1

Next, we define the operator T on X by T x:= f +

m  Ki ψi (x), x ∈ X, i=1

then the Eq. (2.2) can be written as x = T x.

(2.3)

Throughout the paper, the following assumptions are made on f, ki (., .) and ψi (., x(.)) : (i) (ii) (iii) (iv)

f ∈ C[−1, 1],   limt→t  ki (t, .) − ki (t , .)∞ = 0, t, t ∈  [−1, 1], 1 ≤ i ≤ m, m mi , m i = supt, s∈[−1, 1] |ki (t, s)| < ∞ and M = i=1 the nonlinear functions ψi (s, x) are continuous in s ∈ [−1, 1] and Lipschitz continuous in x, i.e., for any x1 , x2 ∈ R, there exists constants ci > 0, i = 1, 2, . . . , m such that |ψi (s, x1 ) − ψi (s, x2 )| ≤ ci |x1 − x2 | ,

and l1 = supi=1,2,...,m ci , (0,1) (s, x) of ψi (s, x) with respect to the second variable (v) the partial derivatives ψi exist and Lipschitz continuous in x, i.e., for any x1 , x2 ∈ R, there exists constants qi > 0, i = 1, 2, . . . , m such that    (0,1)  (0,1) (s, x1 ) − ψi (s, x2 ) ≤ qi |x1 − x2 | , ψi and l2 =

sup i=1,2,...,m

qi , (0,1)

(vi) ψi (s, y) = ψi (s, y0 ) + ψi 0 < θ < 1.

(s, y0 + θ (y − y0 ))(y − y0 ), ∀y, y0 ∈ R, and

The following theorem gives the condition for the existence of unique solution of the Eq. (2.3) in X. Theorem 2.1 Let X = L 2 [−1, 1] or C[−1, 1] and f ∈ X, and let ki (., .) ∈ C([−1, 1] × [−1, 1]) with m i = supt,s∈[−1, 1] |ki (t, s)| < ∞. Let ψi (s, x(s)) ∈ C([−1, 1] × R) satisfy the Lipschitz condition in the second variable, i.e., |ψi (s, y1 ) − ψi (s, y2 )| ≤ ci |y1 − y2 | , y1 , y2 ∈ R, i = 1, 2, . . . , m,

123

P. Das et al.

m with 2Ml1 < 1, where M = i=1 m i and l1 = max{c1 , c2 , . . . , cm }. Then the operator equation x = T x has a unique solution x0 ∈ X, i.e., we have x0 = T x0 . Proof of the above theorem can be easily done using similar technique given in Theorem 2.4 of [9]. We set the following notations. Let m 



(Ki ψi ) (x0 ) x(t) =

i=1

m   i=1

1 −1

(0,1)

ki (t, s)ψi

(s, x0 (s)) x(s)ds.

For the rest of the paper we assume that the kernel ki (., .) ∈ C r ([−1, 1] × [−1, 1]). m  ki  j,∞ , where Let k j,∞ = i=1

ki  j,∞ =

 j  j    ∂ p+q    , t, s ∈ [−1, 1]. k (t, s)  ∂t p ∂s q i  p=0 q=0

For j = 0, 1, . . . , r, we have  m   m      ( j)   1 ∂j      (0,1) (t) =  k (t, s)ψ (s)) x(s)ds x (Ki ψi ) (x0 ) x (s,   i 0 i     ∂t j −1 i=1 i=1    m  1    ∂j    (0,1) ≤ sup  j ki (t, s) sup sup ψi |x(s)|ds (s, x0 (s)) −1 s,t∈[−1,1] ∂t i=1,2,...,m s∈[−1,1] i=1

≤ ≤

√ √

2

sup

i=1,2,...,m

di

m 

ki  j,∞ x L 2

i=1

2dk j,∞ x L 2 ≤ 2dk j,∞ x∞ ,

(2.4)

where di = sups∈[−1,1] |ψi(0,1) (s, x0 (s))| and d = supi=1,2,...,m di . Hence, for j = 0, 1, 2, . . . , r, we have   m   ( j)     (Ki ψi ) (x0 ) x    

≤

√ 2dk j,∞ x L 2 ≤ 2dk j,∞ x∞ ,

(2.5)

∞

i=1

and  m    ( j)     (Ki ψi ) (x0 ) x     i=1

L2

  m  ( j)  √     ≤ 2 (Ki ψi ) (x0 ) x    i=1

∞

≤ 2dk j,∞ x L 2 .

(2.6) Next we prove the following lemma, which we need in our convergence analysis.

123

Convergence analysis of Legendre spectral projection methods

Lemma 2.1 For any x, y ∈ L 2 [−1, 1] or C[−1, 1], the following hold  m  m    √   (Ki ψi ) (x0 ) − (Ki ψi ) (x) ≤ 2Ml1 x0 − x L 2 ,    i=1 i=1 ∞  m

 m     √     (Ki ψi ) (x0 ) − (Ki ψi ) (x) y  ≤ 2Ml2 x0 − x L 2 y∞ .    i=1

∞

i=1

(0,1)

Proof Using Lipschitz’s continuity of ψi ity, we have

(., x0 (.)) and Cauchy–Schwarz inequal-

 m

 m       (Ki ψi ) (x0 ) − (Ki ψi ) (x) (t)    i=1 i=1  m    1    ≤ ki (t, s) [ψi (s, x0 (s)) − ψi (s, x(s))] ds    i=1 −1  1 m  |ψi (s, x0 (s)) − ψi (s, x(s))| ds ≤ sup |ki (t, s)| ≤

−1

i=1 s,t∈[−1,1]  1 m 

mi

i=1

−1

ci |(x0 − x) (s)| ds

 1 |(x0 − x) (s)| ds ≤ Ml1 −1 √ ≤ 2Ml1 x0 − x L 2 . This implies that  m  m      (Ki ψi ) (x0 ) − (Ki ψi ) (x)    i=1

i=1

≤

√ 2Ml1 x0 − x L 2 .

∞ (0,1)

On the similar lines, using Lipschitz’s continuity of ψi Schwarz inequality, we obtain  m

 m         (Ki ψi ) (x0 ) − (Ki ψi ) (x) y     i=1

i=1

(., x0 (.)) and Cauchy–

≤ Ml2 x0 − x L 2 y L 2 ∞

≤ Hence the proof follows.

(2.7)

(2.8)

√ 2Ml2 x0 − x L 2 y∞ . (2.9) 

Next we will apply Legendre Galerkin and Legendre collocation methods to the Eq. (2.1). To do this, we let Xn be the sequence of polynomial subspaces of X of degree ≤n

123

P. Das et al.

and we choose Legendre polynomials {φ0 , φ1 , φ2 , . . . , φn } as an orthonormal basis for the subspace Xn , where, φ0 (x) = 1, φ1 (x) = x, x ∈ [−1, 1], and for i = 1, 2, . . . , n − 1 (i + 1)φi+1 (x) = (2i + 1)xφi (x) − iφi−1 (x), x ∈ [−1, 1].

(2.10)

Orthogonal projection operator let X = L 2 [−1, 1] or C[−1, 1] and let the operator → Xn be the orthogonal projection defined by

PnG : X

PnG x =

n  x, φ j φ j , x ∈ X,

(2.11)

j=0

1 where x, φ j = −1 x(t)φ j (t)dt. We quote the following lemmas which follows from (Canuto et al. [4], pp. 283–287). Lemma 2.2 Let PnG : X → Xn denote the orthogonal projection defined by (2.11). Then the projection PnG satisfies the following properties. (i) {PnG : n ∈ N} is uniformly bounded in L 2 -norm. (ii) There exists a constant c > 0 such that for any n ∈ N and u ∈ X,     G Pn u − u 

L2

≤ c inf u − φ L 2 . φ∈Xn

Lemma 2.3 Let PnG be the orthogonal projection defined by (2.11). Then for any u ∈ C r [−1, 1], there hold         u − PnG u  2 ≤ cn −r u (r )  2 , L  L   3  G  −r  (r )  4 u  2 , u − Pn u  ≤ cn ∞  L 1  G  −r u − Pn u  ≤ cn 2 V u (r ) , ∞

(2.12) (2.13) (2.14)

where c is a constant independent of n and V (u (r ) ) denotes the total variation of u (r ) . Interpolatory projection operator Let {τ0 , τ1 , . . . , τn } be the zeros of the Legendre polynomial of degree n + 1 and define interpolatory projection PnC : X → Xn by PnC u ∈ Xn , PnC u (τi ) = u (τi ) , i = 0, 1, . . . , n, u ∈ X.

(2.15)

According to the analysis of (Canuto et al. [4], p. 289), PnC satisfies the following lemmas.

123

Convergence analysis of Legendre spectral projection methods

Lemma 2.4 Let PnC : X → Xn be the interpolatory projection defined by (2.15). Then there hold (i) {PnC : n ∈ N} is uniformly bounded in L 2 -norm. (ii) There exists a constant c > 0 such that for any n ∈ N and u ∈ X,    C  Pn u − u 

L2

≤ c inf u − φ L 2 → 0, n → ∞. φ∈Xn

Lemma 2.5 Let PnC : X → Xn be the interpolatory projection defined by (2.15). Then for any u ∈ C r [−1, 1], there exists a constant c independent of n such that         (2.16) u − PnC u  2 ≤ cn −r u (r )  2 . L

L

Noting that    C Pn 

∞

23/2 = 1 + √ n 1/2 + B0 + O n −1/2 , π

where B0 is a bounded constant (see Tang et al. [12]), we have         inf u − χ ∞ (I − PnC )u  ≤ 1 + PnC  ∞ ∞ χ ∈Xn     1 1     ≤ cn 2 n −r u (r )  ≤ cn 2 −r u (r )  . ∞

∞

(2.17)

Throughout this paper, we assume that the projection operator Pn : X → Xn is either orthogonal projection PnG defined by (2.11) or interpolatory projection operator PnC defined by (2.15). By Lemmas 2.2 and 2.4, we have that Pn  L 2 is uniformly bounded. We denote, Pn  L 2 ≤ p, for all n ∈ N and Pn x L 2 ≤ p1 x∞ , where p and p1 are constants independent of n. Further, we have from Lemmas 2.3 and 2.5 and estimate (2.17) that     u − Pn u L 2 ≤ cn −r u (r )  2 , (2.18) L     u − Pn u∞ ≤ cn β−r u (r )  , 0 < β < 1, and r = 0, 1, 2, . . . , (2.19) ∞

where c is a constant independent of n, β = 43 for orthogonal projection operators and β = 21 for interpolatory projections. Note that Pn u − u∞  0, as n → ∞ for any u ∈ C[−1, 1]. The projection method for Eq. (2.2) is seeking an approximate solution xn ∈ Xn such that m  xn − Pn Ki ψi (xn ) = Pn f. (2.20) i=1

If Pn is chosen to be PnG , the above scheme (2.20) leads to Legendre Galerkin method, whereas if Pn is replaced by PnC we get the Legendre collocation method.

123

P. Das et al.

Let Tn be the operator defined by Tn u :=

m 

Pn Ki ψi (u) + Pn f, u ∈ X.

(2.21)

i=1

Then the Eq. (2.20) can be written as xn = Tn xn .

(2.22)

In order to obtain more accurate approximate solution, we further consider the iterated projection method for (2.2). To this end, we define the iterated solution as x˜n = f +

m 

Ki ψi (xn ) .

(2.23)

i=1

Applying Pn on both sides of the Eq. (2.23), we obtain Pn x˜n = Pn f +

m 

Pn Ki ψi (xn ) .

(2.24)

i=1

From Eqs. (2.20) and (2.24), it follows that Pn x˜n = xn . Using this, we see that the iterated solution x˜n satisfies the following equation x˜n −

m 

Ki ψi (Pn x˜n ) = f.

(2.25)

i=1

m Letting Tn (u) := f + i=1 Ki ψi (Pn u), u ∈ X, the Eq. (2.25) can be written as x˜n = Tn x˜n . Next we prove one lemma which we will use further in our analysis. m   Frechet derivatives of Lemma 2.6 Let i=1 (Ki ψi ) (x 0 ) and Tn (x 0 ) be the  m n (x), respectively at x0 . Then (I−Pn ) m (Ki ψi ) (x0 ) L 2 → K ψ (x) and T i i i=1 i=1  0 and (I − Pn )Tn (x0 ) L 2 → 0, as n → ∞. Proof Using the estimates (2.6) and (2.18), we have  m

       (Ki ψi ) (x0 ) (x) (I − Pn )   i=1

≤ cn L2

−r

 m    (r )     (Ki ψi ) (x0 ) x     i=1

−r

≤ cn 2dkr,∞ x L 2 → 0, as n → ∞.

123

L2

(2.26)

Convergence analysis of Legendre spectral projection methods

Again using the estimate (2.18), we have      (I − Pn ) Tn (x0 ) (x)

L2

≤ cn

−r

 (r )      T (x0 ) x   n 

L2

.

(2.27)

The Frechet derivative of Tn (x) at x0 is given by  Tn (x0 ) x(t) =

=

m 



(Ki ψi ) (Pn x0 ) Pn x(t)

i=1 m  1  i=1

−1

ki (t, s)ψi(0,1) (s, Pn x0 (s)) Pn x(s)ds.

Consider  m    (r )     (t) (Ki ψi ) (Pn x0 ) Pn x    i=1  m    1 ∂ r    (0,1) = k (t, s)ψ x (s)) P x(s)ds P (s,  i n 0 n i r   ∂t i=1 −1  r   1  m  ∂   (0,1) ≤ sup  r ki (t, s) (s, Pn x0 (s))  ψi ∂t −1 i=1 t,s∈[−1,1]  1        (0,1) (0,1) − ψi (s, x0 (s)) Pn x(s) ds + (s, x0 (s)) Pn x(s) ds ψi ≤

m 

 ki r,∞

i=1

−1

1

−1

qi |(Pn x0 (s) − x0 (s))| |Pn x(s)| ds

   (0,1)  + sup ψi (s, x0 (s))

1

−1

s∈[−1,1]

|Pn x(s)| ds



√ ≤ l2 kr,∞ Pn x0 − x0  L 2 Pn x L 2 + 2d Pn x L 2

√ ≤ l2 p Pn x0 − x0  L 2 + 2dp x L 2 kr,∞ .

(2.28)

Hence, we have  (r )      Tn (x0 ) x 

L2

√  (r )    2  Tn (x0 ) x  ∞   m   (r )   √    = 2 (Ki ψi ) (Pn x0 ) Pn x    i=1 √

∞ ≤ 2l2 p Pn x0 − x0  L 2 + 2dp kr,∞ x L 2 . ≤

123

P. Das et al.

Combining this with the estimate (2.27), we have      (I − Pn ) Tn (x0 )

L2

≤ cn −r

√

2l2 p Pn x0 − x0  L 2 + 2dp kr,∞

→ 0, as n → ∞.

(2.29) 

This completes the proof.

We quote the following theorem from Vainikko [13] which gives us the conditions under which the solvability of one equation leads to the solvability of other equation.  and T  be continuous operators over an open set Ω in a Banach Theorem 2.2 Let T x has an isolated solution x˜0 ∈ Ω and let the space X. Let the equation x = T following conditions be satisfied.  is Frechet differentiable in some neighbourhood of the point x˜0 , (a) The operator T  (x˜0 ) is continuously invertible. while the linear operator I − T (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 x − x˜0  ≤ δ is contained within Ω). 

−1 

         I − T T (x) − T sup  (x˜0 ) (x˜0 )   ≤ q, x−x˜0 ≤δ  

−1       (x˜0 )  (x˜0 ) − T  (x˜0 )  ≤ δ(1 − q). α = I−T T 

(2.30) (2.31)

x has a unique solution xˆ0 in the sphere x − x˜0  ≤ δ. Then the equation x = T Moreover, the inequality   α α ≤ xˆ0 − x˜0  ≤ 1+q 1−q

(2.32)

is valid. Next we discuss the existence of approximate and iterated approximate solutions and their error bounds. To do this, first we recall the following definition of ν-convergence and a theorem from Ahues et al. [1]. Definition 2.1 Let X be Banach space and BL(X) be space of bounded linear operators from X into X. Let Tn , T ∈ BL(X). We say Tn is ν-convergent to T if Tn  ≤ C, (Tn − T ) T  → 0, (Tn − T ) Tn  → 0, as n → ∞. Lemma 2.7 Let X be a Banach space and T , Tn be bounded linear operators on X. If Tn is norm convergent to T or Tn is ν-convergent to T and (I − T )−1 exists, then (I − Tn )−1 exists and uniformly bounded on X, for sufficiently large n.

123

Convergence analysis of Legendre spectral projection methods

Theorem 2.3 Let x0 ∈ C r [−1, 1] be an isolated solutionof the Eq. (2.2). Assume  m (Ki ψi ) (x0 ), where that 1 is not an eigenvalue of the linear operator i=1 m m  i=1 (Ki ψi ) (x 0 ) denotes the Frechet derivative of i=1 Ki ψi (x) at x 0 . Let Pn : X → Xn be either orthogonal or interpolatory projection operator defined by (2.11) or (2.15), respectively. Then the Eq. (2.20) has a unique solution xn ∈ B(x0 , δ) = {x : x − x0  L 2 < 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 ≤ xn − x0  L 2 ≤ , 1+q 1−q 

where αn = (I − Tn (x0 ))−1 (Tn (x0 ) − T (x0 )) L 2 . Proof Using estimate (2.26), we have  m   m     

         Pn (Ki ψi ) (x0 ) − (Ki ψi ) (x0 ) (x)  Tn (x0 ) − T (x0 ) (x) 2 =    L i=1 i=1 L2  m

       = (Pn − I) (Ki ψi ) (x0 ) (x)   i=1

≤ 2dcn Hence

−r

      Tn (x0 ) − T (x0 )

L2

kr,∞ x L 2 .

L2

→ 0, as n → ∞.





This shows that Tn (x0 ) is norm convergent to T (x0 ). Hence by Lemma 2.7, we have 

(I − Tn (x0 ))

−1

exists and uniformly bounded on X, for some sufficiently large n, i.e., −1



there exists some A1 > 0 such that (I − Tn (x0 ))  L 2 ≤ A1 < ∞. Now from the estimate (2.8), we have for any x ∈ B(x0 , δ),  m  m    

         Pn (Ki ψi ) (x0 ) (y) − Pn (Ki ψi ) (x)(y)  Tn (x0 ) − Tn (x) y  2 =    L 2 i=1 i=1  m L

m         ≤ Pn  L 2  (Ki ψi ) (x0 ) − (Ki ψi ) (x) (y)   2 i=1 i=1  m L

m    √      ≤ 2Pn  L 2  (Ki ψi ) (x0 )− (Ki ψi ) (x) (y)   i=1 i=1 ∞ √ ≤ 2 pMl2 δy L 2 . (2.33) This implies       Tn (x0 ) − Tn (x)

L2

≤

√ 2 pMl2 δ.

123

P. Das et al.

Hence, we have 

−1 

     I − Tn  (x0 )  sup T (x) − T (x ) n 0 n  

x−x0  L 2 ≤δ

L2

√ ≤ A1 2 pMl2 δ ≤ q, (say)

where 0 < q < 1, which proves the Eq. (2.30) of Theorem 2.2. Taking use of (2.18), we have  

−1     αn =  I − Tn (x0 ) (Tn (x0 ) − T (x0 )) 

L2

≤ A1 Tn (x0 ) − T (x0 ) L 2   m m      = A1  Pn (Ki ψi x0 + f ) − (Ki ψi x0 + f )   i=1 i=1 L2   m      = A1 (Pn − I) (Ki ψi x0 + f )   i=1

L2

= A1 (Pn − I) x0  L 2    (r )  ≤ A1 cn −r x0  2 → 0, as n → ∞.

(2.34)

L

By choosing n large enough such that αn ≤ δ(1 − q), the Eq. (2.31) of Theorem 2.2 is satisfied. Hence by applying Theorem 2.2, we obtain αn αn ≤ xn − x0  L 2 ≤ . 1+q 1−q 

This completes the proof.

Theorem 2.4 Let x0 ∈ C r [−1, 1], r ≥ 1, be an isolated solution of the Eq. (2.2).  m (Ki ψi ) (x0 ), where Assume that 1 is not an eigenvalue of the linear operator i=1 m m  i=1 (Ki ψi ) (x 0 ) denotes the Frechet derivative of i=1 Ki ψi (x) at x 0 . Let Pn : X → Xn be either orthogonal or interpolatory projection operator defined by (2.11) or (2.15), respectively. Then the Eq. (2.20) has a unique solution xn ∈ B(x0 , δ) = {x : x − x0 ∞ < 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 ≤ xn − x0 ∞ ≤ , 1+q 1−q 

where αn = (I − Tn (x0 ))−1 (Tn (x0 ) − T (x0 ))∞ . Proof Using estimates (2.5) and (2.19), we have  

     Tn (x0 ) − T (x0 ) x 

123

∞



 m m         =  Pn (Ki ψi ) (x0 ) − (Ki ψi ) (x0 ) x    i=1

i=1

∞

Convergence analysis of Legendre spectral projection methods

  m       = (Pn − I) (Ki ψi ) (x0 ) x    i=1

≤ cn

β−r



(r )

∞

[(Ki ψi ) (x0 )x] ∞

≤ 2cn β−r dkr,∞ x∞ .

(2.35)

Since 0 < β < 1, for β < r = 1, 2, 3, . . . , it follows that       Tn (x0 ) − T (x0 )

∞

  = O n β−r → 0, as n → ∞. 

−1

Hence by applying Lemma 2.7, we see that (I − Tn (x0 )) exists and uniformly bounded on X, for some sufficiently large n, i.e., there exists some A2 > 0 such that −1  (I − Tn (x0 )) ∞ ≤ A2 < ∞. Now, we have for any x ∈ B(x0 , δ), ⎡ ⎤     m   m     ⎣ ⎦ ≤ Pn (Ki ψi ) (x0 ) − Pn (Ki ψi ) (x) (y)  ∞  i=1  i=1   ∞   m        ≤ − I ψ ψ (x) (y) − (P (K ) ) (x ) (K ) n 0 i i i i     i=1 ⎡ ⎤ ∞    m   m    ⎣ ⎦ (y) + ψ ψ (x) − (K ) (x ) (K ) 0 i i i i    i=1  i=1 ∞ ⎡⎛ ⎞ ⎤(r )    m m      ⎣⎝  β−r ⎠ ⎦ ≤ cn (Ki ψi ) (x0 ) − (Ki ψi ) (x) (y)       i=1 i=1 ∞ ⎡ ⎤     m m      ⎣ + (Ki ψi ) (x0 ) − (Ki ψi ) (x)⎦ (y)   . (2.36)  i=1  i=1

 

     Tn (x0 ) − Tn (x) y 

∞

Now using the Lipschitz’s continuity of ψi(0,1) (., x0 (.)), we have    (r )     m m        (Ki ψi ) (x0 ) − (Ki ψi ) (x) (y)    i=1  i=1 ∞    (r )    m m      = sup  (t) (Ki ψi ) (x0 ) − (Ki ψi ) (x) (y) t∈[−1,1]   i=1 i=1  m    1 ∂ r      (0,1) (0,1) = sup  ki (t, s) ψi (s, x(s)) y(s)ds  (s, x0 (s)) − ψi r   ∂t −1 t∈[−1,1] i=1

123

P. Das et al.

≤

m 

sup

i=1 t,s∈[−1,1]

 r  ∂     ∂t r ki (t, s)

1 −1

qi |(x0 − x) (s)| |y(s)|ds

≤ 2l2 kr,∞ x0 − x∞ y∞ ≤ 2l2 kr,∞ δy∞ .

(2.37)

Hence combining the estimates (2.8), (2.36) and (2.37), we obtain 



−1       sup Tn (x0 ) − Tn (x)   I − Tn (x0 )  x−x0 ∞ ≤δ ∞  β−r  ≤ A2 cn 2l2 kr,∞ δ + 2l2 Mδ   ≤ 2 A2 l2 M + cn β−r kr,∞ δ ≤ q, (say) where 0 < q < 1, which proves the Eq. (2.30) of Theorem 2.2. Since β < r = 1, 2, . . . , using the estimate (2.19), we have  

−1     αn =  I − Tn (x0 ) (Tn (x0 ) − T (x0 )) 

∞

≤ A2 Tn (x0 ) − T (x0 )∞   m   m        = A2 Pn Ki ψi x0 + f − Ki ψi x0 + f    i=1 i=1 ∞   m       = A2 (Pn − I) Ki ψi x0 + f    i=1  ∞ β−r  (r )  = A2 (Pn − I) x0 ∞ ≤ cn x0  → 0, as n → ∞. ∞

(2.38)

By choosing n large enough such that αn ≤ δ(1 − q), the Eq. (2.31) of Theorem 2.2 is satisfied. Hence by applying Theorem 2.2, we obtain αn αn ≤ xn − x0 ∞ ≤ . 1+q 1−q 

This completes the proof.

Next we discuss the existence and convergence of the iterated approximate solution x˜n to x0 .   Theorem 2.5 Tn (x0 ) is ν-convergent to T (x0 ) in both L 2 -norm and infinity norm.

Proof Consider  m            (Ki ψi ) (Pn x0 ) Pn x(t) Tn (x0 ) x(t) =    i=1

123

Convergence analysis of Legendre spectral projection methods

 m 

m         ≤ (Ki ψi ) (Pn x0 ) − (Ki ψi ) (x0 ) Pn x(t)   i=1 i=1  m       + (2.39) (Ki ψi ) (x0 ) Pn x(t) .   i=1

Now using estimates (2.8), (2.18) and the fact that Pn x L 2 ≤ p1 x∞ , we have  m 

m         (Ki ψi ) (Pn x0 )− (Ki ψi ) (x0 ) Pn x     i=1

i=1

≤ Ml2 Pn x0 −x0  L 2 Pn x L 2 ∞

   (r )  ≤ cMl2 p1 n −r x0  → 0, as n → ∞.

L2

x∞ (2.40)

Again using the assumption (iii), we see that   m      (Ki ψi ) (x0 ) Pn x     i=1

∞

 m       = max  (Ki ψi ) (x0 ) Pn x(t)  t∈[−1,1]  i=1  m    1    (0,1) = max  ki (t, s)ψi (s, x0 (s)) Pn x(s)ds   t∈[−1,1]  −1 i=1  1 |Pn x(s)| ds ≤ M sup di ≤

√

1≤i≤m

−1

2Md Pn x L 2 ≤

√ 2Mdp1 x∞ .

(2.41)

Now combining the estimates (2.39)–(2.41) we obtain      Tn (x0 )

∞

   (r )  ≤ cMl2 p1 n −r x0 

L2

+

√

2Mdp1 < ∞.

 This shows that Tn (x0 )∞ is uniformly bounded. Consider          T (x0 ) − Tn (x0 ) T (x0 ) x(t)  m 

m m           = (Ki ψi ) (x0 ) − (Ki ψi ) (Pn x0 ) Pn (Ki ψi ) (x0 ) x(t)   i=1 i=1 i=1  m m

m        = Pn (Ki ψi ) (x0 ) + (I − Pn ) (Ki ψi ) (x0 ) (Ki ψi ) (x0 ) x(t)  i=1 i=1 i=1  m m       − (Ki ψi ) (Pn x0 ) Pn (Ki ψi ) (x0 ) x(t) 

i=1

i=1

123

P. Das et al.

 m 

m m           ≤ (Ki ψi ) (x0 ) − (Ki ψi ) (Pn x0 ) Pn (Ki ψi ) (x0 ) x(t)   i=1 i=1 i=1  m  m        + (Ki ψi ) (x0 ) (I − Pn ) (Ki ψi ) (x0 ) x(t) .   i=1

(2.42)

i=1

Now for the second term in the above estimate (2.42), we have  m  m        (Ki ψi ) (x0 ) (I − Pn ) (Ki ψi ) (x0 ) x(t)    i=1 i=1  m   m  1      (0,1) = ki (t, s)ψi (s, x0 (s)) (I − Pn ) (Ki ψi ) (x0 ) x(s)ds    i=1 −1 i=1   m       ≤ < gt (.), (I − Pn ) (Ki ψi ) (x0 ) x(.) >   i=1   m    √    ≤ 2 gt  L 2 (I − Pn ) (2.43) (Ki ψi ) (x0 ) x∞ ,   i=1

L2

m where gt (s) = i=1 ki (t, s)ψi(0,1) (s, x0 (s)). Hence we have   m m        (Ki ψi ) (x0 ) (I − Pn ) (Ki ψi ) (x0 )    i=1 i=1 ∞   m    √    ≤ 2 gt  L 2 (I − Pn ) (Ki ψi ) (x0 ) .   i=1

(2.44)

L2

Now for the first term in the estimate (2.42), using (2.8) we have  m 

m m           (Ki ψi ) (x0 ) − (Ki ψi ) (Pn x0 ) Pn (Ki ψi ) (x0 ) x     i=1 i=1 i=1 ∞   m       ≤ Ml2 Pn x0 − x0  L 2 Pn (Ki ψi ) (x0 ) x    2 i=1  L  m      ≤ Ml2 p1 (I − Pn ) x0  L 2  (2.45) (Ki ψi ) (x0 ) x∞ .   i=1

∞

Hence combining estimates (2.18), (2.26), (2.42), (2.44) and (2.45), we have           Tn (x0 ) − T (x0 ) T (x0 )

∞

123

  m      ≤ Ml2 p1 Pn x0 − x0  L 2  (Ki ψi ) (x0 )   i=1

∞

Convergence analysis of Legendre spectral projection methods

+

√

  m       2 gt  L 2 (I − Pn ) (Ki ψi ) (x0 )   i=1

→ 0, as n → ∞.

L2

Next consider          T (x0 ) − Tn (x0 ) Tn (x0 ) x(t)  m 

m          = (Ki ψi ) (x0 ) − (Ki ψi ) (Pn x0 ) Pn Tn (x0 ) x(t)   i=1 i=1  m        = (Ki ψi ) (x0 ) Pn Tn (x0 ) + (I − Pn ) Tn (x0 ) x(t)  i=1  m       − (Ki ψi ) (Pn x0 ) Pn Tn (x0 ) x(t)  i=1  m 

m          ≤ (Ki ψi ) (x0 ) − (Ki ψi ) (Pn x0 ) Pn Tn (x0 ) x(t)   i=1 i=1  m        + (Ki ψi ) (x0 ) (I − Pn ) Tn (x0 ) x(t) .  

(2.46)

i=1

For the second term in the above estimate (2.46), using Cauchy–Schwartz inequality, we get  m        (Ki ψi ) (x0 ) (I − Pn ) Tn (x0 ) x(t)    i=1  m    1     (0,1) = ki (t, s)ψi (s, x0 (s)) (I − Pn ) Tn (x0 ) x(s)ds    i=1 −1      = < gt (.), (I − Pn ) Tn (x0 ) x(.) >   √    ≤ 2 gt  L 2 (I − Pn ) Tn (x0 ) 2 x∞ . L

(2.47)

This implies  m      √        (Ki ψi ) (x0 ) (I − Pn ) Tn (x0 ) x  ≤ 2 gt  L 2 (I − Pn ) Tn (x0 ) 2 x∞ .    L i=1

∞

(2.48)

123

P. Das et al.

Using the estimate (2.8), we have  m 

m          (Ki ψi ) (x0 ) − (Ki ψi ) (Pn x0 ) Pn Tn (x0 ) x     i=1 i=1 ∞      ≤ Ml2 Pn x0 − x0  L 2 Pn Tn (x0 ) x  2   L   ≤ Ml2 p1 (I − Pn ) x0  L 2 Tn (x0 ) x∞ .

(2.49)

∞

Hence combining estimates (2.18), (2.29), (2.46), (2.48) and (2.49) we have                Tn (x0 ) − T (x0 ) Tn (x0 ) ≤ Ml2 p1 Pn x0 − x0  L 2 Tn (x0 ) ∞  ∞ √     + 2 gt  L 2 (I − Pn ) Tn (x0 ) 2 → 0, as n → ∞.

L

  This shows that Tn (x0 ) is ν-convergent to T (x0 ) in infinity norm. On similar lines it can be shown that Tn (x0 ) is ν-convergent to T  (x0 ) in L 2 - norm. This completes the proof . 

The following theorem is obtained by direct application of Lemma 2.7 and Theorem 2.5. r [−1, 1] be an isolated solution of the Eq. (2.2). Assume that Theorem 2.6 Let x0 ∈ C  m (Ki ψi ) (x0 ). Then for sufficiently large n, the operator 1 is not an eigenvalue of i=1  I − Tn (x0 ) is invertible on C[−1, 1] and there exist constants L , L 1 > 0 independent   of n such that (I − Tn (x0 ))−1 ∞ ≤ L and (I − Tn (x0 ))−1  L 2 ≤ L 1 .

Theorem 2.7 Let x0 ∈ C r [−1, 1] be an isolated solution of the Eq. (2.2). Let Pn : X → Xn be either orthogonal or interpolatory projection operator m defined by (2.11) (Ki ψi ) (x0 ), then or (2.15), respectively. Assume that 1 is not an eigenvalue of i=1 for sufficiently large n, the iterated solution x˜n defined by (2.25) is the unique solution in the sphere B(x0 , δ) = {x : x − x0 ∞ < δ}. Moreover, there exists a constant 0 < q < 1, independent of n such that βn βn ≤ x˜n − x0 ∞ ≤ , 1+q 1−q where  

−1        βn =  I − Tn (x0 ) Tn (x0 ) − T (x0 )   . ∞

Proof From Theorem 2.6, there exists a constant L > 0 such that 

−1     I − T (x0 )  ≤ L, n   ∞

123

Convergence analysis of Legendre spectral projection methods

for sufficiently large value of n. Using the estimate (2.8) of Lemma 2.1 and the fact that Pn x L 2 ≤ p1 x∞ , for any x ∈ B(x0 , δ), we have          Tn (x) − Tn (x0 ) y 

∞

 m 

m         = (Ki ψi ) (Pn x) − (Ki ψi ) (Pn x0 ) Pn y    i=1

i=1

≤ Ml2 Pn (x − x0 ) L 2 Pn y L 2 ≤ Ml2 p12 x − x0 ∞ y∞ ≤ Ml2 p12 δy∞ .

∞

(2.50)

This implies sup

x−x0 ∞ ≤δ



−1 

     I − T (x0 )    T (x) − T (x ) 0  n n n 

∞

≤ L Ml2 p12 δ ≤ q, (say)

where 0 < q < 1, which proves the Eq. (2.30) of Theorem 2.2. Now using the estimate (2.7) and (2.18), we have  m

 m          Tn (x0 ) − T (x0 )  ≤  K ψ x K ψ − (P ) (x ) i i n 0 i i 0  ∞   i=1 i=1 ∞ √ ≤ l1 M 2 (I − Pn ) x0  L 2   √  (r )  ≤ l1 M 2cn −r x0  2 → 0, as n → ∞. L

(2.51)

Hence  

−1        βn =  I − Tn (x0 ) Tn (x0 ) − T (x0 )  

∞

  √  (r )  ≤ Ll1 M 2cn −r x0 

L2

→ 0, as n → ∞.

Choose n large enough such that βn ≤ δ(1−q). Then the Eq. (2.31) of Theorem 2.2 is satisfied. Thus by applying Theorem 2.2, we obtain βn βn ≤ x˜n − x0 ∞ ≤ , 1+q 1−q where  

−1    n (x0 ) n (x0 ) − T (x0 )  . βn =  I − T T   ∞

This completes the proof.



Theorem 2.8 Let x0 ∈ C r [−1, 1] be an isolated solution of the Eq. (2.2). Let Pn : X → Xn be either orthogonal or interpolatory projection operator m defined by (2.11) (Ki ψi ) (x0 ), then and (2.15), respectively. Assume that 1 is not an eigenvalue of i=1

123

P. Das et al.

for sufficiently large n, the iterated solution x˜n defined by (2.25) is the unique solution in the sphere B(x0 , δ) = {x : x − x0  L 2 < δ}. Moreover, there exists a constant 0 < q < 1, independent of n such that βn βn ≤ x˜n − x0  L 2 ≤ , 1+q 1−q where

 

−1        Tn (x0 ) − T (x0 )  βn =  I − Tn (x0 )  2. L

 Proof From Theorem 2.6, we have (I − Tn (x0 ))−1 exists and it is uniformly bounded in L 2 -norm on C[−1, 1], i.e., there exists a constant L 1 > 0 such that  (I − Tn (x0 ))−1  L 2 ≤ L 1 . Now using the estimate (2.8) and the fact that Pn  L 2 ≤ p, we have for any x ∈ B(x0 , δ),        √           Tn (x) − Tn (x0 ) (y) 2 ≤ 2  Tn (x) − Tn (x0 ) (y) L ∞  m 

m    √      ≤ 2 (Ki ψi ) (Pn x)− (Ki ψi ) (Pn x0 ) Pn y    i=1 i=1 ∞ √ ≤ 2Ml2 Pn (x − x0 ) L 2 Pn y L 2 √ ≤ 2Ml2 p 2 x − x0  L 2 y L 2 √ ≤ 2Ml2 p 2 δy L 2 .

Thus we obtain 

−1 

         sup Tn (x) − Tn (x0 )    I − Tn (x0 ) x−x0  L 2 ≤δ

L2

√ ≤ L 1 2Ml2 p 2 δ ≤ q, (say)

where 0 < q < 1, which proves the Eq. (2.30) of Theorem 2.2. Now using the estimate (2.51), we have  

−1    n (x0 ) n (x0 ) − T (x0 )  I − T βn =  T   2 L  

−1          ≤  2 Tn (x0 ) − T (x0 ) L 2  I − Tn (x0 ) L √   Tn (x0 ) − T (x0 )∞ → 0, as n → ∞. ≤ L 1 2 Choose n large enough such that βn ≤ δ(1 − q). Hence the Eq. (2.31) of Theorem 2.2 is satisfied. Then applying Theorem 2.2, we get βn βn ≤ x˜n − x0  L 2 ≤ , 1+q 1−q

123

Convergence analysis of Legendre spectral projection methods

where  

−1        βn =  I − Tn (x0 ) Tn (x0 ) − T (x0 )   2. L



This completes the proof.

In the following theorem we give the error bounds for the iterated solutions in both infinity norm and L 2 -norm. Theorem 2.9 Let x0 ∈ C[−1, 1] be an isolated solution of the Eq. (2.2). Let x˜n defined by the iterated scheme (2.25). Then the following hold ! x˜n − x0 ∞ ≤ c x0 − Pn x0 2L 2 + |< gt , (I − Pn ) x0 >| , (2.52) and

! x˜n − x0  L 2 ≤ c x0 − Pn x0 2L 2 + |< gt , (I − Pn ) x0 >| ,

where gt (s) =

m

i=1 ki (t,

(2.53)

s)ψi(0, 1) (s, x0 (s)) and c is a constant independent of n.

Proof From Theorem 2.7, we have βn βn ≤ x˜n − x0 ∞ ≤ , 1+q 1−q where,

 

−1    n (x0 ) n (x0 ) − T (x0 )  . I − T T βn =    ∞

Hence using Theorem 2.6, we get x˜n − x0 ∞

 

−1        Tn (x0 ) − T (x0 )  ≤ βn =  I − Tn (x0 )  ∞  

−1         Tn (x0 ) − T (x0 )∞ ≤  I − Tn (x0 )  ∞   m m      ≤ L Ki ψi (Pn x0 ) − Ki ψi (x0 ) .   i=1

i=1

(2.54)

∞

m (0,1) ki (t, s)ψi (s, x0 (s) + θ (x(s) − x0 (s))) and We denote g(t, s, x0 , x, θ ) = i=1 m (0,1) (s, x0 (s)). gt (s) = i=1 ki (t, s)ψi We have  m      [(Ki ψi ) (Pn x0 ) − (Ki ψi ) (x0 )] (t)    i=1

123

P. Das et al.

 m    1    = ki (t, s) [ψi (s, Pn x0 (s)) − ψi (s, x0 (s))] ds    −1 i=1  m    1    = ki (t, s) [ψi (s, x0 (s) + θ (Pn x0 (s) − x0 (s)))] (x0 − Pn x0 ) (s)ds    i=1 −1  1    =  g (t, s, x0 , Pn x0 , θ ) (x0 − Pn x0 ) (s)ds  −1  1    =  [g (t, s, x0 , Pn x0 , θ ) − gt (s) + gt (s)] (x0 − Pn x0 ) (s)ds  −1  1     ≤ [g (t, s, x0 , Pn x0 , θ ) − gt (s)] (x0 − Pn x0 ) (s)ds  −1  1     (2.55) + gt (s) (x0 − Pn x0 ) (s)ds  . −1

For the first term of the above estimate (2.55), using Lipschitz’s continuity of (0,1) ψi (., x(.)), we have  1     [g (t, s, x0 , Pn x0 , θ ) − gt (s)] (x0 − Pn x0 ) (s)ds   −1  m   1   (0,1) = ki (t, s) ψi (s, x0 (s) + θ (Pn x0 (s) − x0 (s)))  −1 i=1

−ψi (s, x0 (s))] (x0 − Pn x0 ) (s)ds|  1 m  sup |ki (t, s)| qi |(x0 − Pn x0 ) (s)| |(x0 − Pn x0 ) (s)| ds ≤ i=1 s,t∈[−1,1]  1

= Ml2

−1

−1

|(x0 − Pn x0 ) (s)|2 ds ≤ Ml2 x0 − Pn x0 2L 2 .

Now for the second term of (2.55), we have  1     gt (s) (x0 − Pn x0 ) (s)ds  = |< gt (.), (I − Pn ) (x0 ) >| .  −1

(2.56)

(2.57)

Hence combining estimates (2.55)–(2.57), we have   m m      Ki ψi (x0 )  Ki ψi (Pn x0 ) −   i=1

i=1

∞

≤ Ml2 x0 − Pn x0 2L 2 + |< gt (.), (I − Pn ) (x0 ) >| . Therefore from estimates (2.54) and (2.58), we have x˜n − x0 ∞ ≤ L K (Pn x0 ) − K (x0 )∞

123

(2.58)

Convergence analysis of Legendre spectral projection methods

! ≤ c x0 − Pn x0 2L 2 + |< gt (.), (I − Pn ) (x0 ) >| .

(2.59)

where c is a constant independent of n, this proves the estimate (2.52). Similarly for L 2 -norm, we can show that x˜n − x0  L 2 ≤

√ 2 x˜n − x0 ∞

! ≤ c x0 − Pn x0 2L 2 + |< gt (.), (I − Pn ) (x0 ) >| ,

c being a constant independent of n. This completes the proof.

(2.60) 

Now we discuss the convergence rates for the approximate and iterated approximate solutions. To distinguish between the Legendre Galerkin solutions and Legendre collocation solutions, we set the following notations. In case of Legendre Galerkin method, we denote the approximate solution and the iterated approximate solutions by xn = xnG and x˜n = x˜nG , respectively. For Legendre collocation method, we write the approximate solution and iterated approximate solution as xn = xnC and x˜n = x˜nC , respectively. Theorem 2.10 Let x0 ∈ C r [−1, 1] be a isolated solution of the Eq. (2.1) and xn = xnG be the Legendre Galerkin solution or xn = xnC be the Legendre collocation approximation of x0 . Then the following hold           x0 − xnG  2 , x0 − xnC  2 = O n −r , L

and

L

        x0 − xnG  , x0 − xnC  ∞

∞

1 = O n 2 −r .

Proof From Theorem 2.3, we have αn αn ≤ xn − x0  L 2 ≤ , 1+q 1−q 

where αn = (I − Tn (x0 ))−1 (Tn (x0 ) − T (x0 )) L 2 . Hence we have from the estimate (2.34) xn − x0  L 2 ≤ αn ≤ A1 (Pn − I ) x0  L 2      (r )  ≤ A1 cn −r x0  2 = O n −r , L

where c is a constant independent of n. Now for the error in infinity norm, using the estimate (2.38), we have xn − x0 ∞ ≤ A2 (Pn − I ) x0 ∞ .

123

P. Das et al.

Hence for Legendre Galerkin solution xn = xnG , using estimate (2.14) of Lemma 2.3, we have   

1

 1  G    (r ) xn − x0  ≤ A2  PnG − I x0  ≤ A2 cn 2 −r V x0 = O n 2 −r , (2.61) ∞

∞

and for Legendre collocation solution xnC , using estimate (2.17), we have     C xn − x0 

∞



   ≤ A2  PnC − I x0 

∞

  1  (r )  ≤ A2 cn 2 −r x0 

∞

1 = O n 2 −r . (2.62)

Hence the proof follows.

 Next we will discuss the error bounds for the iterated Legendre Galerkin and iterated Legendre collocation solutions separately. Theorem 2.11 Let x0 ∈ C r [−1, 1] be a isolated solution of the equation (2.1) and x˜nG be the iterated Legendre Galerkin approximations of x0 . Then there hold x0 − x˜nG  L 2 = O(n −2r ), x0 − x˜nG ∞ = O(n −2r ). Proof From Theorem 2.9, we have    G  x˜n − x0 

∞

" 2  #

    ≤ c x0 − PnG x0  2 + < gt (.), I − PnG (x0 ) > ,

(2.63)

L

where c is a constant independent of n. Using the orthogonality of the projection operators PnG and Cauchy–Schwarz inequality, we obtain    

    < gt (.), I − PnG (x0 ) > = < I − PnG gt (.), I − PnG (x0 ) >    

    ≤  I − PnG gt (.) 2 x0 − PnG x0  2 . (2.64) L

L

Hence using estimates (2.12), (2.63) and (2.64), we have    G  x˜n − x0 

∞

And also

"    2  #

    G  G G  ≤ c x0 − Pn x0  2 +  I − Pn gt (.) 2 x0 − Pn x0  2

   G  x˜n − x0 

Hence the proof follows.

123

L

L

       (r ) 2  (r )    ≤ cn −2r x0  2 + cn −2r x0  2 (gt (.))(r )  2 L L L

−2r . =O n

L2

≤

 √    2x˜nG − x0 

∞

= O n −2r .

L

(2.65)

(2.66) 

Convergence analysis of Legendre spectral projection methods

Theorem 2.12 Let x0 ∈ C r [−1, 1] be a isolated solution of the Eq. (2.1) and x˜nC be the iterated Legendre collocation approximations of x0 . Then there hold       x0 − x˜nC  2 = O n −r , L       x0 − x˜nC  = O n −r . ∞

Proof Using Theorem 2.9, Lemma 2.5, we have for the interpolatory projection operator PnC    C  x˜n − x0 

∞

and

" 2  #

   C  C ≤ c x0 − Pn x0  2 + < gt (.), I − Pn (x0 ) > L " 2   #    C C  ≤ c x0 − Pn x0  2 + gt  L 2 x0 − Pn x0  2 L L "  2   #     (r ) (r ) ≤ c n −2r x0  2 + n −r gt  L 2 x0  2 L L   (2.67) = O n −r ,

    C x˜n − x0 

L2

≤

 √    2x˜nC − x0 

∞

  = O n −r .

Hence the proof follows.

(2.68) 

Remark From Theorems 2.10–2.12 we observe that the Legendre Galerkin and Legendre collocation solutions of Hammerstein integral equation of mixed type have same 1 order of convergence, O(n −r ) in L 2 -norm and O(n 2 −r ) in infinity norm. The iterated Legendre Galerkin solution converges with the order O(n −2r ) in both L 2 -norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order O(n −r ) in both L 2 -norm and in infinity norm. This shows that iterated Legendre Galerkin method improves over the iterated Legendre collocation method. 3 Numerical example In this section we present the numerical results. To apply Legendre Galerkin and Legendre collocation methods, we choose the approximating subspaces Xn to be the Legendre polynomial subspaces of degree ≤n. Legendre polynomials can be generated by the following three-term recurrence relation φ0 (s) = 1, φ1 (s) = s, s ∈ [−1, 1], and (i + 1)φi+1 (s) = (2i + 1)sφi (s) − iφi−1 (s), s ∈ [−1, 1], i = 1, 2, . . . , n − 1. (3.1)

123

P. Das et al.

We denote, the Galerkin and iterated Galerkin solutions by xnG and x˜nG , respectively and also the collocation and iterated collocation solutions by xnC and x˜nC , respectively, in the tables. We present the errors of the approximation solutions and the iterated approximation solutions in both L 2 -norm and infinity norm . In Tables 1 and 2, n represents the highest degree of the Legendre polynomials employed in the computation. The numerical algorithms are compiled by using Matlab. Example 3.1 We consider the following integral equation  x(t) −

1

 k1 (t, s)ψ1 (s, x(s))ds −

0

1

k2 (t, s)ψ2 (s, x(s))ds = f (t), 0 ≤ t ≤ 1,

0

(3.2) with the kernel function k1 (t, s) = 15 cos(π t) sin(π s), ψ1 (s, x(s)) = [x(s)]3 , k (t, s) = 1, ψ2 (s, x(s)) = [x(s)]2 and the function f (t) = sin(π t) + 20 9 (−20 + √ √2 391), where the exact solution is given by x(t) = sin(π t) + 13 (20 − 391) cos(π t). From Tables 1 and 2, we see that the numerical results agree with the theoretical results. Table 1 Legendre Galerkin method n

x − xnG  L 2

x − xnG ∞

x − x˜nG  L 2

x − x˜nG ∞

2

0.018497926387881

0.061401240718413

0.001732620314859

0.002666926405633

3

0.017336478824659

0.048737889846769

0.001574242749833

0.002453522073331

4

0.000424247321694

0.001851144219817

0.000006496665114

0.000011127957495

5

0.000369076883065

0.001206622182063

0.000005443607765

0.000009310972638

6

0.000005222931138

0.000024729337375

0.000000010339987

0.000000017430265

7

0.000004157449815

0.000014238564338

0.000000007739170

0.000000013043854

8

0.000000474088890

0.000002409744427

0.000000000130864

0.000000000221094

9

0.000000501093684

0.000002898739808

0.000000000121043

0.000000000204675

x − x˜nC  L 2

x − x˜nC ∞

Table 2 Legendre collocation method n

x − xnC  L 2

x − xnC ∞

2

0.052643089675581

0.136549373507889

0.046581704234504

0.055514809817546

3

0.017418602723992

0.047507136655563

0.002282365996117

0.003123012968493

4

0.000565650258793

0.002817858373402

0.000058722738211

0.000066937610850

5

0.000369120217887

0.001204863102227

0.000005861591406

0.000010127658152

6

0.000006674539893

0.000038873663482

0.000000056737713

0.000000098268420

7

0.000004155367261

0.000014621422220

0.000000007524786

0.000000012713376

8

0.000000049276632

0.000000301230662

0.000000000046095

0.000000000078096

9

0.000000028881317

0.000000103466737

0.000000000003588

0.000000000005924

123

Convergence analysis of Legendre spectral projection methods

References 1. Ahues, M., Largillier, A., Limaye, B.V.: Spectral Computations for Bounded Operators. Chapman and Hall/CRC, New York (2001) 2. Atkinson, K.: A survey of numerical methods for solving nonlinear integral equations. J. Integral Equ. Appl. 4(1), 15–46 (1992) 3. Atkinson, K.E., Potra, F.A.: Projection and iterated projection methods for nonlinear integral equations. SIAM J. Numer. Anal. 24(6), 1352–1373 (1987) 4. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods, Fundamentals in Single Domains. Springer, Berlin (2006) 5. Das, P., Sahani, M.M., Nelakanti, G.: Legendre spectral projection methods for Urysohn integral equations. J. Comput. Appl. Math. 263, 88–102 (2014) 6. Ganesh, M., Joshi, M.C.: Discrete numerical solvability of Hammerstein integral equations of mixed type. J. Integral Equ. Appl. 2(1), 107–124 (1989) 7. Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral-equations of mixed type. IMA J. Numer. Anal. 11(1), 21–31 (1991) 8. Guo, B.: Spectral Methods and Their Applications. World Scientific, Singapore (1998) 9. Kaneko, H., Noren, R.D., Xu, Y.: Regularity of the solution of Hammerstein equations with weakly singular kernels. Integral Eqs. Oper. Theory 13(5), 660–670 (1990) 10. Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006) 11. Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics. Springer, New York (2011) 12. Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra type integral equations and the convergence analysis. J. Comput. Math. 26(6), 825–837 (2008) 13. Vainikko, G.M.: A perturbed Galerkin method and the general theory of approximate methods for nonlinear equations. USSR Comput. Math. Math. Phys. 7(4), 1–41 (1967) 14. Wan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4(1), 181–193 (2009) 15. Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53(2), 414–434 (2012)

123