Quadratic Spline Collocation for the Smoothed Weakly Singular Fredholm Integral Equations

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Quadratic Spline Collocation for the Smoothed WeaklySingular Fredholm Integral EquationsRene Pallav a & Arvet Pedas aa Institute of Mathematics, University of Tartu , Tartu, EstoniaPublished online: 22 Dec 2009.

To cite this article: Rene Pallav & Arvet Pedas (2009) Quadratic Spline Collocation for the Smoothed WeaklySingular Fredholm Integral Equations, Numerical Functional Analysis and Optimization, 30:9-10, 1048-1064, DOI:10.1080/01630560903408705

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Numerical Functional Analysis and Optimization, 30(9–10):1048–1064, 2009Copyright © Taylor & Francis Group, LLCISSN: 0163-0563 print/1532-2467 onlineDOI: 10.1080/01630560903408705

QUADRATIC SPLINE COLLOCATION FOR THE SMOOTHED WEAKLYSINGULAR FREDHOLM INTEGRAL EQUATIONS

Rene Pallav and Arvet Pedas

Institute of Mathematics, University of Tartu, Tartu, Estonia

� This article addresses a quadratic spline collocation method for the numerical solutionof weakly singular Fredholm integral equations of the second kind. Using suitable smoothingtechniques, optimal global convergence estimates are derived and a local superconvergence resultis given.

Keywords Quadratic spline collocation method; Smoothing transformation; Weaklysingular integral equation.

AMS Subject Classification 45B05; 65R20.

1. INTRODUCTION

Let � = �1, 2, � � � �, �0 = �0� ∪ �, � = (−∞,∞),

Db = �[0, b] × [0, b]�\�(x , y) ∈ �2 : x = y��

We consider a class of linear weakly singular Fredholm integral equationsof the form

u(x) =∫ b

0K (x , y)u(y)dy + f (x), x ∈ [0, b], (1.1)

where b > 0, K : Db → � and f : [0, b] → � are (at least) continuousfunctions, and K (x , y) may have a weak singularity at x = y. Actually,we assume that K ∈ � m,�(Db), m ∈ �0, m ≤ 4, � ∈ �, −∞< � < 1. Here,� m,�(Db), m ∈ �0, � ∈ �, � < 1 is defined as the collection of m times

Received 20 August 2009; revised 2 October 2009; accepted 9 October 2009Address correspondence to Arvet Pedas, Institute of Mathematics, University of Tartu, J.Liivi

2, Tartu 50409, Estonia; E-mail: [email protected]

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Smoothed Weakly Singular Fredholm Integral Equations 1049

continuously differentiate functions K : Db → � satisfying

∣∣∣∣(

�

�x

)j(�

�x+ �

�y

)k

K (x , y)∣∣∣∣ ≤ c

1 if � + j < 0,

1 + | log |x − y|| if � + j = 0,

|x − y|−�−j if � + j > 0�

(1.2)

with a constant c = c(K ) for all (x , y) ∈ Db and all integers j , k ∈ �0 suchthat j + k ≤ m. It follows from (1.2) that

|K (x , y)| ≤ c

1 if � < 01 + | log |x − y|| if � = 0|x − y|−� if � > 0

, (x , y) ∈ Db � (1.3)

Thus, a kernel K ∈ � m,�(Db) is at most weakly singular for 0 ≤ � < 1. For� < 0, the kernel K ∈ � m,�(Db) is bounded on Db but its derivatives mayhave diagonal singularities.

Equations of this type arise in the potential theory, polymer physics,atmospheric physics, and many other fields (see, e.g., [2, 3, 6, 8, 13]).The main difficulty with this type of equations is that the solution u to(1.1) (if it exists) is generally not a smooth function on the entire interval[0, b] even if f is sufficiently smooth on [0, b]. Instead, we find that thederivatives of u(x) are typically unbounded at x = 0 and x = b (see, e.g.,[11, 36, 37] and Lemma 2.2 below). In collocation methods the possiblesingular behavior of the exact solution can be taken into account by usingpiecewise polynomial functions on special non-uniform grids which areproperly graded in order to compensate the generic boundary singularitiesof the derivatives of the exact solution, see, for example, [1, 5, 12, 15,25, 31, 33, 35, 37]. A similar approach with smooth quadratic and cubicsplines is considered in [20–22] and [23, 24, 34], respectively. However,in practice, the use of strongly graded grids may create unacceptableround-off errors in calculations and therefore lead to unstable behavior ofnumerical results.

To avoid problems associated with the use of strongly graded gridswe first perform in equation (1.1) a change of variables so that thesingularities of the derivatives of the exact solution of the resultingequation will be milder or disappear. After that, we solve the transformedequation by a quadratic spline collocation method on a mildly graded oruniform grid. Our approach is based on the ideas of [26, 28], see also[7, 9, 10, 14, 18, 38, 39]. Optimal global convergence estimates are derived(Theorem 4.2) and the optimal superconvergence rate at the collocationpoints in the case of special collocation and smoothing parameters isestablished (Theorem 5.1). These results refine and complement thecorresponding results of [17, 20–22]. Actually, Theorem 5.1 has particular

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1050 R. Pallav and A. Pedas

importance since in [17, 20–22] the superconvergence properties ofproposed algorithms have not been studied.

2. REGULARITY OF THE SOLUTION AND SMOOTHING

We denote by Cm(�) the set of m times (m ≥ 0) differentiablefunctions on � ⊂ �,C 0(�) = C(�). By C [0, b] the Banach space ofcontinuous functions u : [0, b] → � with the norm ‖u‖∞ = max0≤x≤b |u(x)|will be denoted. Throughout the article c , c0, c1, � � � are positive constantsthat may have different values in different occurrences.

Denote

� (I − TK ) = �u ∈ C [0, b] : u = TKu�

where I is the identity mapping and TK is defined by formula

(TKu)(x) =∫ b

0K (x , y)u(y)dy, 0 ≤ x ≤ b�

If K ∈ � 0,�(Db),−∞< � < 1, then it follows from (1.3) that TK is compactas an operator from C [0, b] into C [0, b]. A consequence of this is thefollowing lemma.

Lemma 2.1. Let f ∈ C [0, b],K ∈ � 0,�(Db), −∞< � < 1, and let � (I −TK ) = �0�. Then equation (1.1) is uniquely solvable and its solution belongs toC [0, b].

Below we always keep in mind that the conditions of Lemma 2.1 arefulfilled.

In order to characterize the smoothness of the solution to (1.1)we introduce the set of functions Cm,�(0, b),m ∈ �, � ∈ �,−∞< � < 1,consisting of all functions u ∈ C [0, b] ∩ Cm(0, b) such that the estimation

|u(j)(x)| ≤ c

1 if j < 1 − �,

1 + | log �(x)| if j = 1 − �,

�(x)1−�−j if j > 1 − �

(2.1)

holds with a constant c = c(u) for all x ∈ (0, b) and j = 1, � � � ,m. Here

�(x) = min�x , b − x�

is the distance from x ∈ (0, b) to the boundary of the interval [0, b]. Clearly,Cm[0, b] ⊂ Cm,�(0, b),m ∈ �, a < 1. Notice also that if u ∈ Cm,�(0, b) and� < 1 − m, then the derivative u(m) is bounded on (0, b) and the derivativesup to order m − 1 can be extended so that u ∈ Cm−1[0, b].

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The regularity of a solution to (1.1) can be characterized by thefollowing lemma (see [27, 29, 35]).

Lemma 2.2. Let K ∈ � m,�(Db), f ∈ Cm,�(0, b),m ∈ �,−∞< � < 1, and let� (I − TK ) = �0�. Then equation (1.1) has a solution u ∈ Cm,�(0, b) which isunique in C [0, b].

The boundary singularities of the derivatives of a solution u to (1.1)allowed by the claim u ∈ Cm,�(0, b) are generic, they occur for majorityof f even if f ∈ C∞[0, b]. To improve the singular behavior of the exactsolution, we perform in equation (1.1) the change of variables (cf. [26, 28,38]) x = �(t), y = �(s), with � defined by formula

�(t) = cp

∫ t

0p−1(b − )p−1d, cp = b

[ ∫ b

0p−1(b − )p−1d

]−1

, p ∈ �,

(2.2)

where 0 ≤ t ≤ b. Clearly, �(0) = 0,�(b) = b and � is strictly increasing on[0, b] implying that � maps [0, b] onto [0, b] and there exists an inverse�−1 ∈ C [0, b] with �−1(0) = 0 and �−1(b) = b. Note that �(t) ≡ t for p = 1.We obtain an equation of the form

v(t) =∫ b

0K�(t , s)v(s)ds + f�(t), 0 ≤ t ≤ b, (2.3)

where K�(t , s) = K (�(t),�(s))�′(s), f�(t) = f (�(t)) and v(t) = u(�(t)) isunknown. It follows from (1.3) and (2.2) that K� ∈ � 0,�(Db), i.e. K�(t , s) iscontinuous for (t , s) ∈ Db and

|K�(t , s)| ≤ c

1 if � < 01 + | log |t − s|| if � = 0|t − s|−� if � > 0

, (t , s) ∈ Db � (2.4)

This together with � < 1 yields that TK� , the integral operator of (2.3), iscompact as an operator from C [0, b] into C [0, b]. Due to � (I − TK ) = �0�also � (I − TK�) = �0�. Now Lemma 2.1 implies that both equations (1.1)and (2.3) are uniquely solvable and their solutions are related by theequalities

u(x) = v(�−1(x)), v(t) = u(�(t))�

If p > 1, then it follows from Lemma 2.3 below, that � has a smoothingproperty for u(�(t)) with respect to the possible singularities of u(j)(x) (j =1, � � � ,m) at x = 0 and x = b. Also the possible boundary singularities of

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f (j)(x) (j = 1, � � � ,m) will be smoothed but not the diagonal singularity ofthe kernel K (x , y).

Lemma 2.3. Assume u ∈ Cm,�(0, b), m ∈ �,−∞< � < 1. Let v(t) =u(�(t)), 0 ≤ t ≤ b. with � defined by the formula (2.2) for an integer p ≥ 1.Then v ∈ Cm,�p (0, b) where �p = 1 − p(1 − �).

Proof. The proof is carried out by using the formula of Faa di Bruno fordifferentiating the composite function v = u ◦ � (cf. [26, 38]). �

Remark 2.4. Instead of (2.2) also other transformations are possible. Werefer to [26, 28] for a discussion in this connection.

3. QUADRATIC SPLINE INTERPOLATION

For given N ∈ � and r ∈ [1,∞) let

rN = �t0, � � � , t2N : 0 = t0 < · · · < t2N = b� (3.1)

be a partition of the interval [0, b] such that

tj = b2

(jN

)r

, j = 0, � � � ,N ; tN+j = b − tN−j , j = 1, � � � ,N � (3.2)

The parameter r characterizes the non-uniformity of the partition (grid)(3.1): if r = 1, then r

N is a uniform grid, if r > 1, then the grid pointst1, � � � , t2N−1 are more densely located near the boundary of the interval[0, b].

Let �2 be the set of all real algebraic polynomials of degree ≤2 and let

S (1)2 (r

N ) = �zN ∈ C 1[0, b] : zN |[tj−1,tj ] ∈ �2, j = 1, � � � , 2N �

be the linear space of C 1-smooth quadratic splines on the grid rN .

We introduce an interpolation operator PN : C [0, b] → C [0, b] (N ∈ �)which assigns to any function v ∈ C [0, b] its quadratic spline interpolantPN v ∈ S (1)

2 (rN ) satisfying

(PN v)(si) = v(si), i = 0, � � � , 2N + 1, (3.3)

with

s0 = t0 = 0, si = ti−1 + �(ti − ti−1), i = 1, � � � , 2N , s2N+1 = t2N = b,(3.4)

where � ∈ (0, 1) is a fixed real number not depending on N .

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It is easy to see that the operator PN is well defined, i.e., the functionPN v ∈ S (1)

2 (rN ) satisfying (3.3) is uniquely determined for any v ∈ C [0, b].

In what follows, for a Banach space E we denote by �(E) the Banachspace of linear bounded operators A : E → E with the norm ‖A‖�(E) =sup�‖Av‖E : v ∈ E , ‖v‖E = 1�.

Lemma 3.1 ([20]). Let the interpolation points (3.4) with � ∈ (0, 1) and gridpoints (3.2) of the grid r

N be used. Then

‖PN v − v‖∞ → 0 as N → ∞ for any v ∈ C [0, b]� (3.5)

A consequence of Lemma 3.1 is that

‖PN ‖�(C [0,b]) ≤ c , N ∈ �, (3.6)

with a constant c which is independent of N .

Lemma 3.2 ([22]). Let the interpolation points (3.4) with � ∈ (0, 1) and gridpoints (3.2) of the grid r

N , be used, and let v ∈ C 3,�(0, b), −∞< � < 1. Then

‖PN v − v‖∞ ≤ c�(�,r )N (3.7)

where c is a positive constant not depending on N and

�(�,r )N =

N −3 for � < −2, r ≥ 1,N −3(1 + logN ) for � = −2, r = 1,N −3 for � = −2, r > 1,N −r (1−�) for � > −2, 1 ≤ r < 3/(1 − �),N −3 for � > −2, r ≥ 3/(1 − �)�

(3.8)

Remark 3.3. Interpolation by quadratic splines is also studied, forexample, in [4, 16, 17, 19, 30].

4. NUMERICAL METHOD AND ERROR ANALYSIS

The approach proposed in this work for the numerical solution ofequation (1.1) can be described in the following two steps.

Step 1. We find an approximation vN for v, the solution to (2.3),determining vN = vN ,�,r from the conditions

vN ∈ S (1)2 (r

N ), (4.1)

vN (si) =∫ b

0K�(si , s)vN (s)ds + f�(si), i = 0, � � � , 2N + 1 (4.2)

with s0, � � � , s2N+1 given by (3.4).

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Step 2. We determine an approximation vN = vN ,�,r for u, thesolution to (1.1), setting

uN (x) = vN (�−1(x)), x ∈ [0, b]� (4.3)

Note that the conditions (4.1) and (4.2) determine a system ofequations whose exact form depends on the choice of a basis in the spaceS (1)2 (r

N ). A thorough study of polynomial spline spaces and their basescan be found in [4, 32, 40]. In particular, a basis for S (1)

2 (rN ) is given

by quadratic B -spline functions Bj : � → �, j = 0, � � � , 2N + 1, defined asfollows:

B0(t) ={(t1 − t)2/(t1 − t0)2 if t ∈ [t0, t1),0 if t � [t0, t1),

B1(t) =

(t − t0)(t1 − t)(t1 − t0)2

+ (t2 − t)(t − t0)(t2 − t0)(t1 − t0)

if t ∈ [t0, t1),(t2 − t)2

(t2 − t0)(t2 − t1)if t ∈ [t1, t2),

0 if t � [t0, t2),

Bj(t) =

(t − tj−2)2/((tj − tj−2)(tj−1 − tj−2)) if t ∈ [tj−2, tj−1),

(t − tj−2)(tj − t)(tj − tj−2)(tj − tj−1)

+ (tj+1 − t)(t − tj−1)

(tj+1 − tj−1)(tj − tj−1)if t ∈ [tj−1, tj),

(tj+1 − t)2/((tj+1 − tj−1)(tj+1 − tj)) if t ∈ [tj , tj+1),

0 if t � [tj−2, tj+1),

with j = 2, � � � , 2N − 1,

B2N (t) =

(t − t2N−2)2

(t2N − t2N−2)(t2N−1 − t2N−2)if t ∈ [t2N−2, t2N−1),

(t − t2N−2)(t2N−t)

(t2N − t2N−2)(t2N − t2N−1)if t ∈ [t2N−1, t2N ),

+ (t2N − t)(t − t2N−1)

(t2N − t2N−1)2

0 if t � [t2N−2, t2N ),

B2N+1(t) ={(t − t2N−1)

2/(t2N − t2N−1)2 if t ∈ [t2N−1, t2N ],

0 if t � [t2N−1, t2N ]�

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Here tj ∈ rN , j = 0, � � � , 2N . Using Bj , j = 0, � � � , 2N + 1, we can present

uN ∈ S (1)2 (r

N ) in the form

vN (t) =2N+1∑j=0

cjBj(t), t ∈ [0, b],

with some constants c0, � � � , c2N+1. The collocation conditions (4.2) nowlead to a linear system of equations with respect to the coefficients �cj�:

2N+1∑j=0

[Bj(si) −

∫ b

0K�(si , s)Bj(s)ds

]cj = f�(si), i = 0, � � � , 2N + 1�

Theorem 4.1. Let f ∈ C [0, b], K ∈ W 0,�(Db), −∞< � < 1, and let � (I −TK ) = �0�. Furthermore, assume that � is defined by the formula (2.2) and theinterpolation nodes (3.4) with � ∈ (0, 1) and grid points (3.2) of the grid r

N areused.

Then equation (1.1) has a unique solution u ∈ C [0, b] and equation (2.3)has a unique solution v ∈ C [0, b]. The settings (4.1), (4.2), and (4.3) determinefor sufficiently large N , say N ≥ N0, unique approximations vN and uN respectivelyfor v and u and

‖uN − u‖∞ = ‖vN − v‖∞ → 0 as N → ∞� (4.4)

Proof. It follows from Lemma 2.1 that equation (1.1) has a uniquesolution u ∈ C [0, b]. We write equation (2.3) in the form v = TK�v + f�,with f� ∈ C [0, b]. Since TK� is compact as an operator from C [0, b] intoC [0, b] and the corresponding homogeneous equation v = TK�v has inC [0, b] only the trivial solution v = 0, equation v = TK�v + f� has a uniquesolution v ∈ C [0, b]. Further, conditions (4.1) and (4.2) have the operatorequation representation

vN = PNTK�vN + PN f�, (4.5)

with PN defined in Section 3. Since TK� : C [0, b] → C [0, b] is compact, weobtain by Lemma 3.1 that ‖TK� − PNTK�‖�(C [0,b]) → 0 as N → ∞. Usingthis convergence and the invertibility of the operator I − TK� : C [0, b] →C [0, b], we get for N > N0 the invertibility of the operator I − PNTK� :C [0, b] → C [0, b], with

‖(I − PNTK�)−1‖�(C [0,b]) ≤ c , N ≥ N0, (4.6)

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where c is a positive constant not depending on N . Therefore, equation(4.5) has a unique solution vN ∈ S (1)

2 (rN ) ⊂ C [0, b] and, since vN − v =

(I − PNTK�)−1(PN v − v),

‖vN − v‖∞ ≤ c‖PN v − v‖∞, N ≥ N0� (4.7)

Due to v(t) = u(�(t)) and (4.3)

max0≤x≤b

|uN (x) − u(x)| = max0≤t≤b

|vN (t) − v(t)|� (4.8)

This together with (3.5) yields the convergence (4.4).

Theorem 4.2. Let K ∈ W 3,�(Db), f ∈ C 3,�(0, b), −∞< � < 1, and let� (I − TK ) = �0�. Assume that � is defined by the formula (2.2) for an integerp ≥ 1 and the interpolation nodes (3.4) with � ∈ (0, 1) and grid points (3.2) ofthe grid r

N are used.Then, in the notation of Theorem 4.1,

‖uN − u‖∞ = ‖vN − v‖∞

≤ c

N −pr (1−�) for 1 ≤ pr <3

1 − �,

N −3(1 + logN ) for pr = 31 − �

, r = 1,

N −3 for pr = 31 − �

, r > 1,

N −3 for pr >3

1 − �,

(4.9)

where N ≥ N0 and c is a positive constant not depending on N .

Proof. It follows from Theorem 4.1 that method (4.1)–(4.3) determinesfor N ≥ N0 a unique approximation vN for v, the solution to (2.3), anda unique approximation uN for u, the solution to (1.1). On the basis ofLemmas 2.2 and 2.3 we find that v ∈ Cm,�p (0, b), where �p = 1 − p(1 − �).Therefore. by Lemma 3.2,

‖PN v − v‖∞ ≤ c�(�p ,r )N

where c is a positive constant not depending on N and �(�,r )N is defined by

formula (3.8). This together with (4.7) and (4.8) yields (4.9).

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5. LOCAL SUPERCONVERGENCE

Theorem 4.2 proposes, in particular, how p and r should be chosen toachieve the highest possible convergence order ‖uN − u‖∞ = ‖vN − v‖∞ ≤cN −3 by using quadratic splines vN ∈ S (1)

2 (rN ). Nevertheless, as shown in

Theorem 5.1 below, the attainable order of local convergence may behigher than �(N −3) for a special choice of parameters p, r , and �.

Theorem 5.1. Let K ∈ W 4,�(Db), f ∈ C 4,�(0, b), −∞< � < 1, and let � (I −TK ) = �0�. Let � be given by the formula (2.2) for p > 4/(1 − �). Finally, assumethat the interpolation nodes (3.4) with � = 1/2 and grid points (3.2) of theuniform grid 1

N are used.Then, for N ≥ N0, in the notation of Theorem 4.1,

maxi=0,��,2N+1

|uN (�(si)) − u(�(si))| = maxi=0,��,2N+1

|vN (si) − v(si)| ≤ cN −3(�)N (5.1)

where c is a constant not depending on N and

(�)N =

N −1 for � < 0,N −1(1 + logN ) for � = 0,N −(1−�) for � > 0�

Proof. We know from the proof of Theorem 4.1 that equation (4.5) hasa unique solution vN for N ≥ N0. We have for it and for v, the solution ofv = TK�v + f�, that

(I − PNTK�)(vN − PN v) = PNTK�(PN v − v), N ≥ N0� (5.2)

As I − PNTK� is invertible in C [0, b] for N ≥ N0, we obtain from (3.6),(4.6), and (5.2) the estimate

‖vN − PN v‖∞ ≤ c‖TK�(PN v − v)‖∞, N ≥ N0� (5.3)

It follows from Lemmas 2.2 and 2.3 that v ∈ C 4,�p (0, b), �p = 1 − p(1 − �).Therefore v ∈ C 3[0, b] and by Lemma 3.2,

‖PN v − v‖∞ ≤ cN −3, N ≥ N0� (5.4)

Fix t ∈ [0, b] and let

h = b2N

,

U (t , h) = (t − h, t + h) ∩ [0, b],J1(t) = �j ∈ �1, � � � , 2N � : [tj−1, tj ] ∩ U (t , h) = ∅�,J2(t) = �j ∈ �1, � � � , 2N � : [tj−1, tj ] ∩ U (t , h) = ∅�,

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With the help of (5.4) we obtain for N ≥ N0 that

max0≤i≤2N+1

|vN (si) − v(si)| ≤ ‖vN − PN v‖∞ ≤ c‖TK�(PN v − v)‖∞

≤ c1

(h3

∑j∈J1(t)

∫ tj

tj−1

|K�(t , s)|ds

+∣∣∣∣ ∑j∈J2(t)

∫ tj

tj−1

|K�(t , s)|[(PN v)(s) − v(s)]ds∣∣∣∣),

(5.5)

with some positive constants c and c1 not depending on N . Due to (2.4),∑j∈J1(t)

∫ tj

tj−1

|K�(t , s)|ds

≤ c2

∫ t+2h

t−2h

1 if � < 01 + | log |t − s|| if � = 0|t − s|−� if � > 0

ds ≤ c3

(�)N (5.6)

where c2 and c3 are some positive constants not depending on N .Further, on every interval [tj−1, tj ] (j = 1, � � � , 2N ) we can present PN v

in the form (see, e.g., [17])

(PN v)(s) = v(sj) +[h8

− (tj − s)2

2h

]mj−1 +

[(s − tj−1)

2

2h− h

8

]mj ,

tj−1 ≤ s ≤ tj , j = 1, � � � , 2N , (5.7)

with

mj = (PN v)′(tj), j = 1, 2, � � � , 2N �

Taking into account the continuity of the function PN v on the interval[0, b] and the conditions

(PN v)(s0) = v(s0)

and

(PN v)(s2N+1) = v(s2N + 1),

we arrive at the following system of equations with respect to m0, � � � ,m2N :

3m0 + m1 = 8h

[v(s1) − v(s0)],mi−1 + 6mi + mi+1 = 8

h[v(si+1) − v(si)], i = 1, � � � , 2N − 1,

m2N−1 + 3m2N = 8h

[v(s2N+1) − v(s2N )],(5.8)

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Due to the diagonal dominance of the matrix of the system (5.8) thissystem is uniquely solvable. Next, we will find another representation forPN v (the argument is similar to that in [30]). Denote

j = mj − v ′(tj) + (h2/12)v ′′′(tj), j = 0, � � � , 2N , (5.9)

where m0, � � � ,m2N is the solution to (5.8). Using (5.8), (5.9) and a Taylorexpansion

v(l)(s) =3−l∑k=0

v(l+k)(ti)k! (s − ti)k + v(4)(�li)

(4 − l)!(s − ti)4−l ,

with �li ∈ (s, ti) (0 ≤ l ≤ 3) for finding v(si), v(si+1), v ′(ti−1), v ′(ti+1),v ′′′(ti−1) and v ′′′(ti+1), we obtain

3 0 + 1 = �0, i−1 + 6 i + i+1 = �i , i = 1, � � � , 2N − 1, 2N−1 + 3 2N = �2N

(5.10)

Here the numbers �0, � � � , �2N ∈ � are such that |�i | ≤ ch3, i = 0, � � � , 2N .This together with the diagonal dominance of the matrix of the system(5.10) yields that

maxj=0,��,2N

| i | ≤ 12

maxj=0,��,2N

|�j | ≤ ch3, (5.11)

with a constant c which is independent of h (of N ). Further, we have

v(sj) =3∑

k=0

v(k)(s)k! (sj − s)k + v(4)(�j)

4! (sj − s)4, �j ∈ (sj , s),

v ′(tj) =2∑

k=0

v(k+1)(s)k! (tj − s)k + v(4)(�′

j)

3! (tj − s)3, �′j ∈ (tj , s),

v ′′′(tj) = v ′′′(s) + v(4)(�′′′j )(tj − s), �′′′ ∈ (tj , s)�

Using these expressions we obtain with the help of (5.7), (5.9), and (5.11)the following representation for PN v:

(PN v)(s) = v(s) + v ′′′(s)[(sj − s)3

6+ h2

24(s − sj)

]+ O(h4),

tj−1 ≤ s ≤ tj , j = 1, � � � , 2N �

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Therefore, by (2.4),

∣∣∣∣ ∑j∈J2(t)

∫ tj

tj−1

K�(t , s)[(PN v)(s) − v(s)]ds∣∣∣∣

≤ ch3(�)N +∣∣∣∣ ∑j∈J2(t)

∫ tj

tj−1

K�(t , s)v ′′′(s)[(sj − s)3

6+ h2

24(s − sj)

]ds

∣∣∣∣, (5.12)

where N ≥ N0. Let us estimate the second term on the right hand side of

the inequality (5.12). We have

K�(t , s)v ′′′(s) = K�(t , sj)v ′′′(sj) + �K�(t , s)�s

∣∣∣∣s=�j

v ′′′(�j)(s − sj)

+ K�(t , �j)v(4)(�j)(s − sj), (5.13)

with �j ∈ (s, sj), tj−1 ≤ s ≤ tj , j ∈ J2(t). Here

�K�(t , s)�s

∣∣∣∣s=�j

= �

�yK (�(t), y)

∣∣∣∣y=�(�j )

[�′(�j)]2 + K (�(t),�(�j))�′′(�j)

Using the identity ��s = (

��t + �

�s

) − ��t we obtain from (1.2) that

∣∣∣∣ ��y K (x , y)∣∣∣∣ ≤ c

1 if � + 1 < 01 + | log |x − y|| if � + 1 = 0|x − y|−�−1 if � + 1 > 0

, (x , y) ∈ Db � (5.14)

On the basis of (2.2) we get (cf. [26, 28]) that

|�(t) − �(s)| ≥ c0|t − s|[t p−1 + sp−1], t , s ∈[0,

b2

],

|�(t) − �(s)| ≥ c0|t − s|[(b − t)p−1 + (b − s)p−1], t , s ∈[b2, b

],

with a constant c0 > 0 which is independent of t , s ∈ [0, b]. An observation

shows that

c1 ≤ |t − �j ||t − s| ≤ c2, �j ∈ (s, sj), s ∈ [tj−1, tj ], j ∈ J2(t),

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with some constants c2 ≥ c1 > 0 which are independent of t , s and �j . Thistogether with (1.2) and (5.14) yields that

|K (�(t),�(�j))�′′(�j)| ≤ c

1 if � < 0,

1 + | log |t − s|| if � = 0,

|t − s|−� if � > 0

(5.15)

and

∣∣∣∣ ��y K (�(t), y)∣∣∣∣y=�(�j )

[�′(�j)]2∣∣∣∣ ≤ c

1 if � + 1 < 0,

1 + | log |t − s|| if � + 1 = 0,

|t − s|−�−1 if � + 1 > 0,

(5.16)

where �j ∈ (s, sj), s ∈ [tj−1, tj ], j ∈ J2(t).Indeed, let us consider the case

t ∈ [0, b/2], s ∈ [tj−1, tj ] ⊂ [0, b/2], �j ∈ (s, sj), j ∈ J2(t)� (5.17)

Then we obtain (5.15) and (5.16) by the following steps:

|K (�(t),�(�j))|�′′(�j)

≤ c

1 if � < 01 + | log |�(t) − �(�j)|| if � = 0|�(t) − �(�j)|−� if � > 0

�

p−2j

≤ c1

�p−2j if � < 0

(1 + | log |t − �j || + | log(t p−1 + �p−1j )|)�p−2

j if � = 0

|t − �j |−�(t p−1 + �p−1j )−��

p−2j if � > 0

≤ c2

1 if � < 0

1 + | log |t − �j || + �p−2j | log �j | if � = 0

|t − �j |−��p(1−�)+�−2j if � > 0

≤ c3

1 if � < 01 + | log |t − �j || if � = 0|t − �j |−� if � > 0

≤ c4

1 if � < 01 + | log |t − s|| if � = 0|t − s|−� if � > 0

;

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1062 R. Pallav and A. Pedas∣∣∣∣ ��y K (�(t), y)∣∣∣∣y=�(�j )

[�′(�j)]2∣∣∣∣

≤ c1

1 if � + 1 < 0

(1 + | log |t − �j || + | log(t p−1 + �p−1j )|)�2(p−1)

j if � + 1 = 0

|t − �j |−�−1(t p−1 + �p−1j )−�−1�

2(p−1)j if � + 1 > 0

≤ c2

1 if � + 1 < 01 + | log |t − �j || if � + 1 = 0|t − �j |−�−1 if � + 1 > 0

≤ c3

1 if � + 1 < 01 + | log |t − s|| if � + 1 = 0|t − s|−�−1 if � + 1 > 0

Thus, (5.15) and (5.16) hold for the case (5.17). The proof of (5.15) and(5.16) for other possible values of t ∈ [0, b], s ∈ [tj−1, tj ] ⊂ [0, b], �j ∈ (s, sj),j ∈ J2(t) is similar.

Since v ∈ C 3[0, b] ∩ C 4(0, b), |v4(s) ≤ c , 0 < s < b, and

∫ tj

tj−1

[(sj − s)3

6+ h2

24(s − sj)

]ds = 0, j ∈ J2(t),

we obtain with the help of (5.13), (5.15), and (5.16) that

∣∣∣∣ ∑j∈J2(t)

∫ tj

tj−1

K�(t , s)v ′′′(s)[(sj − s)3

6+ h2

24(s − sj)

]ds

∣∣∣∣

≤ ch4∑j∈J2(t)

∫ tj

tj−1

1 if � + 1 < 01 + | log |t − s|| if � + 1 = 0|t − s|−�−1 if � + 1 > 0

ds

≤ c1h3(�)N , N ≥ N0,

with some positive constants c and c1 which are independent of h (of N ).This together with (5.5), (5.6), and (5.12) yields (5.1).

ACKNOWLEDGMENTS

This work has been supported by Estonian Science Foundationresearch grant No 7353. We are grateful to the referees for their remarks.

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Quadratic Spline Collocation for the Smoothed Weakly Singular Fredholm Integral Equations