 Applied Mathematics, 2013, 4, 1563-1567 Published Online November 2013 (http://www.scirp.org/journal/am) http://dx.doi.org/10.4236/am.2013.411211 Open Access AM Cubic Spline Approximation for Weakly Singular Integral Models Franca Caliò, Elena Marchetti Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Email: franca.calio@polimi.it, elena.marchetti@polimi.it Received April 10, 2013; revised May 10, 2013; accepted May 17, 2013 Copyright © 2013 Franca Caliò, Elena Marchetti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper we propose a numerical collocation method to approximate the solution of linear integral mixed Volterra- Fredholm equations of the second kind, with particular weakly singular kernels. The collocation method is based on the class of quasi-interpolatory splines on locally uniform mesh. These approximating functions are particularly suitable to tackle on problems with weakly regular solutions. We analyse the convergence problems and we present some numeri-cal results and comparisons to confirm the efficiency of the numerical model. Keywords: Volterra-Fredholm Integral Equations; Collocation Methods; Splines 1. Introduction Splines have been used in numerical integration, with all their well known properties, ever since they entered in the numerical analysis scene . In the nineties, splines have been used in more general aspects in numerical integration such as product integra-tion and numerical approximation of models with Cauchy principal value integrals [2,3]. However these results are not completely satisfactory as they use functional values at equally-spaced nodes, whereas in applications it is desirable to densify points in places where the integrand function is not smooth and use fewer nodes where it is. To tackle on this problem, Rabinowitz  proposed, with respect to numerical inte-gration, the use of an important class of splines, known as variation diminishing splines (VDS), introduced and investigated, as a tool of approximation theory, in the seventies by Schoenberg . Subsequently, to improve the quality of the approxi-mation, the quasi interpolatory (q.i.) splines, proposed and analysed by Lyche and Schumaker, , in different kind of integrals are used, algorithms are given and con-vergence results are proved in [7,8]. From the second half of the nineties, taking advantage of all these results, the use of q.i. splines in different kind of integral equations is suggested and analysed in [9-12]. In this work we apply a numerical model based on cu-bic q.i. splines approximation to special mixed Volterra- Fredholm integral equations of second kind with particu-lar convolution kernels. In Section 2 we present the mathematical model, in Section 3 we recall the background on q.i. spline space, in Section 4 the numerical method is described, Section 5 is devoted to convergence analysis, finally in Section 6 we show numerical results to complete the theoretical statements and to emphasize the efficiency of the method in the case of solution with discontinuity from the first derivative. 2. Volterra-Fredholm Integral Equations In this paper we consider the following Volterra-Fred- holm integral equation:   11200,d ,dxuxfxkxsus skxsus s (1) where :0,1u is the unknown function, fx is a known function such that 0,1 .fC The kernels ,,s1kx 2,kxs are of the form: 01andlogsxsx  (2) if  and 0, there exists a unique function 0,1uC solution of (1). 3. On the q.i. Splines In the following we recall the necessary background on q.i. splines space. F. CALIÒ, E. MARCHETTI 1564 Let 0,1,, 1,:mmm mmmmXxaxx xb  be a partition of the interval :,Jab:max, 0HxxHm with 1, ,0mjmjmmjm  as and let :0,,1jdj m01mdd be a vector of positive integers where (p ≥ 2) and . pm,jdp1, ,jmWe set 10:jjnp d and define :1,,nitin p m as the nondecreasing sequence obtained from X by repeating ,jmx exactly jd times, 0, ,1.jmn is the set of knots defining the p-order poly- nomial spline space ,np Any spline space .S,npS based on the set mX is said to be locally uniform if: 1, ,1, ,,1,1,,jm jmkm kmxxAkjj mxx1 where 1A does not depend on nor m. jLet consider as a basis for the spline space ,npS the set of the normalized B-splines ,ip of order defined by the following recurrence relation: ,n1,Bip  ,,111ipiipipi pipiip itxxtBxB xBxtt tt1,1 1,11, .0, otherwiseiiitxtBx  To the aim to define q.i. spline operators we consider a set of nodes Tij belonging 1, ,;1, ,injpnfor each to a subset of and such that 1, ,i,iipttij ih for . jhIn  and in  the following sets are suggested: 1::1,1,, ,1,,1ipiij ittTtjj pip n 21::,1,,, 1,,2ipiij ittTtjj pip  np 31: :,1,,,1,,iji jTt j pipnp 4: :,1,,,1,,iji jTt j pipn 15::, 1,,,1,,2ip ijijttTjpipn p 162 13112:,1,,:: ,: ,,:,1,,22iiiiiiip ppiinTppin  (3) where 11,1iipittpn 1,,,i with a suitable choice of the nodes for the remaining values of : in i3TT5 as in  and in as in . 6Let now consider the operator T,:, nnpCabS so defined:  ,11:,ppijijgxB xvgnni ij (4) where 1:pjsiijij issjv (5)  ,1,11! !:1 1!j,ij i kijkkkpkcdp jk (6) with ,11 ,,,ik kcsymm t1it1i p ,1,1dsymmp,,ijk j kiij In the following we use in (4) (see ). 4, 1jd, ,jm1, 014,mdd and 7ij T, where 7T is defined as in (3) with the remaining nodes suitably 6Tchosen as: 1,21,31,42,41:,:,2 ,2 ,3,4 1,4:,:2nnn nn. Consequently we obtain that the following properties for the operator hold: n1p-n reproduces exactly a polynomial of  degree that is : =,npPPP -as ij chosen in T belong to a proper subinterval of p7,iitt, for all i and then is a projection operator , that is: 1,, ,jppn,,nSS. (7) nSS4. The Numerical Model The Equation (1) can be reduced to the following compact form ,Iuf (8) where: -I is the identity operator; - is the following operator: 12ggg:,x:,x where:  1110d,gxksgssx 0,10,1  1220d,gxksgs sx  1,if0,: .0ifss xkxs sx1kx Open Access AM F. CALIÒ, E. MARCHETTI 1565Let nnrI u  fnn (9) where n is in (4). If we collocate (9) in a set of points, we could completely define n. Neverthless the choice of the set 7 of the nodes and the definition of ij as in (5) allow, by the algorithm, to reduce the dimension of the collocation system. Consequently the collocation system on a set of distinct collocation points chosen in is the following one u1, 2,knpuTv, ,k0,1 ,0,1, 2,,nkn kkrIuf k  (10) We assume as an approximation of the solution of (1) the following function belonging to spline space ,npS ,11,pnnipijwxBx vuiji where the iu are the approximated values of function in u1i, obtained from the collocation system (10). Finally we observe that to complete the algorithm we must to compute the coefficients of the collocation system and then to evaluate the following integrals: 11 ,011,dkpnnkijiji pijuksvuBs ssdsdsddu (11) 122 ,011,dpnnkijiji pijuksvu Bs (12) which lead to the determination of 11,1 1,10,kpikiBksBss (13) and 112,1 2,10,pikiBksBsss (14) with . 1, ,inThe computation of (13) and (14) is carried out through a closed analytical form, when possible. Otherwise we substitute (11) and (12) with: 11 1,011,kpnnkijijiji pijkukv uBss  and 122 2,011,pnnkijijiji pijkukvuBs s  respectively. 5. On the Convergence In this Section we study the convergence of for nw.nLet 0,1 ,EC be a Banach space on  with :max;0 10,1yyytt yC and the norm of the operator :EE1supyy Lemma 1: Let n be a sequence of l.u. partitions. The operator ,1nS:0,nCp is a bounded compact operator and such that for 0,1gC 0as .ngg n (15) Proof: As ij in (6) for all are bounded and ,ij1ij isssj in (5) has a minimum (see ), then ndefined in (4) is a bounded operator. Moreover, as 7ij T (, for all i), the thesis follows (see Theorem A in ). 1, ,jpFurthermore it can be noted that the kernel 12,,kxsk xskxs, satisfies the following properties: 1) ,kxs is Riemann-integrable as function of s, for all 0,x1. 2) 1 for ,0,1xx. 0lim,, d0,xx kxskxs s3) 100,1max, d.xkxs s Consequently the operator in (8) is a bounded compact operator. uMoreover this condition states the existence and unique-ness of the solution of (1) (see ), that is the existence of 1I. Lemma 2: Let n be a sequence of l.u. partitions. Let consider the sequence of bounded and projection operators in (7): n,0,1 npCS, it follows that: 0as .nn (16) Proof: As :0,10,1CC is a compact opera-tor and since (15) holds, then (16) is proved. Theorem 1: Let n be a sequence of l.u. partitions. Let consider the bounded and projection operator ,:0,1CnSnp. For all sufficiently large (n ≥ N) the operator nn 1,1 0,1ICC:0 exists. Moreover it is uniformly bounded, that is: 1sup nnNIM (17) and 1nnuwIuu n (18) Open Access AM F. CALIÒ, E. MARCHETTI Open Access AM 1566 uthat is for nProof: from (10) (see ) w.n1111nNIIMI 0nnrx (19) As is bounded projection operator (19) becomes: nthen (17) is proved. From (21) it follows (18), that is 0,nuw exactly with the same rate of convergence as ,nnuu does (see Lemma 1). 0nnnIw Iu    (20) From (20) we can easily obtain 1nnuwIu u  n (21) Remark 1. The assumption of the ij points in 7T (1, ,jNow we must prove the existence and boundedness of . 1nIBy simple algebraic steps it follows that 1nnIIII     (22) p for all ) is decisive for the convergence. Moreover we underline that the choice of the 7ijiT arises from a compromise between two practical different constraints: maximizing the polynomial precision of the approximation and minimizing the collocation system order (see ). 6. Numerical Results It is necessary to ensure that has an inverse bounded operator. 1nII In what follows we present some numerical results for some Volterra-Fredholm integral Equations (1), by using the numerical method presented above. The algorithm is implemented by MATLAB 7.3. As, from Lemma 2, 0n as , we can find an integer such that nN11supNnnN I We consider the following equation:  11200,d ,d0,1 .xuxfxkxsus skxsus sx , Consequently, adapting Theorem 3.1.1. in , we obtain for nNIn Tables 1 and 2 we show the results obtained with the choice  1212,,kxsk xssx and 1,logkxs sx, 2,1kxs 11111nNIII and we can conclude that  11nII , respectively, and λ = 1. In particular, the polynomial exactness of the method till third degree is tested. In the interval [0,1] we choose points 11mjx 1,, 9j, all simple. exists and it is bounded. Considering that from (22)  1111nnIIII The unknown function is approximated in 13 nodes belonging to 0,1 . For brevity in Table 1 we indicate the mean of the absolute values of the errors evaluated in the interval. In Table 3 we show the results obtained with the choice 121,,2,0kxs kxs sx  ,,uxx 2fxxx with different number of nodes in and consequently Table 1.  kxs kxssx1212,, . fx ux ERR 322311 24xxxx x 164.45 10 322711 65 81 25325xxxxxx72 x3 179.08 10 423291187 483564123525635xxxxxxx  72 x4 51.88 10 F. CALIÒ, E. MARCHETTI 1567 Table 2. 1,logkxs sx, 2,1kxs. fx ux ERR 2logxxx 1 154.44 1023log11 631 3xx x x2 151.02 10 45log13760515xx x x4 57.41 10 Table 3. 121,,2,0kxskxs sx  . x 11m 21m 41m 101m 0.1 37.6 10 31.1 10 57.1 10 66.9 10 0.5 56.6 10 51.1 10 63.4 10 78.4 10 1 51.8 10 63.0 10 61.2 10 73.2 10 [0,1] . 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