Applied Mathematics, 2013, 4, 15631567 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 quasiinterpolatory 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: VolterraFredholm 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 [1]. 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 equallyspaced 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 [4] 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 [5]. Subsequently, to improve the quality of the approxi mation, the quasi interpolatory (q.i.) splines, proposed and analysed by Lyche and Schumaker, [6], 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 [912]. 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. VolterraFredholm Integral Equations In this paper we consider the following VolterraFred holm integral equation: 1 12 00 ,d ,d x uxfxkxsus skxsus s (1) where :0,1u is the unknown function, x is a known function such that 0,1 .fC The kernels ,,s 1 kx 2,kxs are of the form: 01andlog xs 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 xaxx xb be a partition of the interval :, ab :max, 0HxxHm with 1, , 0 mjmjmm jm as and let :0,,1 j dj m 01m dd be a vector of positive integers where ( p ≥ 2) and . p m , j dp1, ,jm We set 1 0 : j np d and define :1,, ni tin p m as the nondecreasing sequence obtained from by repeating , m exactly d times, 0, ,1.jm n is the set of knots defining the porder poly nomial spline space ,n p Any spline space .S,n p S based on the set m is said to be locally uniform if: 1, , 1, , ,1,1,, jm jm km km xx Akjj m xx 1 where 1 does not depend on nor m. j Let consider as a basis for the spline space ,n p S the set of the normalized Bsplines ,ip of order defined by the following recurrence relation: ,n 1,Bi p ,,1 11 ip i ipipi p ipiip i tx xt BxB xBx tt tt 1,1 1 ,1 1, . 0, otherwise ii i txt Bx To the aim to define q.i. spline operators we consider a set of nodes Tij belonging 1, ,;1, ,injp n for each to a subset of and such that 1, ,i, iip tt ij ih for . jh In [7] and in [13] the following sets are suggested: 1::1,1,, ,1,, 1 ipi ij i tt Ttjj pi p n 2 1 ::,1,,, 1,, 2 ipi ij i tt Ttjj pi p n p 31 : :,1,,,1,, iji j Tt j pipn p 4: :,1,,,1,, iji j Tt j pipn 1 5::, 1,,,1,, 2 ip ij ij tt Tjpipn p 1 62 131 12 :,1,, :: ,: ,, :,1,, 22 ii iiii ip p p i in T pp in (3) where 11 , 1 iip i tt p n 1,,,i with a suitable choice of the nodes for the remaining values of : in i 3 TT 5 as in [7] and in as in [13]. 6 Let now consider the operator T , :, n np CabS so defined: , 11 :, p pij ij gxB xvg n ni ij (4) where 1 : p j s i ij ij is sj v (5) ,1, 1 1! ! :1 1! j , ij i kijk k kpk cd p jk (6) with ,11 ,,, ik k csymm t 1i t1i p ,1,1 dsymm p ,, ijk j kiij In the following we use in (4) (see [6]). 4, 1 j d , ,jm1, 01 4, m dd and 7ij T , where 7 T is defined as in (3) with the remaining nodes suitably 6 T chosen as: 1,21,3 1,42,41 :,:, 2 ,2 ,3 ,4 1,4 :,: 2 nn n n n . Consequently we obtain that the following properties for the operator hold: n 1p n reproduces exactly a polynomial of degree that is [6]: =, np PPP as ij chosen in T belong to a proper subinterval of p 7 , ii tt , for all i and then is a projection operator [14], that is: 1,, ,jp p n , , n SS . (7) n SS 4. The Numerical Model The Equation (1) can be reduced to the following compact form , uf (8) where:  is the identity operator;  is the following operator: 12 gg :,x :,x where: 1 11 0d,gxksgssx 0,1 0,1 1 22 0d,gxksgs sx 1 ,if0 ,: . 0if ss x kxs sx 1 kx Open Access AM
F. CALIÒ, E. MARCHETTI 1565 Let nn rI u f n n (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 np u T v , , k 0,1 , 0,1, 2,, nkn kk rIuf k (10) We assume as an approximation of the solution of (1) the following function belonging to spline space ,n p S , 11 , p n nip ij wxBx vu iji where the i u 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 ,d k p n nkijiji p ij uksvuBs s s ds ds d d u (11) 1 22 , 011 ,d p n nkijiji p ij uksvu Bs (12) which lead to the determination of 1 1,1 1,1 0, kp iki BksBss (13) and 11 2,1 2,1 0, p iki BksBsss (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 , k p n nkijijiji p ij kukv uBss and 1 22 2, 011 , p n nkijijiji p ij kukvuBs s respectively. 5. On the Convergence In this Section we study the convergence of for n w .n Let 0,1 ,EC be a Banach space on with :max;0 10,1yyytt yC and the norm of the operator :EE 1 sup y Lemma 1: Let n be a sequence of l.u. partitions. The operator , 1n S :0, n Cp 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 is s sj in (5) has a minimum (see [7]), then n defined in (4) is a bounded operator. Moreover, as 7ij T (, for all i), the thesis follows (see Theorem A in [7]). 1, ,jp Furthermore it can be noted that the kernel 12 ,,kxsk xskxs , satisfies the following properties: 1) ,kxs is Riemannintegrable as function of , for all 0,x1. 2) 1 for ,0,1xx . 0 lim,, d0, xx kxskxs s 3) 1 0 0,1 max, 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 [15]), that is the existence of 1 I . 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 n p CS , 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 , :0,1Cn S np . For all sufficiently large (n ≥ N) the operator n n 1,1 0,1ICC:0 exists. Moreover it is uniformly bounded, that is: 1 sup n nN IM (17) and 1 nn uwIuu n (18) Open Access AM
F. CALIÒ, E. MARCHETTI Open Access AM 1566 u that is for n Proof: from (10) (see [15]) w.n 1 1 1 1 n N I IM I 0 nn rx (19) As is bounded projection operator (19) becomes: n then (17) is proved. From (21) it follows (18), that is 0, n uw exactly with the same rate of convergence as ,n n uu does (see Lemma 1). 0 nnn Iw Iu (20) From (20) we can easily obtain 1 nn uwIu u n (21) Remark 1. The assumption of the ij points in 7 T (1, ,j Now we must prove the existence and boundedness of . 1 n I By simple algebraic steps it follows that 1 nn IIII (22) p for all ) is decisive for the convergence. Moreover we underline that the choice of the 7ij i T arises from a compromise between two practical different constraints: maximizing the polynomial precision of the approximation and minimizing the collocation system order (see [13]). 6. Numerical Results It is necessary to ensure that has an inverse bounded operator. 1 n II In what follows we present some numerical results for some VolterraFredholm integral Equations (1), by using the numerical method presented above. The algorithm is implemented by MATLAB 7.3. As, from Lemma 2, 0 n as , we can find an integer such that n N 1 1 sup Nn nN I We consider the following equation: 1 12 00 ,d ,d 0,1 . x uxfxkxsus skxsus s x , Consequently, adapting Theorem 3.1.1. in [15], we obtain for nNIn Tables 1 and 2 we show the results obtained with the choice 12 12 ,,kxsk xssx and 1,logkxs sx , 2,1kxs 1 1 1 1 1 n N II I and we can conclude that 1 1 n II , 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 1,, 9j, all simple. exists and it is bounded. Considering that from (22) 1 1 11 n n I III 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 12 1,, 2,0kxs kxs sx , ,uxx 2 xxx with different number of nodes in and consequently Table 1. kxs kxssx 12 12 ,, . x ux RR 32 2311 24 xxx x 16 4.45 10 32 2711 65 81 25325xxxxxx 72 x3 17 9.08 10 423 291187 483564123525635xxxxxxx 72 x4 5 1.88 10
F. CALIÒ, E. MARCHETTI 1567 Table 2. 1,logkxs sx, 2,1kxs. x ux ERR 2log xx 1 15 4.44 10 23 log11 631 3xx x x2 15 1.02 10 45 log13760515xx x x4 5 7.41 10 Table 3. 12 1,, 2,0kxskxs sx . x 11m 21m 41m 101m 0.1 3 7.6 10 3 1.1 10 5 7.1 10 6 6.9 10 0.5 5 6.6 10 5 1.1 10 6 3.4 10 7 8.4 10 1 5 1.8 10 6 3.0 10 6 1.2 10 7 3.2 10 [0,1] . The results denote that the use of the cubic q.i. splines with a suitable densification of the nodes near using the graded mesh in [16], leads to absolute errors of the same order as the results obtained in [16] with the choice of quadratic nodal spline and the same meshes. 0, REFERENCES [1] T. N. E. Greville, “Spline Functions, Interpolation and Numerical Quadrature,” In: Mathematical Methods for Digital Computers, Wiley, New York, 1967, pp. 156168. [2] C. Dagnino, “Product Integration of Singular Integrals Based on Cubic Spline Interpolation at Equally Spaced Nodes,” Numerical Mathematics, Vol. 57, No. 1, 1990, pp. 97104. http://dx.doi.org/10.1007/BF01386400 [3] A. P. Orsi, “Spline Approximation for Cauchy Principal Value Integrals,” JCAM, Vol. 30, No. 1, 1990, pp. 191 201. [4] P. Rabinowitz, “Numerical Integration Based on Approxi mating Splines,” Journal of Computational and Applied Mathematics, Vol. 33, No. 1, 1990, pp. 7383. http://dx.doi.org/10.1016/03770427(90)90257Z [5] I. J. Schoenberg, “On Spline Functions,” Inequalities (Symposium at WritePatterson Air Force Base), Aca demic Press, New York, 1967, pp. 255291. [6] T. Lyche and L. L. Schumaker, “Local Spline Approxi mation Methods,” Journal of Approximation Theory, Vol. 15, No. 4, 1975, pp. 294325. http://dx.doi.org/10.1016/00219045(75)90091X [7] C. Dagnino and P. Rabinowitz, “Product Integration of Singular Integrands Using QuasiInterpolatory Splines,” Computers & Mathematics with Applications, Vol. 33, No. 1, 1997, pp. 5967. http://dx.doi.org/10.1016/S08981221(96)002192 [8] C. Dagnino and E. Santi, “Quadrature Based on Quasi Interpolating SplineProjectors for Product Singular Inte gration,” BabesBolyai, Mathematica, Vol. XLI, 1996, pp. 3547. [9] F. Caliò, E. Marchetti and P. Rabinowitz, “On the Nu merical Solution of the Generalized Prandtl Equation Us ing VariationDiminishing Splines,” Journal of Computa tional and Applied Mathematics, Vol. 60, No. 3, 1995, pp. 297307. http://dx.doi.org/10.1016/03770427(94)00024U [10] F. Caliò and E. Marchetti, “An Algorithm Based on q.i. Modified Splines for Singular Integral Models,” Com puters & Mathematics with Applications, Vol. 41, No. 12, 2001, pp. 15791588. http://dx.doi.org/10.1016/S08981221(01)001237 [11] F. Caliò, M. V. Fernandéz Muñoz and E. Marchetti, “Di rect and Iterative Methods for the Numerical Solution of Mixed Integral Equations,” Applied Mathematics and Com putation, Vol. 216, No. 12, 2010, pp. 37393746. http://dx.doi.org/10.1016/j.amc.2010.05.032 [12] F. Caliò, A. I. GarraldaGuillem, E. Marchetti and M. R. Galán, “About Some Numerical Approaches for Mixed Integral Equations,” Applied Mathematics and Computation, Vol. 219, No. 2, 2012, pp. 464474. http://dx.doi.org/10.1016/j.amc.2012.06.013 [13] V. Demichelis, “QuasiInterpolatory Splines Based on Schoenberg Points,” Mathematics of Computation, Vol. 65, No. 215, 1996, pp. 12351247. http://dx.doi.org/10.1090/S0025571896007284 [14] C. de Boor, “Spline Approximation by Quasi Interpo lants,” Journal of Approximation Theory, Vol. 8, No. 1, 1973, pp. 1945. http://dx.doi.org/10.1016/00219045(73)900294 [15] K. E. Atkinson, “The Numerical Solution of Integral Equations of the Second Kind,” Cambridge University Press, Cambridge, 1997. [16] C. Dagnino, V. Demichelis and E. Santi, “A Nodal Spline Collocation Method for Weakly Singular Volterra Inte gral Equations,” BabesBolyai, Mathematica, Vol. XLVIII, 2003, pp. 7181. Open Access AM
