About some numerical approaches for mixed ... - Semantic Scholar
Applied Mathematics and Computation 219 (2012) 464–474
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About some numerical approaches for mixed integral equations Franca Caliò a,⇑, A.I. Garralda-Guillem b, E. Marchetti a, M. Ruiz Galán b a b
Politecnico di Milano-Dipartimento di Matematica, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Universidad de Granada-E.T.S. Ingeniería de Edificación, Departamento de Matemática Aplicada, c/Severo Ochoa s/n, 18071 Granada, Spain
a r t i c l e
i n f o
Keywords: Volterra–Fredholm integral equations Collocation methods Fixed point methods Spline Schauder bases
a b s t r a c t In this paper we tackle on mixed Volterra–Fredholm integral equations, as in linear as in non linear cases. To the aim to obtain numerical solutions of these models, the authors propose in the linear case the direct collocation method using a p-order quasi interpolating spline class and in the nonlinear case the fixed point method based on polynomial approximation built by Schauder tensor bases. The advantages of both methods are outlined and their convergence is studied. Numerical results confirm the theoretical statements. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Integral Volterra–Fredholm equations of first and second kind are models for many real problems in different areas of science as mathematical physics, biology, economy and mechanics. For example, particular models describe the development of an epidemic or contact problems in mechanics of continuous. In particular mixed Volterra–Fredholm models are related to spatio/temporal situations, where the Fredholm component is considered in the space, while the integral Volterra component is considered in the time. Many references can be found in [1]. In this work we consider the following mixed Volterra–Fredholm integral models:
kuðxÞ ¼ f ðxÞ þ
Z a
x
k1 ðx; y; uðyÞÞdy þ
Z
b
k2 ðx; y; uðyÞÞdy;
x 2 ½a; b;
ð1Þ
a
where k 2 Rnf0g; f : ½a; b ! R; k1 ; k2 : ½a; b ½a; b R ! R and u : ½a; b ! R, with f and ki ði ¼ 1; 2Þ known functions and u an unknown function satisfying (1). To be more precise, let f : ½a; b ! R be a continuous function and let k1 ; k2 : ½a; b ½a; b R ! R be continuous functions satisfying a global Lipschitz condition at their third variables, i.e., there exist M 1 ; M 2 > 0 such that
9 x; y 2 ½a; b > =
v; w 2 R j ¼ 1; 2
> ;
) jkj ðx; y; v Þ kj ðx; y; wÞj 6 M j jv wj
ð2Þ
and jkj > ðM 1 þ M 2 Þðb aÞ. Our aim is to provide numerical methods which approximate a continuous function u : ½a; b ! R solution of (1). The linear case of (1),
⇑ Corresponding author. E-mail addresses: [email protected] (F. Caliò), [email protected] (A.I. Garralda-Guillem), [email protected] (E. Marchetti), [email protected] (M. Ruiz Galán). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.06.013
F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
kuðxÞ ¼ f ðxÞ þ
Z
x
h1 ðx; yÞuðyÞdy þ a
Z
465
b
h2 ðx; yÞuðyÞdy;
x 2 ½a; b;
ð3Þ
a
has been numerically dealt in a recent work [2]. Specifically in [2], taking into account the peculiarity of the mixed integral equation, we tried to afford the solution of (3) through a direct as well as an iterative numerical method, respectively: collocation and fixed point method. Both kinds of methods were proposed for a particular class of approximating functions. Namely, the first is a collocation method based on a linear spline class approximation; the second one is a fixed point method built on Schauder piecewise linear bases. In [2] we have also outlined pro and con of both methods and some open problems. One of them experimentally arises: by increasing in both methods respectively the evaluation and iteration numbers, the result’s precision does not improve. Surely predictable conditioning problems of the collocation method and algorithmic instability related to fixed point method appear. Furthermore, in [2] we underlined that the linearity of the model is a restriction to our proposals. In this work we tackle on above mentioned problems using suitable modification and generalization of two methods in [2]: (i) To overcome the first problem we propose for the model (3) a particular collocation method based on p-order (p > 2) quasi interpolating (q.i.) spline approximation. The method allows to improve the numerical solution precision while maintaining a low dimension of collocation system. (ii) To effectively face the nonlinear problem we propose an iterative method of fixed point based on tensor Schauder bases. The paper is organized as it follows: the first Subsection of Section 2 summarizes the collocation method described in [2] and analyses the related conditioning number, the second Subsection of Section 2 describes and analyses the collocation method based on the q.i. spline of order p=q2; Section 3 states the fixed point method related to Schauder bases for the nonlinear case; finally, Section 4 presents some numerical results for both methods to confirm the theoretical statements. 2. P-order spline direct method Let us consider the linear model (3). 2.1. Conditioning number related to spline linear collocation method In this section we summarize the genesis and peculiarity of the direct method in [2] built on linear B-spline approximation, the so called Variation-Diminishing Schoenberg (VDS) splines and we analyse the related conditioning problem. Firstly, let us recall some background on linear VDS splines (see for example [2]). Let n > 1 and Pn :¼ fa ¼ t 1 ¼ t 2 < t 3 < . . . < t n < t nþ1 ¼ t nþ2 ¼ bg be a partition of the interval [a,b] with Hn :¼ max26j6n ðtjþ1 t j Þ; Hn ! 0 as n ! 1. Pn is assumed as mesh of the set of normalized B-splines Bi;p ði ¼ 1; 2; . . . ; nÞ of order p ¼ 2 defined by the following relation:
Bi;2 ðxÞ ¼
Bi;1 ðxÞ ¼
x ti tiþ2 x Bi;1 ðxÞ þ Biþ1;1 ðxÞ; t iþ2 t iþ1 tiþ1 ti
ð4Þ
1; ti 6 x < t iþ1 ; 0; otherwise
according that, for i ¼ 1 and i ¼ n, the second and the first term in (4) hold respectively. The set of Bi;2 ðxÞ ði ¼ 1; 2; . . . ; nÞ can be considered as basis of S2;Pn (order 2 spline space, associated to Pn ). In particular, the VDS splines S n g 2 S2;Pn , with g 2 C½a; b, are defined as:
S n g :¼
n X gðt iþ1 ÞBi;2 ðxÞ:
ð5Þ
i¼1
S n is a projector operator that reproduces exactly a polynomial of first degree (S n P ¼ P; P 2 P 2 ) and also all linear spline functions (S n S ¼ S; S 2 S2;Pn ) [3]. Rewriting (3) as
e ¼ f; ½kI þ Ku where: -I is the identity operator in C½a; b e is the following bounded and compact operator: -K
e ¼K e 1 g þ K2 g: Kg
ð6Þ
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e 1 and K2 are defined as follows: -K
Z
e 1 gðxÞ :¼ K K2 gðxÞ :¼
Z
b
e 1 ðx; yÞgðyÞdy; h
x 2 ½a; b;
h2 ðx; yÞgðyÞdy;
x 2 ½a; b
a b a
and
e h 1 ðx; sÞ :¼
h1 ðx; sÞ; if 0; if
a6s6x ; s>x
then the collocation system on a set of distinct collocation points nk ðk ¼ 1; 2; . . . ; nÞ, chosen in ða; bÞ, is the following one
e n ðnk Þ ¼ f ðnk Þ; ½kI þ Kw
k ¼ 1; 2; . . . ; n
ð7Þ
being
wn ðxÞ ¼
n X i Bi;2 ðxÞ u i¼1
i are the approximated values of function u in t iþ1 ði ¼ 1; 2; . . . ; nÞ. and the u Using the norm of operator as in [5] it follows: Theorem 1. Denoting by A the coefficients matrix in system (7) it follows that
nþ1 e 1 jjðk þ jj KjjÞ: e jjðkI þ KÞ condðAÞ 6 1 þ 2 2
ð8Þ
Proof. System (7) can be rewritten as
e n ¼ Sn f : S n ðkI þ KÞw Taking into account the S n operator peculiarities and easily adapting some results in [5], the following relation holds:
e 1 jjC1 jj; jjA1 jj 6 jjS n jjðkI þ S n KÞ n
where Cn is the following matrix:
n
Cn ¼ Bi;2 ðnk Þ
i;k¼1
with nk belonging to the interval ða; bÞ. e ! ðkI þ KÞ, e Since for n sufficiently large ðkI þ S n KÞ the original problem and its approximating equation are approximately the same in their conditioning. So it is significant to analyse the behavior of jjS n jj and jjC1 n jj. Precisely it follows that jjS n jj ¼ 1 and assuming for a given matrix M 2 Rnn the row norm so defined n X kMkt :¼ max jmi;j j; 16i6n
j¼1
it can be easily shown that
jjCn jj ¼ 1;
jjC1 n jj ¼ 1 þ 2
nþ1 : 2
Moreover it follows that
e jjAjj 6 ðk þ jj KjjÞ; where
e :¼ max jj Kjj a6t6b
Z a
b
e ð h 1 ðt; sÞ þ h2 ðt; sÞÞds:
So it follows that
nþ1 e 1 jjðk þ jj KjjÞ: e jjðkI þ KÞ condðAÞ 6 jjA1 jjjjAjj 6 1 þ 2 j 2 Consequently, an obvious dependence of the conditioning number on system dimension arises.
F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
467
2.2. Direct model spline cubic approximation Taking into account (8) and experimental results in [2], we intend to improve the method’s precision with lower dimension of collocation system. Hereafter we present the numerical model suitable for (3) based on the collocation method using approximating splines, in particular the so called modified q.i. splines of order p > 2 (see for example [8,10]). In the following subsection we recall the definitions and peculiarities of p-order q.i. spline bases. 2.2.1. About p-order q.i. B-spline Let X m :¼ fx0 ¼ a < x1 < . . . < xm < xmþ1 ¼ bg be a partition of the interval ½a; b with Hm :¼ max06j6m ðxjþ1 xj Þ; Hm ! 0 as m ! 1 and let fdj : j ¼ 0; 1; . . . ; m þ 1g be a vector of positive integers where d0 ¼ dmþ1 ¼ p and dj 6 p 1; j ¼ 1; 2; . . . ; m. Pmþ1 We set n þ p :¼ j¼0 dj and define Pn ¼ fti : i ¼ 1; 2; . . . ; n þ pg as the nondecreasing sequence obtained from X m by repeating xj exactly dj times, j ¼ 0; 1; . . . ; m þ 1. The ti are assumed as nodes of the locally uniform spline space [3]: Sp;Pn :¼ fg : gjðxj ; xjþ1 Þ; 2 P p1 ; j ¼ 0; . . . ; m; (where P p1 is the set of polynomials of degree less than p) and ðiÞ g ðiÞ ðxþ j Þ ¼ g ðxj Þ; i ¼ 0; 1; . . . ; p dj 1; j ¼ 1; 2; . . . ; mg and the end points are pfold nodes. Thus Sp;Pn is the set of polynomials splines of order p with nodes at xj ðj ¼ 0; 1; . . . ; m þ 1Þ of multiplicity dj 6 p 1 ðj ¼ 1; 2; . . . ; mÞ, consequently every spline in Sp;Pn is in C½a; b. The set of the normalized B-splines Bi;p ði ¼ 1; 2; . . . ; nÞ of order p defined by the following recurrence relation:
x ti tiþp x Bi;p1 ðxÞ þ Biþ1;p1 ðxÞ; tiþp t iþ1 tiþp1 t i
Bi;p ðxÞ ¼
Bi;1 ðxÞ ¼
ð9Þ
1; ti 6 x < t iþ1 ; 0;
ð10Þ
otherwise
is considered as a basis for the spline space Sp;Pn . Let dj ¼ 1 and T be a set of sij ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; pÞ q.i. points belonging for each i ¼ 1; 2; . . . ; n to a subset of ½t i ; tiþp and such that sij – sih for j – h. The q.i. spline operator S n g 2 Sp;Pn , with g 2 C½a; b, is defined as
S n gðxÞ :¼
p n X X Bi;p ðxÞ v ij gðsij Þ;
ð11Þ
j¼1
i¼1
where
v ij :¼
p X
ail
Ql ; ðs sis Þ l¼j s ¼ 1 ij s–j
aij :¼
j X ðk 1Þ!ðp kÞ! ci;k1 di;jk ; ð1Þjk ðp 1Þ! k¼1
ð12Þ
with ci;k1 ¼ symmk1 ðtiþ1 ; . . . ; tiþp1 Þ; di;jk ¼ symmjk ðsi1 ; . . . ; si;j1 Þ (see [3]). S n is a projector operator [6] that reproduces exactly a polynomial of p 1 degree and also all p order spline functions, that is,
S n P ¼ P;
P 2 Pp;
S n S ¼ S;
S 2 Sp;Pn :
ð13Þ
e ¼ KS e n g with g 2 C½a; b, it follows that Moreover (see for example [7]), as we assume S n Kg
e S n Kjj e ! 0 as n ! 1: jj K
ð14Þ
2.2.2. Collocation method Applying to (6) the collocation method on a set of distinct collocation points nk ðk ¼ 1; 2; . . . ; NÞ in ða; bÞ, where N is the number of distinct elements of the set T, and using q.i. spline approximation, we obtain (7), where in this case
wn ðxÞ ¼
n X
p X i Bi;p ðxÞ v ij u
i¼1
ð15Þ
j¼1
i are the approximated values of function u in sij 2 T ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; pÞ. and the u A choice of sij , so that (13) and (14) are satisfied, is proposed in [7]. The optimal algorithmically choice (see [9]) of q.i. points set:
( T :¼ where gi :¼
)
si1 :¼ g1 ; i ¼ 1; . . . ; n si2 :¼ gi1 ; si3 :¼ giþ1 ; . . . ; sip :¼ gið1Þl ½2p ; i ¼ ½2p þ 1; . . . ; n ½2p ;
t iþ1 þ...þt iþp1 p1
ð16Þ
; i ¼ 1; 2; . . . ; n, with a suitable choice of the remaining nodes of T, also satisfies (13) and (14) (see [6]).
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2.2.3. Remarks about the convergence It is possible to generalize to the case p > 2 the remarks in [2] about the convergence. To be more precise, applying the bounded projector operator 11 to 6 and to (7) with wn as in (15) we obtain
e ¼ Sn f ; S n ðkI þ KÞu e n ¼ Sn f S n ðkI þ KÞw and consequently
e ðkI þ S n KÞðu wn Þ ¼ kðI S n Þu:
ð17Þ
e S n Kjj e ! 0 as n ! 1, and following the proof of Theorem 1 in [4], we can state from (17) that As jj K
e Þ1 jj 6 M < 1; supnPN jjðkI þ S n K for n sufficiently large. It follows that jju wn jj ! 0, as n ! 1, exactly with the same rate of convergence as jju S n ujj does. 3. Schauder bases iterative method In this section, we consider the nonlinear integral equation (1). As the notations in the previous sections to this more general nonlinear framework, we write K1 and K2 for the operators on C½a; b
K1 v ðxÞ :¼
Z
x
k1 ðx; y; v ðyÞÞdy;
v 2 C½a; b;
x 2 ½a; b
k2 ðx; y; v ðyÞÞdy;
v 2 C½a; b;
x 2 ½a; b;
a
and
K2 v ðxÞ :¼
Z
b a
respectively, and K :¼ K1 þ K2 . For a compact and Hausdorff topological space X we denote by CðXÞ the Banach space of those real valued functions defined on X and endowed with its maximum norm. Then, it is clear that u 2 C½a; b is a solution of the nonlinear integral Eq. (1) if, and only if, it is a fixed point of the operator N : C½a; b ! C½a; b given, for each v 2 C½a; b, by
N v :¼
1 ðf Kv Þ: k
As a straightforward consequence of the Banach fixed point theorem, if
q :¼
ðM 1 þ M 2 Þðb aÞ < 1; jkj
where M 1 ; M 2 are the Lipschitz constants in (2), then N has one and only one fixed point u, and for each m P 1 we have that
jjN
m
v ujj 6
qm jjN v v jj 1q
v 2 C½a; b and ð18Þ
and thus,
limjjN n
n
v ujj ¼ 0: n
Easy examples show that it is not always possible to calculate the sequence of iterations fN v gnP1 , and for this very reason, a numerical method is needed in order to approximate the fixed point of N . We develop such an iterative method by means of the use of Schauder bases in an adequate Banach space associated in a natural way with the nonlinear mixed integral Eq. (1). 3.1. Elementary facts on Schauder bases The proposed method is based on the use of Schauder bases and extends that developed in [2] for the linear mixed Volterra–Fredholm integral equation: on the one hand, the linear case satisfies clearly the Lispschitz conditions (2); on the other hand, in [2] a concrete Schauder basis is considered, unlike our analysis, which works for any Schauder basis. In fact, the basis in [2] is univariate, what restricts its application to a specific kind of equation. Let us also point out that Schauder bases have been successfully used in the numerical treatment of integral, integro–differential or differential equations (see [11–14]). Let us recall that a Schauder basis in a Banach space E is a sequence fvn gnP1 in E such that any x 2 E has a unique representation of the form
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F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
n X
lim x tk vk ¼ 0; n
k¼1 where ftn gnP1 is a sequence of real numbers. Given m P 1, the m–th biorthogonal functional m ðxÞ
v
vm : E ! R is defined as
:¼ tm ;
where x 2 E and limn jjx
Q m ðxÞ :¼
Pn
k¼1 t k
vk jj ¼ 0, and the mth projection is the finite–rank linear operator on E given by
m X
vk ðxÞvk ; x 2 E:
k¼1
It follows from the Baire category theorem [15, Theorem 2.1] that for all m P 1 the biorthogonal functional projection Q m are continuous, and from the definition of these functionals and projections, that for all x 2 E
limjjQ n ðxÞ xjj ¼ 0:
vm and the ð19Þ
n
3.2. The iterative numerical method Let us now introduce our numerical method for approximating the unique solution of the mixed nonlinear integral Eq. (1). Our approach combines an iterative scheme with some facts related to an arbitrary Schauder basis in the Banach space C½a; b2 . We describe in a concise way the proposed numerical method. We start from three data: a Schauder basis fvn gnP1 in C½a; b2 , with sequence of associated projections fQ n gnP1 , a sequence of positive integers fjn gnP1 and a function v 2 C½a; b. We define recursively the sequence of approximate functions:
AF0 ðxÞ :¼ v ðxÞ;
x 2 ½a; b
ð20Þ
and if m P 1
AFm ðxÞ :¼
1 k
Z a
1 f ðxÞ; k
x
ð1Þ
Q jm ð m1 ðx; yÞÞdy þ
Z a
!
b
ð2Þ
Q jm ð m1 ðx; yÞÞdy ;
x 2 ½a; b;
ð21Þ
where for j ¼ 1; 2, ðjÞ
m1 ðx; yÞ :¼ kj ðx; y; AFm1 ðyÞÞ;
x; y 2 ½a; b:
ð22Þ
By the triangle inequality we have that
jjAFm ujj 6 jjAFm N
m
v jj þ jjN m v ujj
and since (18) gives an upper bound for the second summand in the right–hand side, it remains to bound the first one in m order to control the error jjAF m ujj. With this in mind, we deal with estimating the norm jjAF m N v jj. Lemma 2. Let k – 0; f 2 C½a; b and let k1 ; k2 2 C½a; b2 satisfying the Lipschitz conditions (2), with Lipschitz constants M 1 and M 2 , respectively, and let
q :¼ ðM1 þ M2 Þðb aÞ=jkj: Assume in addition that v 2 C½a; b; fjn gnP1 is a sequence of positive integers and that fvn gnP1 is a Schauder basis in C½a; b2 whose sequence of associated projections is fQ n gnP1 . If for each m P 1; AF m is the approximate function defined by (20)–(22), then
jjAFm N
m
v jj 6
m ðb aÞ X ð1Þ ð1Þ ð2Þ ð2Þ jj k1 Q jk ð k1 Þjj þ jj k1 Q jk ð k1 Þjj qmk : jkj k¼1
Proof. We proceed inductively on m. For m ¼ 1, given x 2 ½a; b it is clear that
jAF1 ðxÞ N v ðxÞj 6 6
1 jkj
Z a
x
ð1Þ
ð1Þ
j 0 ðx; yÞ Q j1 ð 0 ðx; yÞÞjdy þ
1 jkj
Z a
b
ð2Þ
ð2Þ
j 0 ðx; yÞ Q j1 ð 0 ðx; yÞÞjdy
b a ð1Þ ð1Þ ð2Þ ð2Þ jj 0 Q j1 ð 0 Þjj þ jj 0 Q j1 ð 0 Þjj jkj
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F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
so,
jjAF1 N v jj 6
b a ð1Þ ð1Þ ð2Þ ð2Þ jj 0 Q j1 ð 0 Þjj þ jj 0 Q j1 ð 0 Þjj : jkj
Now, let the announced inequality holds for m 1. Then it also is valid for m, because taking into account the induction hypothesis as well as the fact that for all g; h 2 C½a; b; jjN g N hjj 6 qjjg hjj, we arrive, for all x 2 ½a; b, at m
m
jAFm ðxÞ N v ðxÞj 6 jAFm ðxÞ N AFm1 ðxÞj þ jjN AFm1 N v jj Z x Z b 1 1 m1 ð1Þ ð1Þ ð2Þ ð2Þ j m1 ðx; yÞ Q jm ð m1 ðx; yÞÞjdy þ j m1 ðx; yÞ Q jm ð m1 ðx; yÞÞjdy þ qjjAFm1 N v jj 6 jkj a jkj a ðb aÞ ð1Þ ð1Þ ð2Þ ð2Þ 6 jj k1 Q jm ð m1 Þjj þ jj m1 Q jm ð m1 Þjj jkj X ð1Þ ðb aÞ m1 ð1Þ ð2Þ ð2Þ þ q jj k1 Q jk ð k1 Þjj þ jj k1 Q jk ð k1 Þjj qm1k jkj k¼1 m ðb aÞ X ð1Þ ð1Þ ð2Þ ð2Þ ¼ jj k1 Q jk ð k1 Þjj þ jj k1 Q jk ð k1 Þjj qmk jkj k¼1 and thus the arbitrariness of x 2 ½a; b implies the announced inequality. h Let us observe that in the preceding lemma, the condition q < 1 is not required. Now we are in a position to guarantee that for a suitable choice of m P 1 and j1 ; . . . ; jm P 1; AFm is as closed as desired to the fixed point of the operator N : Theorem 3. Suppose that k – 0; f 2 C½a; b and that k1 ; k2 2 C½a; b2 satisfy the Lipschitz conditions (2), with respective Lipschitz constants M 1 and M 2 , and in such a way that
ðM 1 þ M 2 Þðb aÞ=jkj < 1: Let u be the unique solution of Eq. (1) and let fvn gnP1 be a Schauder basis in C½a; b2 with sequence of associated projections fQ n gnP1 . Then for each e > 0 and each v 2 C½a; b, there exist m P 1 and j1 ; . . . ; jm P 1 such that
jjAFm ujj < e; where AFm is the approximate function defined by (20)–(22). Proof. Since
q :¼ ðM1 þ M2 Þðb aÞ=jkj < 1; let m P 1 such that
qm 1q
jjN v v jj
2, we give some results related to the same test functions as in [2], comparing them in terms of precision of results, using the same number of elements of T. Precisely we propose two examples: (1) h1 ðx; sÞ ¼ h2 ðx; sÞ ¼ eðxþsÞ for Example 4. (2) h1 ðx; sÞ ¼ h2 ðx; sÞ ¼ sinðxÞ cosðsÞ for Example 5. For the collocation method we consider:
the mesh points as equispaced points in ½0; 1 with nodes of p-multiplicity in 0 and 1, the mesh points all simple in ð0; 1Þ, the q.i. points as (16), the number of the collocation points equal to Òthat of the distinct q.i. points. The algorithm is implemented in MATLAB 7:3 . In Tables 1–4 we compare for p ¼ 2 and p ¼ 4 the maximun absolute value of the errors related to the approximation of unknown function in a suitable subset of n points in ð0; 1Þ.
Example 1. Fork ¼ 15,uðxÞ ¼ x3 and
f ðxÞ ¼ 15x3 ex ð2e þ 12 þ ex ðx3 3x2 þ 6x 6ÞÞ;
x 2 ½0; 1;
the numerical results appears in Table 1. Analogously, Table 2 includes the numerical experiments when k ¼ 15,uðxÞ ¼ ex and
f ðxÞ ¼
1 x e ðe2 e2x þ 32Þ; 2
x 2 ½0; 1:
If we would obtain 1010 precision order with p ¼ 2, in the case of Table 2, the rounding error does not make reliable the solution of the collocation system, while, in the case p ¼ 4, we achieve the goal by n ¼ 101.
Table 1 Example 1 u(x) = x3. n
p¼2
p¼4
11
7.5 E3
2.5 E15
Table 2 Example 1 u(x) = ex. n
p¼2
p¼4
11 101
3.3 E3 3.2 E5
6.6 E5 8.0 E10
Table 3 Example 2 u(x) = x2. n
p¼2
p¼4
11
2.7 E3
5.5 E16
n
p¼2
p¼4
11 101 201
3.1 E3 3.1 E5 7.6 E6
6.6 E6 7.9 E10 5.0 E11
Table 4 Example 2 u(x) = ex.
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F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
Example 2. Now k ¼ 1, uðxÞ ¼ x2 and
f ðxÞ ¼ x2 sin xðx2 sin x þ 2x cos x 2 sin x sin 1 þ 2 cos 1Þ;
x 2 ½0; 1;
x
with the corresponding results in Table 3, and k ¼ 1, uðxÞ ¼ e and
f ðxÞ ¼ ex sin xð ex ðsin x þ cos xÞ þ eðsin 1 þ cos 1Þ 2Þ=2;
x 2 ½0; 1;
4.2. Numerical testing for the Schauder bases iterative method Let us consider the non linear case in (1). We include an example to ensure the efficiency and simplicity of the proposed method. We emphasize that the approximation results for the fixed point method in Section 3.2 are stated for an arbitrary Schauder basis in C½a; b2 . In order to obtain concrete numerical approximations of the solution of a nonlinear mixed Volterra–Fredholm integral equation, we shall fix a concrete bivariate Schauder basis. The Schauder basis considered here is that obtained as the bivariate basis derived from the usual Schauder basis in C½a; b, also known as the Faber-Schauder system. To be more precise, let us recall that given a sequence ft n gnP1 of distinct points in ½a; b such that t1 ¼ a and t2 ¼ b, the usual Schauder basis ffn gnP1 in C½a; b is defined by
f1 ðtÞ :¼ 1;
t 2 ½a; b;
while for n P 1; fn is the piecewise linear continuous function on ½a; b with nodes at ftk : 1 6 k 6 ng, given by
fn ðt n Þ ¼ 1 and for k < n
fn ðt k Þ ¼ 0: We denote by ffn gnP1 and fP n gnP1 its associated sequences of biorthogonal and projections, respectively. The usual Schauder basis fvn gnP1 in C½a; b2 , which is what we set in our numerical experiments, is the corresponding bivariate tensor basis of ffn gnP1 ([16] and [17]): if [] denotes ‘‘integer part’’ and r : N ! N N is the bijective mapping given by
pffiffiffi pffiffiffi 8 pffiffiffi pffiffiffi if ½ n ¼ n > < ð n; pnffiffiffiÞ; pffiffiffi pffiffiffi pffiffiffi rðnÞ :¼ ðn ½ n2 ; ½ n þ 1Þ; if 0 < n ½ n2 6 ½ n ; > p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi : ð½ n þ 1; n ½ n2 ½ nÞ; if ½ n < n ½ n2 then
vn ðt; sÞ :¼ fp ðtÞfq ðsÞ; t; s 2 ½a; b; whenever rðnÞ ¼ ðp; qÞ. fvn gnP1 and fQ n gnP1 stand for the respective sequences of biorthogonal functionals and projections. It is a well-known fact that the sequence fvn gnP1 is easily determined by finite linear combinations of evaluations at adequate known functions, and therefore so does fQ n gnP1 . More specifically, for each g 2 C½a; b2 ,
v1 ðgÞ ¼ gðt1 ; t1 Þ and for n P 2 and
rðnÞ ¼ ðp; qÞ,
vn ðgÞ ¼ gðtp ; tq Þ
n1 X
vk ðgÞvk ðtp ; tq Þ:
ð25Þ
k¼1
Now we describe the parameters of the numerical method. In what follows we assume that ½a; b ¼ ½0; 1, because such is the case in the concrete nonlinear example below. The subset ftn gnP1 chosen for constructing the usual Schauder basis in C½0; 12 is defined by
t0 ¼ 0;
t1 ¼ 1
and for n P 1,
tnþ1 ¼
2k þ 1 2lþ1
provided that n ¼ 2l þ k þ 1, where 0 6 k < 2l are positive integers. Once we have one of the data defining the approximating sequence fAFm gmP1 given by (20)–(22), we choose the other two: the initial function is
AF0 ¼
f jkj 2
and for all m P 1, we take jm ¼ j for simplicity in the implementation, where j is a given integer.
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F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474 Table 5 Example 3 u(x) = 1 x. x
j¼9
j ¼ 17
jAF 3 ðxÞ uðxÞj 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1
2.9 3.1 3.4 3.6 3.9 4.2 4.5 4.9 5.3
jAF 3 ðxÞ uðxÞj
E4 E4 E4 E4 E4 E4 E4 E4 E4
7.4 7.7 8.1 8.6 9.1 9.7 1.0 1.1 1.2
E5 E5 E5 E5 E5 E5 E4 E4 E4
We include in the following numerical Example 6, for j = 9 and j = 17, the absolute errors committed in certain representative points x 2 ½0; 1, when we approximate the exact solution u by the approximate function AFh where h ¼ 3. Example 3. Let us consider the nonlinear mixed integral equation, similar to that in [18, Example 2],
4 uðxÞ ¼ f ðxÞ þ
Z 0
x
sinðx yÞ cos uðyÞdy þ
Z
1
ðx yÞð1 þ uðyÞ2 Þdy;
x 2 ½0; 1;
0
where for x 2 ½0; 1,
f ðxÞ ¼
1 ð55 64x þ 6ðsin 1Þx cos x 6ðcos 1Þx sin x 6ðsin 1Þ sin xÞ: 12
Its exact solution is uðxÞ ¼ 1 x and the numerical results obtained from the iterative method are taken in Table 5 for the parameters previously commented. The algorithms associated with the numerical method have been performed using Mathematica 7. Let us emphasize that the choice for AF 0 seems to be irrelevant, since after some iterations its influence in the approximations disappears. Let us also point out that this iterative numerical method has the advantage of being very easy to implement. The approximating functions are the sum of a known function and integrals of piecewise bivariate polynomials of degree 2. As immediately follows from (25), the calculation of the coefficients of such polynomials just requires linear combinations of several evaluations of the basic functions at adequate points.
5. Final remarks In this work we presented two numerical methods for solving mixed Volterra–Fredholm integral problems. In particular we proposed, to tackle on the linear cases, a direct collocation method based on q.i. spline of order of precision p P q2. In this way a high degree of accuracy is easily ensured using collocation systems of minimum order, overcoming consequently rounding errors problems. In the nonlinear cases we suggested a fixed point method derived from suitable properties of a Schauder basis in the Banach space of continuous functions C½a; b2 , well suited to define and apply tensorial schemes, that in addition is very easily implemented. The tables of the numerical results confirm the theoretical statements. Acknowledgements The research was partially supported by Prin MURST 200834WK7H_005 for the first and third authors and by Junta de Andalucía Grant FQM359 for the second and fourth authors. References [1] B. Basirat, K. Maleknejad, E. Hashemizadeh, Operational matrix approach for the nonlinear Volterra-Fredholm integral equations: arising in physics and engineering, Int. J. Phys. Sci. 7 (2012) 226–233. [2] F. Caliò, M.V. Fernández Muñoz, E. Marchetti, Direct and iterative methods for the numerical solution of mixed integral equations, Appl. Math. Comput. 216 (2010) 3739–3746. [3] T. Lyche, L.L. Schumaker, Local spline approximation methods, J. Approx. Theory 15 (1975) 294–325. [4] C. Dagnino, V. Demichelis, E. Santi, A nodal spline collocation method for weakly singular Volterra integral equations, Babe ßs-Bolyai, Mathematica, XLVIII, 2003, pp. 71–81. [5] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997. [6] C. De Boor, G.J. Fix, Spline approximation by quasi interpolants, J. Approx. Theory 8 (1973) 19–45. [7] C. Dagnino, V. Demichelis, E. Santi, Numerical integration based on quasi interpolating spline, Computing 50 (1993) 146–163. [8] F. Caliò, E. Marchetti, An algorithm based on q.i. modified splines for singular integral models, Comput. Math. Appl. 41 (2001) 1579–1588.
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[9] V. Demichelis, Quasi interpolatory splines based on Schoenberg points, Math. Comput. 65 (1996) 1235–1247. [10] F. Caliò, E. Marchetti, P. Rabinowitz, On the numerical solution of the generalized Prandtl equation using variation-diminishing splines, J. Comput. Appl. Math 60 (1995) 297–307. [11] M.I. Berenguer, D. Gámez, A.I. Garralda-Guillem, M. Ruiz Galán, M.C. Serrano Pérez, Biorthogonal systems for solving Volterra integral equation systems of the second kind, J. Comput. Appl. Math. 235 (2011) 1875–1883. [12] M.I. Berenguer, M.A. Fortes, A.I. Garralda-Guillem, M. Ruiz Galán, Linear Volterra integro-differential equation and Schauder bases, Appl. Math. Comput. 159 (2004) 495–507. [13] M.I. Berenguer, A.I. Garralda-Guillem and M. Ruiz Galán, Biorthogonal systems approximating the solution of the nonlinear Volterra integro– differential equation, Fixed Point Theory Appl., vol. 2010, Article ID 470149, 2010, 9p. [14] D. Gámez, A.I. Garralda-Guillem, M. Ruiz Galán, High order nonlinear initial–value problems countably determined, J. Comput. Appl. Math. 228 (2009) 77–82. [15] H. Brezis, Functional Analysis, Spaces and Sobolev Partial Differential Equations, Universitext, Springer, New York, 2011. [16] B. Gelbaum, J. Gil de Lamadrid, Bases on tensor products of Banach spaces, Pacific J. Math. 11 (1961) 1281–1286. [17] Z. Semadeni, Product Schauder bases and approximation with nodes in spaces of continuous functions, Bull. Acad. Polon. Sci. 11 (1963) 387–391. [18] M. Cui, H. Du, Representation of exact solution for the nonlinear Volterra—Fredholm integral equations, Appl. Math. Comput. 182 (2006) 1795–1802.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
About some numerical approaches for mixed integral equations Franca Caliò a,⇑, A.I. Garralda-Guillem b, E. Marchetti a, M. Ruiz Galán b a b
Politecnico di Milano-Dipartimento di Matematica, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Universidad de Granada-E.T.S. Ingeniería de Edificación, Departamento de Matemática Aplicada, c/Severo Ochoa s/n, 18071 Granada, Spain
a r t i c l e
i n f o
Keywords: Volterra–Fredholm integral equations Collocation methods Fixed point methods Spline Schauder bases
a b s t r a c t In this paper we tackle on mixed Volterra–Fredholm integral equations, as in linear as in non linear cases. To the aim to obtain numerical solutions of these models, the authors propose in the linear case the direct collocation method using a p-order quasi interpolating spline class and in the nonlinear case the fixed point method based on polynomial approximation built by Schauder tensor bases. The advantages of both methods are outlined and their convergence is studied. Numerical results confirm the theoretical statements. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Integral Volterra–Fredholm equations of first and second kind are models for many real problems in different areas of science as mathematical physics, biology, economy and mechanics. For example, particular models describe the development of an epidemic or contact problems in mechanics of continuous. In particular mixed Volterra–Fredholm models are related to spatio/temporal situations, where the Fredholm component is considered in the space, while the integral Volterra component is considered in the time. Many references can be found in [1]. In this work we consider the following mixed Volterra–Fredholm integral models:
kuðxÞ ¼ f ðxÞ þ
Z a
x
k1 ðx; y; uðyÞÞdy þ
Z
b
k2 ðx; y; uðyÞÞdy;
x 2 ½a; b;
ð1Þ
a
where k 2 Rnf0g; f : ½a; b ! R; k1 ; k2 : ½a; b ½a; b R ! R and u : ½a; b ! R, with f and ki ði ¼ 1; 2Þ known functions and u an unknown function satisfying (1). To be more precise, let f : ½a; b ! R be a continuous function and let k1 ; k2 : ½a; b ½a; b R ! R be continuous functions satisfying a global Lipschitz condition at their third variables, i.e., there exist M 1 ; M 2 > 0 such that
9 x; y 2 ½a; b > =
v; w 2 R j ¼ 1; 2
> ;
) jkj ðx; y; v Þ kj ðx; y; wÞj 6 M j jv wj
ð2Þ
and jkj > ðM 1 þ M 2 Þðb aÞ. Our aim is to provide numerical methods which approximate a continuous function u : ½a; b ! R solution of (1). The linear case of (1),
⇑ Corresponding author. E-mail addresses: [email protected] (F. Caliò), [email protected] (A.I. Garralda-Guillem), [email protected] (E. Marchetti), [email protected] (M. Ruiz Galán). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.06.013
F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
kuðxÞ ¼ f ðxÞ þ
Z
x
h1 ðx; yÞuðyÞdy þ a
Z
465
b
h2 ðx; yÞuðyÞdy;
x 2 ½a; b;
ð3Þ
a
has been numerically dealt in a recent work [2]. Specifically in [2], taking into account the peculiarity of the mixed integral equation, we tried to afford the solution of (3) through a direct as well as an iterative numerical method, respectively: collocation and fixed point method. Both kinds of methods were proposed for a particular class of approximating functions. Namely, the first is a collocation method based on a linear spline class approximation; the second one is a fixed point method built on Schauder piecewise linear bases. In [2] we have also outlined pro and con of both methods and some open problems. One of them experimentally arises: by increasing in both methods respectively the evaluation and iteration numbers, the result’s precision does not improve. Surely predictable conditioning problems of the collocation method and algorithmic instability related to fixed point method appear. Furthermore, in [2] we underlined that the linearity of the model is a restriction to our proposals. In this work we tackle on above mentioned problems using suitable modification and generalization of two methods in [2]: (i) To overcome the first problem we propose for the model (3) a particular collocation method based on p-order (p > 2) quasi interpolating (q.i.) spline approximation. The method allows to improve the numerical solution precision while maintaining a low dimension of collocation system. (ii) To effectively face the nonlinear problem we propose an iterative method of fixed point based on tensor Schauder bases. The paper is organized as it follows: the first Subsection of Section 2 summarizes the collocation method described in [2] and analyses the related conditioning number, the second Subsection of Section 2 describes and analyses the collocation method based on the q.i. spline of order p=q2; Section 3 states the fixed point method related to Schauder bases for the nonlinear case; finally, Section 4 presents some numerical results for both methods to confirm the theoretical statements. 2. P-order spline direct method Let us consider the linear model (3). 2.1. Conditioning number related to spline linear collocation method In this section we summarize the genesis and peculiarity of the direct method in [2] built on linear B-spline approximation, the so called Variation-Diminishing Schoenberg (VDS) splines and we analyse the related conditioning problem. Firstly, let us recall some background on linear VDS splines (see for example [2]). Let n > 1 and Pn :¼ fa ¼ t 1 ¼ t 2 < t 3 < . . . < t n < t nþ1 ¼ t nþ2 ¼ bg be a partition of the interval [a,b] with Hn :¼ max26j6n ðtjþ1 t j Þ; Hn ! 0 as n ! 1. Pn is assumed as mesh of the set of normalized B-splines Bi;p ði ¼ 1; 2; . . . ; nÞ of order p ¼ 2 defined by the following relation:
Bi;2 ðxÞ ¼
Bi;1 ðxÞ ¼
x ti tiþ2 x Bi;1 ðxÞ þ Biþ1;1 ðxÞ; t iþ2 t iþ1 tiþ1 ti
ð4Þ
1; ti 6 x < t iþ1 ; 0; otherwise
according that, for i ¼ 1 and i ¼ n, the second and the first term in (4) hold respectively. The set of Bi;2 ðxÞ ði ¼ 1; 2; . . . ; nÞ can be considered as basis of S2;Pn (order 2 spline space, associated to Pn ). In particular, the VDS splines S n g 2 S2;Pn , with g 2 C½a; b, are defined as:
S n g :¼
n X gðt iþ1 ÞBi;2 ðxÞ:
ð5Þ
i¼1
S n is a projector operator that reproduces exactly a polynomial of first degree (S n P ¼ P; P 2 P 2 ) and also all linear spline functions (S n S ¼ S; S 2 S2;Pn ) [3]. Rewriting (3) as
e ¼ f; ½kI þ Ku where: -I is the identity operator in C½a; b e is the following bounded and compact operator: -K
e ¼K e 1 g þ K2 g: Kg
ð6Þ
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F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
e 1 and K2 are defined as follows: -K
Z
e 1 gðxÞ :¼ K K2 gðxÞ :¼
Z
b
e 1 ðx; yÞgðyÞdy; h
x 2 ½a; b;
h2 ðx; yÞgðyÞdy;
x 2 ½a; b
a b a
and
e h 1 ðx; sÞ :¼
h1 ðx; sÞ; if 0; if
a6s6x ; s>x
then the collocation system on a set of distinct collocation points nk ðk ¼ 1; 2; . . . ; nÞ, chosen in ða; bÞ, is the following one
e n ðnk Þ ¼ f ðnk Þ; ½kI þ Kw
k ¼ 1; 2; . . . ; n
ð7Þ
being
wn ðxÞ ¼
n X i Bi;2 ðxÞ u i¼1
i are the approximated values of function u in t iþ1 ði ¼ 1; 2; . . . ; nÞ. and the u Using the norm of operator as in [5] it follows: Theorem 1. Denoting by A the coefficients matrix in system (7) it follows that
nþ1 e 1 jjðk þ jj KjjÞ: e jjðkI þ KÞ condðAÞ 6 1 þ 2 2
ð8Þ
Proof. System (7) can be rewritten as
e n ¼ Sn f : S n ðkI þ KÞw Taking into account the S n operator peculiarities and easily adapting some results in [5], the following relation holds:
e 1 jjC1 jj; jjA1 jj 6 jjS n jjðkI þ S n KÞ n
where Cn is the following matrix:
n
Cn ¼ Bi;2 ðnk Þ
i;k¼1
with nk belonging to the interval ða; bÞ. e ! ðkI þ KÞ, e Since for n sufficiently large ðkI þ S n KÞ the original problem and its approximating equation are approximately the same in their conditioning. So it is significant to analyse the behavior of jjS n jj and jjC1 n jj. Precisely it follows that jjS n jj ¼ 1 and assuming for a given matrix M 2 Rnn the row norm so defined n X kMkt :¼ max jmi;j j; 16i6n
j¼1
it can be easily shown that
jjCn jj ¼ 1;
jjC1 n jj ¼ 1 þ 2
nþ1 : 2
Moreover it follows that
e jjAjj 6 ðk þ jj KjjÞ; where
e :¼ max jj Kjj a6t6b
Z a
b
e ð h 1 ðt; sÞ þ h2 ðt; sÞÞds:
So it follows that
nþ1 e 1 jjðk þ jj KjjÞ: e jjðkI þ KÞ condðAÞ 6 jjA1 jjjjAjj 6 1 þ 2 j 2 Consequently, an obvious dependence of the conditioning number on system dimension arises.
F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
467
2.2. Direct model spline cubic approximation Taking into account (8) and experimental results in [2], we intend to improve the method’s precision with lower dimension of collocation system. Hereafter we present the numerical model suitable for (3) based on the collocation method using approximating splines, in particular the so called modified q.i. splines of order p > 2 (see for example [8,10]). In the following subsection we recall the definitions and peculiarities of p-order q.i. spline bases. 2.2.1. About p-order q.i. B-spline Let X m :¼ fx0 ¼ a < x1 < . . . < xm < xmþ1 ¼ bg be a partition of the interval ½a; b with Hm :¼ max06j6m ðxjþ1 xj Þ; Hm ! 0 as m ! 1 and let fdj : j ¼ 0; 1; . . . ; m þ 1g be a vector of positive integers where d0 ¼ dmþ1 ¼ p and dj 6 p 1; j ¼ 1; 2; . . . ; m. Pmþ1 We set n þ p :¼ j¼0 dj and define Pn ¼ fti : i ¼ 1; 2; . . . ; n þ pg as the nondecreasing sequence obtained from X m by repeating xj exactly dj times, j ¼ 0; 1; . . . ; m þ 1. The ti are assumed as nodes of the locally uniform spline space [3]: Sp;Pn :¼ fg : gjðxj ; xjþ1 Þ; 2 P p1 ; j ¼ 0; . . . ; m; (where P p1 is the set of polynomials of degree less than p) and ðiÞ g ðiÞ ðxþ j Þ ¼ g ðxj Þ; i ¼ 0; 1; . . . ; p dj 1; j ¼ 1; 2; . . . ; mg and the end points are pfold nodes. Thus Sp;Pn is the set of polynomials splines of order p with nodes at xj ðj ¼ 0; 1; . . . ; m þ 1Þ of multiplicity dj 6 p 1 ðj ¼ 1; 2; . . . ; mÞ, consequently every spline in Sp;Pn is in C½a; b. The set of the normalized B-splines Bi;p ði ¼ 1; 2; . . . ; nÞ of order p defined by the following recurrence relation:
x ti tiþp x Bi;p1 ðxÞ þ Biþ1;p1 ðxÞ; tiþp t iþ1 tiþp1 t i
Bi;p ðxÞ ¼
Bi;1 ðxÞ ¼
ð9Þ
1; ti 6 x < t iþ1 ; 0;
ð10Þ
otherwise
is considered as a basis for the spline space Sp;Pn . Let dj ¼ 1 and T be a set of sij ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; pÞ q.i. points belonging for each i ¼ 1; 2; . . . ; n to a subset of ½t i ; tiþp and such that sij – sih for j – h. The q.i. spline operator S n g 2 Sp;Pn , with g 2 C½a; b, is defined as
S n gðxÞ :¼
p n X X Bi;p ðxÞ v ij gðsij Þ;
ð11Þ
j¼1
i¼1
where
v ij :¼
p X
ail
Ql ; ðs sis Þ l¼j s ¼ 1 ij s–j
aij :¼
j X ðk 1Þ!ðp kÞ! ci;k1 di;jk ; ð1Þjk ðp 1Þ! k¼1
ð12Þ
with ci;k1 ¼ symmk1 ðtiþ1 ; . . . ; tiþp1 Þ; di;jk ¼ symmjk ðsi1 ; . . . ; si;j1 Þ (see [3]). S n is a projector operator [6] that reproduces exactly a polynomial of p 1 degree and also all p order spline functions, that is,
S n P ¼ P;
P 2 Pp;
S n S ¼ S;
S 2 Sp;Pn :
ð13Þ
e ¼ KS e n g with g 2 C½a; b, it follows that Moreover (see for example [7]), as we assume S n Kg
e S n Kjj e ! 0 as n ! 1: jj K
ð14Þ
2.2.2. Collocation method Applying to (6) the collocation method on a set of distinct collocation points nk ðk ¼ 1; 2; . . . ; NÞ in ða; bÞ, where N is the number of distinct elements of the set T, and using q.i. spline approximation, we obtain (7), where in this case
wn ðxÞ ¼
n X
p X i Bi;p ðxÞ v ij u
i¼1
ð15Þ
j¼1
i are the approximated values of function u in sij 2 T ði ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; pÞ. and the u A choice of sij , so that (13) and (14) are satisfied, is proposed in [7]. The optimal algorithmically choice (see [9]) of q.i. points set:
( T :¼ where gi :¼
)
si1 :¼ g1 ; i ¼ 1; . . . ; n si2 :¼ gi1 ; si3 :¼ giþ1 ; . . . ; sip :¼ gið1Þl ½2p ; i ¼ ½2p þ 1; . . . ; n ½2p ;
t iþ1 þ...þt iþp1 p1
ð16Þ
; i ¼ 1; 2; . . . ; n, with a suitable choice of the remaining nodes of T, also satisfies (13) and (14) (see [6]).
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2.2.3. Remarks about the convergence It is possible to generalize to the case p > 2 the remarks in [2] about the convergence. To be more precise, applying the bounded projector operator 11 to 6 and to (7) with wn as in (15) we obtain
e ¼ Sn f ; S n ðkI þ KÞu e n ¼ Sn f S n ðkI þ KÞw and consequently
e ðkI þ S n KÞðu wn Þ ¼ kðI S n Þu:
ð17Þ
e S n Kjj e ! 0 as n ! 1, and following the proof of Theorem 1 in [4], we can state from (17) that As jj K
e Þ1 jj 6 M < 1; supnPN jjðkI þ S n K for n sufficiently large. It follows that jju wn jj ! 0, as n ! 1, exactly with the same rate of convergence as jju S n ujj does. 3. Schauder bases iterative method In this section, we consider the nonlinear integral equation (1). As the notations in the previous sections to this more general nonlinear framework, we write K1 and K2 for the operators on C½a; b
K1 v ðxÞ :¼
Z
x
k1 ðx; y; v ðyÞÞdy;
v 2 C½a; b;
x 2 ½a; b
k2 ðx; y; v ðyÞÞdy;
v 2 C½a; b;
x 2 ½a; b;
a
and
K2 v ðxÞ :¼
Z
b a
respectively, and K :¼ K1 þ K2 . For a compact and Hausdorff topological space X we denote by CðXÞ the Banach space of those real valued functions defined on X and endowed with its maximum norm. Then, it is clear that u 2 C½a; b is a solution of the nonlinear integral Eq. (1) if, and only if, it is a fixed point of the operator N : C½a; b ! C½a; b given, for each v 2 C½a; b, by
N v :¼
1 ðf Kv Þ: k
As a straightforward consequence of the Banach fixed point theorem, if
q :¼
ðM 1 þ M 2 Þðb aÞ < 1; jkj
where M 1 ; M 2 are the Lipschitz constants in (2), then N has one and only one fixed point u, and for each m P 1 we have that
jjN
m
v ujj 6
qm jjN v v jj 1q
v 2 C½a; b and ð18Þ
and thus,
limjjN n
n
v ujj ¼ 0: n
Easy examples show that it is not always possible to calculate the sequence of iterations fN v gnP1 , and for this very reason, a numerical method is needed in order to approximate the fixed point of N . We develop such an iterative method by means of the use of Schauder bases in an adequate Banach space associated in a natural way with the nonlinear mixed integral Eq. (1). 3.1. Elementary facts on Schauder bases The proposed method is based on the use of Schauder bases and extends that developed in [2] for the linear mixed Volterra–Fredholm integral equation: on the one hand, the linear case satisfies clearly the Lispschitz conditions (2); on the other hand, in [2] a concrete Schauder basis is considered, unlike our analysis, which works for any Schauder basis. In fact, the basis in [2] is univariate, what restricts its application to a specific kind of equation. Let us also point out that Schauder bases have been successfully used in the numerical treatment of integral, integro–differential or differential equations (see [11–14]). Let us recall that a Schauder basis in a Banach space E is a sequence fvn gnP1 in E such that any x 2 E has a unique representation of the form
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F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474
n X
lim x tk vk ¼ 0; n
k¼1 where ftn gnP1 is a sequence of real numbers. Given m P 1, the m–th biorthogonal functional m ðxÞ
v
vm : E ! R is defined as
:¼ tm ;
where x 2 E and limn jjx
Q m ðxÞ :¼
Pn
k¼1 t k
vk jj ¼ 0, and the mth projection is the finite–rank linear operator on E given by
m X
vk ðxÞvk ; x 2 E:
k¼1
It follows from the Baire category theorem [15, Theorem 2.1] that for all m P 1 the biorthogonal functional projection Q m are continuous, and from the definition of these functionals and projections, that for all x 2 E
limjjQ n ðxÞ xjj ¼ 0:
vm and the ð19Þ
n
3.2. The iterative numerical method Let us now introduce our numerical method for approximating the unique solution of the mixed nonlinear integral Eq. (1). Our approach combines an iterative scheme with some facts related to an arbitrary Schauder basis in the Banach space C½a; b2 . We describe in a concise way the proposed numerical method. We start from three data: a Schauder basis fvn gnP1 in C½a; b2 , with sequence of associated projections fQ n gnP1 , a sequence of positive integers fjn gnP1 and a function v 2 C½a; b. We define recursively the sequence of approximate functions:
AF0 ðxÞ :¼ v ðxÞ;
x 2 ½a; b
ð20Þ
and if m P 1
AFm ðxÞ :¼
1 k
Z a
1 f ðxÞ; k
x
ð1Þ
Q jm ð m1 ðx; yÞÞdy þ
Z a
!
b
ð2Þ
Q jm ð m1 ðx; yÞÞdy ;
x 2 ½a; b;
ð21Þ
where for j ¼ 1; 2, ðjÞ
m1 ðx; yÞ :¼ kj ðx; y; AFm1 ðyÞÞ;
x; y 2 ½a; b:
ð22Þ
By the triangle inequality we have that
jjAFm ujj 6 jjAFm N
m
v jj þ jjN m v ujj
and since (18) gives an upper bound for the second summand in the right–hand side, it remains to bound the first one in m order to control the error jjAF m ujj. With this in mind, we deal with estimating the norm jjAF m N v jj. Lemma 2. Let k – 0; f 2 C½a; b and let k1 ; k2 2 C½a; b2 satisfying the Lipschitz conditions (2), with Lipschitz constants M 1 and M 2 , respectively, and let
q :¼ ðM1 þ M2 Þðb aÞ=jkj: Assume in addition that v 2 C½a; b; fjn gnP1 is a sequence of positive integers and that fvn gnP1 is a Schauder basis in C½a; b2 whose sequence of associated projections is fQ n gnP1 . If for each m P 1; AF m is the approximate function defined by (20)–(22), then
jjAFm N
m
v jj 6
m ðb aÞ X ð1Þ ð1Þ ð2Þ ð2Þ jj k1 Q jk ð k1 Þjj þ jj k1 Q jk ð k1 Þjj qmk : jkj k¼1
Proof. We proceed inductively on m. For m ¼ 1, given x 2 ½a; b it is clear that
jAF1 ðxÞ N v ðxÞj 6 6
1 jkj
Z a
x
ð1Þ
ð1Þ
j 0 ðx; yÞ Q j1 ð 0 ðx; yÞÞjdy þ
1 jkj
Z a
b
ð2Þ
ð2Þ
j 0 ðx; yÞ Q j1 ð 0 ðx; yÞÞjdy
b a ð1Þ ð1Þ ð2Þ ð2Þ jj 0 Q j1 ð 0 Þjj þ jj 0 Q j1 ð 0 Þjj jkj
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so,
jjAF1 N v jj 6
b a ð1Þ ð1Þ ð2Þ ð2Þ jj 0 Q j1 ð 0 Þjj þ jj 0 Q j1 ð 0 Þjj : jkj
Now, let the announced inequality holds for m 1. Then it also is valid for m, because taking into account the induction hypothesis as well as the fact that for all g; h 2 C½a; b; jjN g N hjj 6 qjjg hjj, we arrive, for all x 2 ½a; b, at m
m
jAFm ðxÞ N v ðxÞj 6 jAFm ðxÞ N AFm1 ðxÞj þ jjN AFm1 N v jj Z x Z b 1 1 m1 ð1Þ ð1Þ ð2Þ ð2Þ j m1 ðx; yÞ Q jm ð m1 ðx; yÞÞjdy þ j m1 ðx; yÞ Q jm ð m1 ðx; yÞÞjdy þ qjjAFm1 N v jj 6 jkj a jkj a ðb aÞ ð1Þ ð1Þ ð2Þ ð2Þ 6 jj k1 Q jm ð m1 Þjj þ jj m1 Q jm ð m1 Þjj jkj X ð1Þ ðb aÞ m1 ð1Þ ð2Þ ð2Þ þ q jj k1 Q jk ð k1 Þjj þ jj k1 Q jk ð k1 Þjj qm1k jkj k¼1 m ðb aÞ X ð1Þ ð1Þ ð2Þ ð2Þ ¼ jj k1 Q jk ð k1 Þjj þ jj k1 Q jk ð k1 Þjj qmk jkj k¼1 and thus the arbitrariness of x 2 ½a; b implies the announced inequality. h Let us observe that in the preceding lemma, the condition q < 1 is not required. Now we are in a position to guarantee that for a suitable choice of m P 1 and j1 ; . . . ; jm P 1; AFm is as closed as desired to the fixed point of the operator N : Theorem 3. Suppose that k – 0; f 2 C½a; b and that k1 ; k2 2 C½a; b2 satisfy the Lipschitz conditions (2), with respective Lipschitz constants M 1 and M 2 , and in such a way that
ðM 1 þ M 2 Þðb aÞ=jkj < 1: Let u be the unique solution of Eq. (1) and let fvn gnP1 be a Schauder basis in C½a; b2 with sequence of associated projections fQ n gnP1 . Then for each e > 0 and each v 2 C½a; b, there exist m P 1 and j1 ; . . . ; jm P 1 such that
jjAFm ujj < e; where AFm is the approximate function defined by (20)–(22). Proof. Since
q :¼ ðM1 þ M2 Þðb aÞ=jkj < 1; let m P 1 such that
qm 1q
jjN v v jj
2, we give some results related to the same test functions as in [2], comparing them in terms of precision of results, using the same number of elements of T. Precisely we propose two examples: (1) h1 ðx; sÞ ¼ h2 ðx; sÞ ¼ eðxþsÞ for Example 4. (2) h1 ðx; sÞ ¼ h2 ðx; sÞ ¼ sinðxÞ cosðsÞ for Example 5. For the collocation method we consider:
the mesh points as equispaced points in ½0; 1 with nodes of p-multiplicity in 0 and 1, the mesh points all simple in ð0; 1Þ, the q.i. points as (16), the number of the collocation points equal to Òthat of the distinct q.i. points. The algorithm is implemented in MATLAB 7:3 . In Tables 1–4 we compare for p ¼ 2 and p ¼ 4 the maximun absolute value of the errors related to the approximation of unknown function in a suitable subset of n points in ð0; 1Þ.
Example 1. Fork ¼ 15,uðxÞ ¼ x3 and
f ðxÞ ¼ 15x3 ex ð2e þ 12 þ ex ðx3 3x2 þ 6x 6ÞÞ;
x 2 ½0; 1;
the numerical results appears in Table 1. Analogously, Table 2 includes the numerical experiments when k ¼ 15,uðxÞ ¼ ex and
f ðxÞ ¼
1 x e ðe2 e2x þ 32Þ; 2
x 2 ½0; 1:
If we would obtain 1010 precision order with p ¼ 2, in the case of Table 2, the rounding error does not make reliable the solution of the collocation system, while, in the case p ¼ 4, we achieve the goal by n ¼ 101.
Table 1 Example 1 u(x) = x3. n
p¼2
p¼4
11
7.5 E3
2.5 E15
Table 2 Example 1 u(x) = ex. n
p¼2
p¼4
11 101
3.3 E3 3.2 E5
6.6 E5 8.0 E10
Table 3 Example 2 u(x) = x2. n
p¼2
p¼4
11
2.7 E3
5.5 E16
n
p¼2
p¼4
11 101 201
3.1 E3 3.1 E5 7.6 E6
6.6 E6 7.9 E10 5.0 E11
Table 4 Example 2 u(x) = ex.
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Example 2. Now k ¼ 1, uðxÞ ¼ x2 and
f ðxÞ ¼ x2 sin xðx2 sin x þ 2x cos x 2 sin x sin 1 þ 2 cos 1Þ;
x 2 ½0; 1;
x
with the corresponding results in Table 3, and k ¼ 1, uðxÞ ¼ e and
f ðxÞ ¼ ex sin xð ex ðsin x þ cos xÞ þ eðsin 1 þ cos 1Þ 2Þ=2;
x 2 ½0; 1;
4.2. Numerical testing for the Schauder bases iterative method Let us consider the non linear case in (1). We include an example to ensure the efficiency and simplicity of the proposed method. We emphasize that the approximation results for the fixed point method in Section 3.2 are stated for an arbitrary Schauder basis in C½a; b2 . In order to obtain concrete numerical approximations of the solution of a nonlinear mixed Volterra–Fredholm integral equation, we shall fix a concrete bivariate Schauder basis. The Schauder basis considered here is that obtained as the bivariate basis derived from the usual Schauder basis in C½a; b, also known as the Faber-Schauder system. To be more precise, let us recall that given a sequence ft n gnP1 of distinct points in ½a; b such that t1 ¼ a and t2 ¼ b, the usual Schauder basis ffn gnP1 in C½a; b is defined by
f1 ðtÞ :¼ 1;
t 2 ½a; b;
while for n P 1; fn is the piecewise linear continuous function on ½a; b with nodes at ftk : 1 6 k 6 ng, given by
fn ðt n Þ ¼ 1 and for k < n
fn ðt k Þ ¼ 0: We denote by ffn gnP1 and fP n gnP1 its associated sequences of biorthogonal and projections, respectively. The usual Schauder basis fvn gnP1 in C½a; b2 , which is what we set in our numerical experiments, is the corresponding bivariate tensor basis of ffn gnP1 ([16] and [17]): if [] denotes ‘‘integer part’’ and r : N ! N N is the bijective mapping given by
pffiffiffi pffiffiffi 8 pffiffiffi pffiffiffi if ½ n ¼ n > < ð n; pnffiffiffiÞ; pffiffiffi pffiffiffi pffiffiffi rðnÞ :¼ ðn ½ n2 ; ½ n þ 1Þ; if 0 < n ½ n2 6 ½ n ; > p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi p ffiffiffi : ð½ n þ 1; n ½ n2 ½ nÞ; if ½ n < n ½ n2 then
vn ðt; sÞ :¼ fp ðtÞfq ðsÞ; t; s 2 ½a; b; whenever rðnÞ ¼ ðp; qÞ. fvn gnP1 and fQ n gnP1 stand for the respective sequences of biorthogonal functionals and projections. It is a well-known fact that the sequence fvn gnP1 is easily determined by finite linear combinations of evaluations at adequate known functions, and therefore so does fQ n gnP1 . More specifically, for each g 2 C½a; b2 ,
v1 ðgÞ ¼ gðt1 ; t1 Þ and for n P 2 and
rðnÞ ¼ ðp; qÞ,
vn ðgÞ ¼ gðtp ; tq Þ
n1 X
vk ðgÞvk ðtp ; tq Þ:
ð25Þ
k¼1
Now we describe the parameters of the numerical method. In what follows we assume that ½a; b ¼ ½0; 1, because such is the case in the concrete nonlinear example below. The subset ftn gnP1 chosen for constructing the usual Schauder basis in C½0; 12 is defined by
t0 ¼ 0;
t1 ¼ 1
and for n P 1,
tnþ1 ¼
2k þ 1 2lþ1
provided that n ¼ 2l þ k þ 1, where 0 6 k < 2l are positive integers. Once we have one of the data defining the approximating sequence fAFm gmP1 given by (20)–(22), we choose the other two: the initial function is
AF0 ¼
f jkj 2
and for all m P 1, we take jm ¼ j for simplicity in the implementation, where j is a given integer.
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F. Caliò et al. / Applied Mathematics and Computation 219 (2012) 464–474 Table 5 Example 3 u(x) = 1 x. x
j¼9
j ¼ 17
jAF 3 ðxÞ uðxÞj 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1
2.9 3.1 3.4 3.6 3.9 4.2 4.5 4.9 5.3
jAF 3 ðxÞ uðxÞj
E4 E4 E4 E4 E4 E4 E4 E4 E4
7.4 7.7 8.1 8.6 9.1 9.7 1.0 1.1 1.2
E5 E5 E5 E5 E5 E5 E4 E4 E4
We include in the following numerical Example 6, for j = 9 and j = 17, the absolute errors committed in certain representative points x 2 ½0; 1, when we approximate the exact solution u by the approximate function AFh where h ¼ 3. Example 3. Let us consider the nonlinear mixed integral equation, similar to that in [18, Example 2],
4 uðxÞ ¼ f ðxÞ þ
Z 0
x
sinðx yÞ cos uðyÞdy þ
Z
1
ðx yÞð1 þ uðyÞ2 Þdy;
x 2 ½0; 1;
0
where for x 2 ½0; 1,
f ðxÞ ¼
1 ð55 64x þ 6ðsin 1Þx cos x 6ðcos 1Þx sin x 6ðsin 1Þ sin xÞ: 12
Its exact solution is uðxÞ ¼ 1 x and the numerical results obtained from the iterative method are taken in Table 5 for the parameters previously commented. The algorithms associated with the numerical method have been performed using Mathematica 7. Let us emphasize that the choice for AF 0 seems to be irrelevant, since after some iterations its influence in the approximations disappears. Let us also point out that this iterative numerical method has the advantage of being very easy to implement. The approximating functions are the sum of a known function and integrals of piecewise bivariate polynomials of degree 2. As immediately follows from (25), the calculation of the coefficients of such polynomials just requires linear combinations of several evaluations of the basic functions at adequate points.
5. Final remarks In this work we presented two numerical methods for solving mixed Volterra–Fredholm integral problems. In particular we proposed, to tackle on the linear cases, a direct collocation method based on q.i. spline of order of precision p P q2. In this way a high degree of accuracy is easily ensured using collocation systems of minimum order, overcoming consequently rounding errors problems. In the nonlinear cases we suggested a fixed point method derived from suitable properties of a Schauder basis in the Banach space of continuous functions C½a; b2 , well suited to define and apply tensorial schemes, that in addition is very easily implemented. The tables of the numerical results confirm the theoretical statements. Acknowledgements The research was partially supported by Prin MURST 200834WK7H_005 for the first and third authors and by Junta de Andalucía Grant FQM359 for the second and fourth authors. References [1] B. Basirat, K. Maleknejad, E. Hashemizadeh, Operational matrix approach for the nonlinear Volterra-Fredholm integral equations: arising in physics and engineering, Int. J. Phys. Sci. 7 (2012) 226–233. [2] F. Caliò, M.V. Fernández Muñoz, E. Marchetti, Direct and iterative methods for the numerical solution of mixed integral equations, Appl. Math. Comput. 216 (2010) 3739–3746. [3] T. Lyche, L.L. Schumaker, Local spline approximation methods, J. Approx. Theory 15 (1975) 294–325. [4] C. Dagnino, V. Demichelis, E. Santi, A nodal spline collocation method for weakly singular Volterra integral equations, Babe ßs-Bolyai, Mathematica, XLVIII, 2003, pp. 71–81. [5] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997. [6] C. De Boor, G.J. Fix, Spline approximation by quasi interpolants, J. Approx. Theory 8 (1973) 19–45. [7] C. Dagnino, V. Demichelis, E. Santi, Numerical integration based on quasi interpolating spline, Computing 50 (1993) 146–163. [8] F. Caliò, E. Marchetti, An algorithm based on q.i. modified splines for singular integral models, Comput. Math. Appl. 41 (2001) 1579–1588.
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