 Open Journal of Applied Sciences, 2012, 2, 115-121 doi:10.4236/ojapps.2012.22016 Published Online June 2012 (http://www.SciRP.org/journal/ojapps) Collocation Method for Nonlinear Volterra-Fr edholm Integral Equations Jafar Ahmadi Shali1, Parviz Darania2, Ali Asgar Jodayree Akbarfam1 1Department of Mathematics and Computer Science, University of Tabriz, Tabriz, Iran 2Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran Email: {j_ahmadishali, pdarania}@tabrizu.ac.ir, Akbarfam@yahoo.com Received April 18, 2012; revised May 14, 2012; accepted May 25, 2012 ABSTRACT A fully discrete version of a piecewise polynomial collocation method based on new collocation points, is constructed to solve nonlinear Volterra-Fredholm integral equations. In this paper, we obtain existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of Volterra- Fredholm integral equations. Keywords: Collocation Method; Nonlinear Volterra-Fredholm Integral Equations; Convergence Analysis; Chelyshkov Polynomials 1. Introduction We shall consider the nonlinear Volterra-Fredholm inte- gral equation   12,0,ytgty tFy ttIT  . (1) The Volterra integral operators given by   10:,, d,tCI CIytktsyss (2) where and and Fredholm integral operators given by 1kCD,:0Dts stT  20:,, d,TFCI CIFytktsys s (3) where ,1,rr2kCIIdenotes (real or complex) parameters and and let be a given function. 2IgCThe mentioned equations are characterized by the presence of a linear functional argument and play an im- portant role in explaining many different phenomena. In particular, they turn out to be fundamental when or- dinary differential equations based model fail. These equations arise in industrial applications and in studies based on biology, economy, control and electro-dynamic. Collocation method is a widely popular numerical technique in solving integral equations, differential equa- tions, etc. When collocation method is used to solve complicated engineering problems, it has several disad- vantages, that is, low efficiency, ill-conditioned, etc. Thus, different types of techniques were proposed to improve the computational performance of collocation method. Recently, Chelyshkov has introduced sequences of polynomials in , which are orthogonal over the inter- val 0,1 with the weight function 1. These polynomials are explicitly defined by   101,mk jmkmkjk jmkjm kjPtt km 0,1,,. (4) The polynomials mkPtmk have properties, which are analogous to the properties of the classical orthogonal polynomials. These polynomials can also be connected to a fixed set of Jacobi polynomials . Precisely ,mPt 0,2 112kkmk n kPttP t1. Investigating more on (4), we deduce that in the family of orthogonal polynomials have k multiple 0mkmkPtzeros 0t and mk distinct real zeros in the inter- val 0,1 . Hence, for every m the polynomial 0mPt has exactly m simple roots in 0,1 . Following , it can be shown that the sequence of polynomials 00mmPt generate a family of orthogonal polynomials on 0,1 which possesses all the properties of other classic or- thogonal polynomials e.g. Legendre or Chebyshev poly- nomials. Therefore, if the roots of are chosen as collocation points, then we can obtain an accurate nu- merical quadrature. 0mPtIn the present paper, we further develop the works car- ried out in [2-6]. Copyright © 2012 SciRes. OJAppS J. A. SHALI ET AL. 116 We discuss existence and uniqueness results and ana- lyze the convergence properties of the collocation method when used to approximate smooth solutions of linear Volterra-Fredholm integral equations and finally, some numerical results are presented in the final section, which support the theoretical results obtained in this paper. 2. Existence and Uniqueness Results Let denote the Banach space continuous real- valued functions, such that CIgCI with max .tIgg (5) Lemma 2.1. Assume H is a nonempty closed set in a Banach space V, and that is continuous. Suppose is a contraction for some positive integer m. Then, T has a unique fixed-point in H. :TH HmTProof: For proof see . Here, in integral Equation (1), we assume that for some constants iikM, satisfies a Lipschitz condition with respect to its third argument 121,, ,,, 0,,1,2.ii iiktsyktsyMy ystT yi 2 (6) Theorem 2.2. Assume g and i satisfy the condition (6) and given functions k,,igk are continuous on their domains. Moreover, assume 12.MT (7) Then the integral Equation (1) has a unique solution .tIyC Proof: We define the nonlinear integral operator   12:,,0,TCI CITy tgty tFy ttI T   (8) Let us show that for m sufficiently large, the operator is a contraction on . For mTCI12,yyCI  1211 12021 220 ,,,, d ,,,,dtTTy tTytktsys ktsyssktsysktsyss  (9) Then    12 112021 201212 d d..tTTy tTytMysyssMysyssMtMTyy   (10) Since 2212111 20212 20 ,,,, d,,,, d,tTTy tTytktsTys ktsTyssktsTysktsTyss (11) we get 2212212 212.2!MtTy tTytMTyy (12) By a mathematical induction, we obtain 121212 .!mmmmTyt TytMt MTy ym  (13) Thus 121212 .!mmmmTyt TytTM TMy ym  (14) Since, 201TM then 2lim 0,mmTM  and 1lim 0,!mmMm  the operator is a contraction on mTIC when m is chosen sufficiently large. By the Lemma 2.1, the operator T has a unique fixed-point in IC. 3. Collocation Method Let ,0,,1,ntnhn NtTN define a uniform partition for 0, ,TI and let 011:0 ,:, 01.NNnnnttt Tttn N  The mesh N is constrained in the following sense: ThN with a given mesh N we associate the set of its interior points, 1,,1n N::NnZt .1d For a fixed and, for given integers and the piecewise polynomial space 1N1,mdNmdSZ is defined by ::;,01n,NmdmdSZ uCIunNdd  where πmd denotes the set of (real) polynomials of a degree not exceeding md. The dimension of this space is given by dim 1.NmNddmdSZ Copyright © 2012 SciRes. OJAppS J. A. SHALI ET AL. 117For integral equation, we have hence, the collocation space will be 1,d11NmSZ. Let 11,NmnuSZ for all nt we have 1, 0,1,,1mnrnrututshL sutchnN r (15) From (15) we see that an element 11,NmuSZ is well defined when we know the coefficients nrut ch for all In order to compute these coef- ficients, we consider the set of collocation parameters 0, ,1.nNjc:, where and define the set of collocation points by 10mcc,10jn1,1.,1,mNNnjXt,:,1,,,0,,njn jttchj mnN  The collocation solution will be de- termined by imposing the condition that satisfies the integral Equation (1) on the finite set 11,mNuS ZuNX  12,,Nutgtu tFuttX  (16) Thus, for ,nj the collocation Equation (16) assumes the form n jttt ch ,,,11,022,0,, d ,,d.njtnj njnjTnjutgtk tsusskt suss 17) From this equation and after some computations, we obtain  11,,11,0011,01122,00,, ,,d ,,djnnjnjnj iiicnj nnNnj iiiutgthktt shut shshkttshutshshkttshutshs d (18) Now, by using the local Lagrange basis functions 11,,1,,mrrrllrscLsl mcc, (19) for approximating the integral terms, we use the La- grange interpolating polynomial to approximate and , we obtain 1,,,njkt sus2,,,njkt sus1,11,11, 1100111, 1,011122,1 001 ,, ,,d ,,d.jnjnmnjnj iiilmcnj nnljllNmnj iillilutdgthkttchutchLshktt chut chLsshkttchu tchLss  s,,,,m(20) Defining the quadrature weights 10:d,1,llwLssl m (21) and ,,0:d,,1,jcjl jlwLsslj (22) the fully discretized collocation equation corresponding to (20)-(22) is thus given by ,1,1 1,011,1,1122,01 ,, ,, ,,.njnmnjlnj ilililmjln jnlnllNmlnjil ililutgthwktt chutchhwktt chut chhwkttchutch  (23) Note that, and Equation (23) represent for each 11mNuS Z0,1, ,1,nN a recursive system of m nonlinear algebraic equations with the unknowns ,njut. 4. Global Convergence Let denote the (exact) collocation solu- tion to (1) defined by (16). In our convergence analysis we examine the linear test equation 11mNuS Z, 110220,d ,d,0,tTytgtktsyssktsyss tIT (24) where 1,kCD 2.kCII We will assume that 12 is not in the spectrum F of the Fredholm in- tegral operator F. A comment of the convergence results to the nonlinear Equation (1) can be found at the end of this section. Theorem 4.1. Assume that the given function in (24) satisfy 12,,mm m.gCkC DkC II  Then for all sufficiently small hTN the constrained mesh collo-cation solution to (24), for all 11mNuS Z1,N0,1, ,n satisfies ,mmmCMh (25) Copyright © 2012 SciRes. OJAppS J. A. SHALI ET AL. 118 where m are positive constants not depending on h. This estimate holds for all collocation parameters Cjc with 1Proof: In each interval 1ii, the exact solution y of (24) is m times continuously differentiable. This fol- lows from the smoothness hypotheses we have imposed on 1201mcc .tt,,gkk and from the expressions for yt. From this it is obvious that both the left and right limits of yt, as t tends to , exist and are finite. We will prove the estimate (25) by using the Peano’s Theorem to write nh,1, 0,1.mmnrnrmnrytshLsytchhR ss  (26) Here, we have  1,0:,mmn mnRs Kszytzhzd, (27) and  1111,,1! 0,1mmmmkkKszszLsc zmzk (28) Thus, it follows from (15) that the collocation error :yu possesses to the local representation  ,,1,(0,1mmnrnrmnrtshLshRss ], (29) with and it satisfies the equation ,,njnjnjytch utch,,11,022,0,d ,d.njtnj njTnjtktsskt ssss (30) By substituting the (29) in the (30) and after some computations, we obtain 11,1 1,00111111,,0011,01111,,0122,0,d ,d ,d ,d ,d jjnmnjnj iririrnmnj imiimcnj nrnrrcmnj nmnnj irhkttshLsshkttshRsshkttshLsshkttshRsshkttshLss  10111122, ,00 ,d.NmirirNmnj imiihkttshRss (31) Define the matrices in ,mL 11,01,,d,,1,2,,01inj irnkttshLssij minN , (32) 1,02,,d,,1,2,,jcnj nrnkttshLssrjm (33) 12,03,,,,1,2,,01inj irnktt shLsdsrjminN m, (34) and the vectors in by 11,1,,0, 1,2,,,,Tinnjimid,AkttshRssjmin (35) 2,1 ,,0,d1, 2,,,jTcnnjnmn,AkttshRs sjm (36) 13,2 ,,0,d1, 2,,,,Tinnjimi,Aktt shRssjmin (37) ,1, 2,0,1,,Tnnn nmnN 1, (38) by substituting the Equations (32)-(38) in Equation (31) we obtain   11111,11,12,00111112,23,23,00 ,nniimnninnniiNNiimmnniiihhAhhAhh A n1,0NiiA (39) this linear algebraic system may be written more con- cisely as  112,23,01111111, 11,12,2300 Ninnnniinniimmmnin nniihhhhAhAh     (40) Now, let 12,012,112,12B 0002B 000 000 2BmmmNIhIhIh (41) Copyright © 2012 SciRes. OJAppS J. A. SHALI ET AL. 11901 12 3,023,02 3,00123,12 3,12 3,122 222 2NmNNNmIh hhQhhIh      (42) 1N011N0i (43) Then we have  11111,11,011112, 23,022 22.nniimni niNimmnniQh hAhAh A    (44) Since the kernel iK is continuous on their domains, the elements of the matrixes 2, ,0,1,,nnN12,nh are all bounded. By using the Neumann Lemma the inverse of the matrix 12nmI  exists whenever 12,12nh for some matrix norm. This clearly holds whenever h is sufficiently small. In other words, there is an 0h so that for any mesh N with ,hh each matrix n has a uniformly bounded inverse. Therefore, matrix  has a uniformly bounded inverse. Also, the invertibility of the mm block matrix now depends not only on h but also on NNQ2 it is guar- anteed if 21,2F where *220:max, d,TtI 2FktsskTk (45) assuming that 2*2,kts k and the elements of the matrixes Q are all bounded. Thus from (44), we get mIn  (46) where 1Q (47) and  11111,11,0011112,23,022 +22.nnimnniiiNimmnihhhAh A 1innA (48) It is clear that, matrix has a uniformly bounded in- verse and the elements of the matrixes are all bounded. Note that, from these assumptions and 11,Q there exists a constant so that for all mesh di- ameters 0D0, ,hh the uniform bound 101,mIP (49) holds. Here, for ,mBL 1(operator) norm induced by the -norm in Assume B denotes the matrix 1l.mthat 1 and 1, 11inBP for 0 and 1,inN 121From (46) and (48) we have .P 10111,mnnIP  (50)  111111, 11,10011112,23,0111101 1011011211222 22 22 ,nniimnniniiNimmnninmimimmmm mmhhAhAh APPhPhnmkM KhmkMKhNmkMK   m(51) and hence 1101110,nmimiMh (52) where   00111011221, 12, 1113, 21111100[0,1]2,21,, ,,,: ,:max, d,:max,d.minmmnmmimnmmmmmstIPP hPmK knNkAmkKMinAmkKMAmkKMinMykKszzksvv  k (53) Also, 111101nmNmnj ljjlj  (54) Then, from (52) and (54), we have 1101110, 0,1,,1.nmnimiMh nN Now, by using the discrete Gronwall inequality, we have 121, 0,1,,1,mnmMh nN where exp .n 21 0 Now, by using the local error representation (29) this yields, setting :max ,mjjWL 1,mmnmnmmmtshWhKM CMhm Copyright © 2012 SciRes. OJAppS J. A. SHALI ET AL. 120 uniformly for 0, 1s and 01 where ,nN 2.mm mCW K The is equivalent to the estimate .mmmCy h We conclude this section with a comment regarding the extension of the results of Theorem 1 to the nonlinear Equation (1). Under the assumption of the existence of a (unique) solution yt on I, the nonlinear analogue of the error Equation (30) is ,,1 1,1,022, 2,0,,,, d ,,,, d.njtnj njnjTnj njtktsysktsuskt sysktsusss (55) If the partial derivatives , 1,2ikiy are continuous and bounded on with 1DD1:: ,Dy yysMsI , for some and if is sufficiently small, then (55) may again be written in the form (30). The roles of are now assumed by ,Mi0hk ,,,:, 1,2iiiktszsHts iy where   :1,0iiiizs ysuss 1.Hence, the above proof is easily adapted to deal with the nonlinear case (1), and so the convergence results of Theorem 1 remain valid for nonlinear Volterra-Fredholm integral equations. 5. Presentation of Results In this section, we report on the numerical result of test problem solved by the proposed method of this article. Typical forms of collocation parameters jc are: Gauss points: Zeros of 21;mPt Radou I points: Zeros of 121 21;mmPtP t 1; Radou II points: Zeros of 121 21;mmPtP t Chelyshkov points: Zeros of  0,1012mmmPtP t where and mPt,mPt are Legendre and Jacobi polynomials, respectively. Example 5.1. The nonlinear Volterra-Fredholm inte- gral equation in 0,1  120012d3d,01tytttys sys sst Table 1. Error for example 1. m N Guasse Radaue Radaue Chelyshkove3 2 2 × 10–30 2 × 10–30 2 × 10–30 1 × 10–30 3 4 2 × 10–30 2 × 10–30 3 × 10–30 1 × 10–30 3 8 2 × 10–30 2 × 10–30 2 × 10–30 2 × 10–30 has the following analytical solution ytt therefore, provides an example to verify the accuracy of this meth- od. Table 1 shows the maximum errors involved pre- sented method with 111,,,248h along with the exact solution. For computational purposes, in the test problem dif- ferent forms of kernels are considered. All the computa- tions were carried out with Maple. In each cases of Ex- ample the obtained nonlinear equations was solved by the Newton’s method. The result for collocation points jc are presented in Table 1 which indicates that the numerical solutions ob- tained from (56) and step sizes equal to 11, 24 and 18 are nearly identical. These results indicate that, if we use the Chelyshkov points, then we obtain the numerical so- lutions of minimum error. 6. Conclusion We have shown that the collocation method yields an efficient and very accurate numerical method for the ap- proximation of solutions to Volterra-Fredholm integral equations. Also we have shown that, if the roots of 0mPt are chosen as collocation points, then we can obtain an accurate numerical quadrature. 7. Acknowledgements The authors truly appreciate the comments made by re- ferees. REFERENCES  V. S. Chelyshkov, “Alternative Orthogonal Polynomials and Quadratures,” Electronic Transactions on Numerical Analysis, Vol. 25, No. 7, 2006, pp. 17-26.  H. Brunner, “Collocation Methods for Volterra Integral and Related Functional Equations (Cambridge Mono- graphs on Applied and Computational Mathematics),” Vol. 15, Cambridge University Press, Cambridge, 2004. 2, (56)  H. Brunner, “Implicitly Linear Collocation Methods for Nonlinear Volterra Equations,” Applied Numerical Mathe- matics, Vol. 9, No. 3-5, 1992, pp. 235-247. Copyright © 2012 SciRes. OJAppS J. A. SHALI ET AL. Copyright © 2012 SciRes. OJAppS 121doi:10.1016/0168-9274(92)90018-9  H. Brunner, “High-Order Collocation Methods for Sin- gular Volterra Functional Equations of Neutral Type,” Applied Numerical Mathematics, Vol. 57, No. 5-7, 2007, pp. 533-548. doi:10.1016/j.apnum.2006.07.006  H. 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