 American Journal of Computational Mathematics, 2011, 1, 134-138 doi:10.4236/ajcm.2011.12014 Published Online June 2011 (http://www.scirp.org/journal/ajcm) Copyright © 2011 SciRes. AJCM Numerical Solution of Nonlinear Fredholm-Volterra Integtral Equations via Piecewise Constant Function by Collocation Method Ahmad Shahsavaran Faculty of Science, Islamic Azad University, Borujerd Branch, Borujerd, Iran E-mail: a.shahshavaran@iaub.ac.ir Received March 26, 2011; revised April 17, 2011; accepted May 10, 2011 Abstract In this work, we present a computational method for solving nonlinear Fredholm-Volterra integral equations of the second kind which is based on replacement of the unknown function by truncated series of well known Block-Pulse functions (BPfs) expansion. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples. Keywords: Nonlinear Fre dh ol m- Vo lte rr a In te gral Equat io n, Bl ock- Pulse F unction, Error Analysis, Collocation Points 1. Introduction The integral equation method is widely used for solving many problems in mathematical physics and engineering. This article proposes a computational method for solving nonlinear Fredholm-Volterra integral equations. Several numerical methods for approximating the solution of linear and nonlinear integral equations and specially Fredholm- Volterra integral equations are known [1-10]. Also, Block- Pulse functions are studied by many authors and applied for solving different problems. In presented paper, by using vector forms of BPfs, the main problem can be easily re-duced to a nonlinear system of algebraic equations which can be solved by Newton’s iterative method. 2. Review of Some Related Papers Some computational methods for approximating the so-lution of linear and nonlinear integral equations are known. The classical method of successive approxima-tion for Fredholm-Hammerstein integral equations was introduced in . Brunner in  applied a collocation type method and Ordokhani in  applied rationalized Haar function to nonlinear Volterra-Fredholm-Hammer-stein integral equations. A variation of the Nystrom me- thod was presented in . A collocation type method was developed in . The asymptotic error expansion of a collocation type method for volterra-Hammerstein in-tegral equations h as been considered in . Yousefi in  applied Legendre wavelets to a special type of nonlinear Volterra-Fredholm integral equations of the form. 110120()()(,)( ( ))d (,)(())d, 0, 1,tutf tKtxFuxxKtxGux xxt2 (1) where ()ft, and 1(, )Ktx and 2(, )Ktx are assumed to be in on the interval 2(LR)0x1t, . Yalcinbas in  used Taylor polynomials for solving Equation (1) with ()pFuu and () qGu u. Orthogonal functions and polynomials receive attention in dealing with various problems that one of those in integral equation. The main characteristic of using orthogonal basis is that it reduces these problems to solving a system of nonlinear algebraic equations. The aim of this work is to present a numerical method for approximating the solution of nonlinear Fred-holm-Volterra integral equation of the form : 110120()()(,)( ( ))d (,)(())d, 0, 1,tmnutf tKtxuxxKtxux xxt2 (2) where and are nonnegative integers and 1m n and 2 are constants. For this purpose we define a k-set of BPfs as 11,, forall1,2,,()0, elsewhereiiitiBt kk k (3) A. SHAHSAVARAN ET AL. 135jThe functions are disjoint and orthogonal. That is, ()rBt0,() ()(),ji iiBtBt Bti j (4) 0,() ()1,ijijBtBt ijk (5) A function defined over the interval [0, 1) may be expanded as: ()ut1() ()iiiutuB t. (6) In practice, only k-term of (6) are considered, where k is a power of 2, that is, 1() ()()kkiiututuB tit), (7) with matrix from: () ()()tkututuBtt, (8) where, and 12[, ,,]kuu uu12()[(), (),,()]tktBtBt BtBm. In a similar manner, [( can be approximated in term of BPfs )]ut[()] ()mtut tuBu, that we need to calculate vector whose elements are nonlinear combination of the elements of the vector u For this purpose, we can write () ()ut ttuB and [( . )] ()mut tuBtSo, () [()]mtuB uBtt (9) now using (4) leads to 12() 0()() ()0(kBtBtttBttBB also from (3) we get 10tk implies that and 1() 1Bt() 0iBt for . 2, ,ik1tkk2 implies that and 2() 1Bt() 0iBt for and . 1, ,ik 2i11ktk implies that and () 1kBt() 0iBt for . Therefore, simply we obtain 1, ,1ik 1() ()dtttkIBB10t, (10) where, is the identity matrix of order k. By incorporat-ing these results we have I1100() ()d[()dttt tttktttk t I uuuBBuB( Bt)mt. Hence, 1011111()] ()d ()]()d ()][()()]dttmiktmkiikiktm tkiikktttktttkttttuuBBuB BuBu BBt[[[tt. (11) So using (11) leads to 1112010[, ,, ]0mtkkkuuuu. 1212121100[, ,,]d00001 [, ,,]00kmkkkuu utkuuu . 1211112001[, ,,]d00000 [, ,,]01kmkkkuu utkuuu  . 12000[, ,,]d001kuu ut  Copyright © 2011 SciRes. AJCM 136 A. SHAHSAVARAN ET AL. 2111112 21011112[0,,0]d[0, , ,0]d [0,,0, ]d[, ,,]mmkkkmmmkmm mkku utkuutkuutuu umt Now for evaluating the integral at the collocation point s 0() ()dttttBB12jjtk, 1, 2,,jk , (12) we may proceed as follows 1/2 100 02111/2211201() ()() ()d() ()d ()()d ()()d()()d001d0 1d000000jjtttkktkkjjttkkjjkkkkkttdtttt tttttttttttttt BBBB BBBBBB BBt121/21000 1d00000001 1d000jkjkjjkjkttkD (13) where, 1[1, 1,,, 0,,0]2jkkDiag  D, in fact, the diagonal matrix jD, is defined as follows : 1, 2,,j1,1, 2,1,1,,20,1,, .jmnmn jmnjmn jk2 Also, may be approxim ated as: 2(, )[0, 1)Kxt Lkk11(, )()()ij i jijKxt KBxBt , or in matrix form (, )()()tKxtx tBKB, (14) where 1,[]iji jkKK and 11200 (, )()iji jKkKxtBx B . ()ddtxt3. Solution of the Nonlinear Fredholm- Volterra Integral Equations In order to use BPfs for solving nonlinear Fredholm- Voterra integral equations given in Equation (2), we first approximate the , ()ut ()ft, , , (())mux (())nux 1(,Kt )x and 2(, )Ktx with respect to BPfs () ()ut tButt (15) () ()fttBf(()) (mtux xuB (16) 1)(()) ()ntux xuB (17) 2(, )()t (18) 1( )1Ktx tBKxB(, )()()t (19) 22KtxtxBKB (20) where k-vectors , , 1, 2, and matrices and are BPfs coefficients of , ufuukk()ut1K2K()ft, , , 1(()ux)m(()ux)n(, )Ktx and 2(, )Ktx respectively. For solving Equation (2), we substitute (15-20) into (2), therefore 1122 2()()()( )( )d ()()()dtt tttttttxxtxxxtBuBf BKBBuBKBB u0101xk, (21) We now collocate Equation (21) at k points , it1, 2,,j defined by (12) as 1122 2()()()() ()d ()()()dtt ttjj jttjttt xxtxxxjtBuBfBKBB uBKBB u0101x (22) by using (10) and (13) and the fact that ()jjtBe where, je is the j-th column of the identity matrix of order k, Equati on (2 2) may then be restated as 121122 1, 2,,jktj tji jjufkk eKDueKu , . (23) Equation (23) gi ves nonlinear equations w hi ch can kkCopyright © 2011 SciRes. AJCM A. SHAHSAVARAN ET AL. 137Table 1. t Exact A pp roxima t e for 8k Approximate for 16k0.1 –1.99 –1.9847 –1.9876 0.2 –1.96 –1.9505 –1.9532 0.3 –1.91 –1.8857 –1.8905 0.4 –1.84 –1.7905 –1.8122 0.5 –1.75 –1.7650 –1.7666 0.6 –1.64 –1.6650 –1.6589 0.7 –1.51 –1.5091 –1.5080 0.8 –1.36 –1.3205 –1.3342 0.9 –1.19 –1.1103 –1.1297 Table 2. t Exact Approxima t e f o r 8k Approximate for 16k0.1 0.0998 0.0625 0.0936 0.2 0.1986 0.1866 0.2070 0.3 0.2955 0.3078 0.2776 0.4 0.3894 0.4242 0.3952 0.5 0.4794 0.5139 0.5067 0.6 0.5646 0.5339 0.5596 0.7 0.6442 0.6353 0.6514 0.8 0.7173 0.7268 0.7243 0.9 0.7833 0.8069 0.7874 be solved for the elements using Newton’s iterative method. 1u4. Error in BPfs Approximation Theorem. If a differentiable function with bounded first derivative on (0, 1) is represented in a se-ries of BPfs over subinterval ()ut1[, )iikk, we have 1()( )et Ok, where . ()() ()ketututProof. See . 5. Illustrative Examples Consider the following nonlinear volterra-Fredholm in-tegral equations. Example 1. 642120011 55() 3033 4 ()[()]()[()],tuttt tttx uxdxtx uxdx 0t, 1x. We applied the method presented in this paper for solving Equation (2) with and . 8k16kThe computational results together with the exact so-lution are given in Table 1. 2() 2ut tExample 2. 20111() sinsin2[()]d842tuttttux x ,0t,1x. The computational results with and 8k16k together with the exact so lution are given in Table 2. ()ut sint6. Conclusions The aim of present work is to apply a method for solv ing the nonlinear Volterra-Fredholm integral equations. The properties of the Block Pulse functions together with the collocation method are used to reduce the problem to the solution of nonlinear algebraic equations. Example 1 is solved in  using Chebyshev expansion method (Cem), comparing the results shows Cem is more accurate than BPfs method But, it seems the number of calculatio ns of BPfs method is lower. Also, the benefits of this method are low cost of setting up the equations due to properties of BPfs mentioned in Section 2. In addition, the nonlin-ear system of algebraic equations is sparse. Finally, this method can be easily extended and applied to nonlinear Volterra-Fredholm integral equations of the form Equa-tion (1). Illustrative examples are included to demon-strate the validity and app licability of the technique. 7. References  A. Shahsavaran, E. Babolian, “Numerical Implementation of an Expansion Method for Linear Volterra Integral Equations of the Second Kind with Weakly Singular Kernels,” International Journal of Applied Mathematics, Vol. 3, No. 1, 2011, pp. 1-8.  E. Babolian, F. Fattahzadeh and E. G. Raboky, “A Che-byshev Approximation for Solving Nonlinear Integral Equations of Hammerstein Type,” Applied Mathematics and Computation Vol. 189, No. 1, 2007, pp. 641-646. doi:10.1016/j.amc.2006.11.181  F. G. Tricomi, “Integral equations”, Dover, 1982.  H. Brunner, “Implicity Linear Collocation Method for Nonlinear Volterra Equations,” Applied Numerical Ma- thematics, Vol. 9, No. 3-5, 1982, pp. 235-247. doi:10.1016/0168-9274(92)90018-9  L. J. Lardy, “A Variation of Nysrtom’s Method for Hammerstein Integral Equations,” Journal of Integral Equations, Vol. 3, No. 1, 1982, pp. 123-129.  S. Kumar, I. H. Sloan, “A New Collocation—Type Method for Hammerstein Integral Equations,” Journal of Computational Mathematics, Vol. 48, No. 178, 1987, 585-593.  H. Guoqiang, “Asymptotic Error Expansion Variation of A Collocation Method for Volterra—Hammerstein equa-tions,” Applied Numerical Mathematics, Vol. 13, No. 5, 1993, pp. 357-369. doi:10.1016/0168-9274(93)90094-8  Y. Ordokhani, “Solution of Nonlinear Volterra-Fred-Copyright © 2011 SciRes. AJCM A. SHAHSAVARAN ET AL. Copyright © 2011 SciRes. AJCM 138 holm-Hammerstein Integral Equations Via Rationalized Haar Functions,” Applied Mathematics and Computation, Vol. 180, No. 2, 2006, pp. 436-443. doi:10.1016/j.amc.2005.12.034  S. Yousefi and M. Razzaghi, “Legendre Wavelet Method for the Nonlinear Volterra-Fredholm Integral Equations,” Mathematics and Computers in Simulation, Vol. 70, No. 1, 2005, pp. 1-8. doi:10.1016/j.matcom.2005.02.035  S. Yashilbas, “Taylor Polynomial Solution of Non- linear Volterra-Fredholm Integral Equations,” Ap-plied Mathematics and Computation, Vol. 127 No. 2-3, 2002, pp. 195-200. doi:10.1016/S0096-3003(00)00165-X