Solution of First kind Fredholm Integral Equation by Sinc Function

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1 World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Solution of First kind Fredholm Integrl Eqution y Sinc Function Khosrow Mleknejd, Rez Mollpoursl,Prvin Tori nd Mhdiyeh Alizdeh, Interntionl Science Index, Mthemticl nd Computtionl Sciences wset.org/puliction/12240 Astrct Sinc-colloction scheme is one of the new techniques used in solving numericl prolems involving integrl equtions. This method hs een shown to e powerful numericl tool for finding fst nd ccurte solutions. So, in this pper, some properties of the Sinc-colloction method required for our susequent development re given nd re utilized to reduce integrl eqution of the first kind to some lgeric equtions. Then convergence with exponentil rte is proved y theorem to gurntee pplicility of numericl technique. Finlly, numericl exmples re included to demonstrte the vlidity nd pplicility of the technique. Keywords Integrl eqution, Fredholm type, Colloction method, Sinc pproximtion. I. ITRODUCTIO The purpose of this pper is to develop high order numericl methods for Fredholm integrl equtions of the first kind defined y k(s, tf(tdt = g(s < s < (1 where k(s, t nd g(s re known functions nd f(t is the solution to e determined. This type of equtions pper in mny science nd engineering fields, nd in mny cses, we cn not solve this eqution nlyticlly to find n exct solution. So tht, y using numericl methods we try to estimte solution for this eqution. umericl nd theoreticl methods for solving integrodifferentil nd integrl equtions hve een studied y mny uthors so fr [1-9]. Some of them usully use techniques sed on n expnsion in terms of some sis functions or use some qudrture formuls, nd the convergence rte of these methods re usully of polynomil order with respect to, where represents the numer of terms of the expnsion or the numer of points of the qudrture formul. On the other hnd, in [10] it is shown tht if we use the Sinc method the convergence rte is O(exp( c with some c>0. Although this convergence rte is much fster thn tht of polynomil order. So, in the present pper, we pply the Sinc-colloction method which hs exponentil pproximtion rte for solving Eq. (1. Our method consists of reducing the solution of Eq. (1 to set of lgeric equtions y expnding f(t s K. Mleknejd is with the Deprtment of Mthemtics, Irn University of Science & Technology, rmk, Tehrn , Irn e-mil: mleknejd@iust.c.ir (see R. Mollpoursl nd P. Tori re with the Deprtment of Mthemtics, Irn University of Science & Technology, rmk, Tehrn , Irn. M. Alizdeh is with Deprtment of Mthemtics, Islmic Azd University (Krj Brnch, Krj, Irn Sinc functions with unknown coefficients. The properties of the Sinc function re then utilized to evlute the unknown coefficients [1,11-12]. In some ppers such s [13-15] integrl eqution of first nd second kind hve een studied y some uthors, ut in some of these ppers there is no error nlysis which gurntees the convergence of the mentioned scheme. So, in this study theorem is prepred to show convergence nlysis of Sinc colloction method, then some numericl illustrtion exmples re presented to show ccurcy of this technique. II. PRELIMIARIES In this section, we introduce the crdinl function nd some of its properties. For this result sinc(x definition is followed y { sin(πx sinc(x = πx, x 0 1, x =0. ow, for h>0 nd integer k, we define k th Sinc function with step size h y S(k, h(x = sin(π(x kh/h π(x kh/h. The Sinc pproximtion on the entire intervl (, is defined s f(x f(khs(k, h(x. ow, the following Definition nd Theorem will gurntee the pproximtion uthority of Sinc functions on the rel line. Definition 1. Let H 1 (D d denote the fmily of ll functions nlytic in D d defined y D d = {z C: Im(z <d} such tht for 0 <ɛ<1, D d (ɛ is defined y D d (ɛ ={z C: Im(z <d(1 ɛ, Re(z < 1 ɛ } then (f,d d < with (f,d d = lim( f(z dz ɛ 0 D d Theorem 1. Let α, β nd d s positive constnts, tht 1 f H 1 (D d 2 f decys exponentilly on the rel line such tht f(x α exp( β x, x R Interntionl Scholrly nd Scientific Reserch & Innovtion 4( scholr.wset.org/ /12240

2 World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Interntionl Science Index, Mthemticl nd Computtionl Sciences wset.org/puliction/12240 then we hve <x< f(x f(khs(k, h(x C 1/2 exp[ (πdβ 1/2 ] for some C nd step size h is tken s h =( πd β 1/2. Proof: [11], [12] Let t = φ(z denote conforml mp which mps the simply connected domin D with oundry D onto strip region D d such tht φ((, = (,, lim φ(t =, t lim φ(t =. t ow, in order to hve the Sinc pproximtion on finite intervl (, conforml mp is employed s follow φ(x = ln( x x. This mp crries the eye-shped complex domin { z = x + iy : rg(z z <d π }, 2 onto the infinite strip D d = {μ = α + βi : β <d< π 2 }. The sis function on finite intervl (, re given y S(k, h φ(x = sin(π(φ(x kh/h π(φ(x kh/h, lso, Sinc function for interpoltion points x k = kh is given y { S(k, h(jh=δ (0 1, k = j; kj = 0, k j. So, S(k, h φ(x exhiits Kronecker delt ehvior on the grid points x k = φ 1 (kh = + ekh 1+e kh nd interpoltion nd qudrture formuls for f(x over [, ] re f(x f(x k S(k, h φ(x, f(xdx h f(x k φ (x k. Theorem 2. Assume tht, for vrile trnsformtion z = φ 1 (ξ, the trnsformtion function f(φ 1 (ξ stisfies ssumptions (1 nd (2 in Theorem 1. with some α, β nd d. Then we hve f(x f(φ 1 (khs(k, h φ(x <x< C 1/2 exp[ (πdβ 1/2 ] for some C, where the step size h is tken s Proof: [11], [12] h =( πd β 1/2. III. SIC-COLLOCATIO METHOD ow, for solving integrl eqution of the first kind denoted y k(s, tf(tdt = g(s < s < with Sinc pproximtion, we need to chose method to find unknown coefficients in this expnsion. Colloction method is one of the projection methods tht is used s follow: By sustituting Sinc pproximtion expnsion of unknown function f(t in the ove eqution, we hve k(s, t f(φ 1 (khs(k, h φ(t dt = g(s then, define residul function s follow R (s = k(s, t f(φ 1 (khs(k, h φ(t dt g(s. So, to find f(φ 1 (kh in Sinc pproximtion expnsion, there re some techniques such s projection methods like Glerkin nd colloction [16-18]. In this study, colloction method which hs less computtions thn Glerkin is pplied with some colloction points in intervl [, ] for residul function s follows R (s i =0; i =, +1,..., 1, s i = φ 1 (ih = + eih ; 1+eih i =, +1,..., 1, So tht k(s i,t f(φ 1 (khs(k, h φ(t dt = g(s i Then integrl eqution of the first kind is converted to system of liner lgeric equtions A X = where [ ] A = k(s i,ts(k, h φ(tdt, X T = [ f(φ 1 (kh ], = [g(s i ], i =, +1,..., 1,. ow, for evluting mtrix elements of lgeric equtions we hve k(s i,t j S(k, h φ(t j k(s i,ts(k, h φ(tdt h φ, (t j where j= t j = φ 1 (jh= + ejh ; j =, +1,..., 1,. 1+ejh Interntionl Scholrly nd Scientific Reserch & Innovtion 4( scholr.wset.org/ /12240

3 World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Interntionl Science Index, Mthemticl nd Computtionl Sciences wset.org/puliction/12240 IV. COVERGECE AALYSIS In this section, we discuss out convergence of Sinc- Colloction for Fredholm integrl eqution of the first kind. For this result, consider the following theorem. Theorem 3. In Eq. (1 ssume tht, k(s, t is continuous on squre [, ] 2, nd let φ 1 (ξ = + eξ 1+e ξ such tht, f(φ 1 (ξ stisfies in ssumptions (1 nd (2 in Theorem 1. where f is solution of integrl eqution. Also let P num (f(t = f num (φ 1 (khs(k, h φ(t s Sinc pproximtion of f with step size h nd f num (φ 1 (kh re unknown coefficients which will e determined y solving system of lgeric eqution A X =. ow, if A is nonsingulr then f P num (f [c 1 +c 3 A 1 ] 1/2 exp{ c 2 1/2 } for some positive constnt c 1,c 2,c 3. Proof: Let f(t P num (f(t = f num (φ 1 (khs(k, h φ(t where f num (φ 1 (kh re unknown coefficients which re found y solving liner system of equtions [19]. Also, ssume P (f(t = f(φ 1 (khs(k, h φ(t nd in this pproximtion f(φ 1 (kh is the exct vlue of f t φ 1 (kh. If sustitute P num (f(t s n pproximtion of f(t in Eq. (1 then g(s = ut if use P (f(t then otin ĝ(s = k(s, tp num (f(tdt (2 k(s, tp (f(tdt. (3 ow, y converting Eq. (2 to liner system then y solving this system we hve [f num (φ 1 (kh] = A 1 [g(s i] i= ut, y pplying numericl scheme which ws discussed in previous section to Eq. (3 we hve So tht [f(φ 1 (kh] = A 1 [ĝ(s i] i=. ow, we hve then let k(s, tp (f(tdt = g(s ĝ(s =g(s so tht we hve k(s, t[f(t P (f(t]dt k(s, t[f(t P (f(t]dt ĝ(s i g(s i = i S s [,] k(s, t[f(t P (f(t]dt ( k(s, t f P (f t,s [,] Since k(s, t is continuous on [, ] 2 so tht let M = k(s, t t,s [,] lso regrding to Theorem (2 we hve so, f P (f c 1 1/2 exp{ c 2 1/2 } g(s i ĝ(s i ( Mc 1 1/2 exp{ c 2 1/2 }. (5 i S Finlly, y sustituting Eq. (5 in Eq. (4 we cn derive f num (φ 1 (kh f(φ 1 (kh k S c 3 A 1 1/2 exp{ c 2 1/2 }. Also, we need to determine ound for P (f(t (f(t hence P num P (f(t P num (f(t t [,] = [f(φ 1 (kh f num (φ 1 (kh]s(k, h φ(t t [,] Also, in [10] f(φ 1 (kh f num (φ 1 (kh A 1 c 3 1/2 exp{ c 2 1/2 } t [,] t [,] t [,] S(k, h φ(t 2 {3 + log(} π S(k, h φ(t S(k, h φ(t for sufficiently lrge so, it is possile to replce 2 π {3+ log(} y so tht t [,] finlly, f num (φ 1 (kh f(φ 1 (kh A 1 g(s i ĝ(s i k S i S (4 where S is ll integers elong to [,]. P (f(t P num (f(t c 3 A 1 3/2 exp{ c 2 1/2 } f(t P num (f(t f(t P (f(t + P (f(t P num (f(t c 1 1/2 exp{ c 2 1/2 } + c 3 A 1 3/2 exp{ c 2 1/2 } = 1/2 exp{ c 2 1/2 }[c 1 + c 3 A 1 ], Interntionl Scholrly nd Scientific Reserch & Innovtion 4( scholr.wset.org/ /12240

4 World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Interntionl Science Index, Mthemticl nd Computtionl Sciences wset.org/puliction/12240 nd proof of this theorem is completed. V. UMERICAL EXAMPLES Aim of this section is to show efficiency nd ccurcy of numericl method which is discussed in previous sections. So tht, Eq. (1 is discredited y Sinc-colloction method nd converted to system of lgeric equtions nd then it is solved to find numericl solution for Eq. (1. Exmple1. In this exmple, let π 2 0 Fig. 2. Error Function for =25. (sin(s t+cos(t sf(tdt = cos(s sin(s, where the exct solution is f(t = sin(t(1 sin(t. Tht is esy to show tht this exct solution stisfies ssumptions (1 nd (2 in Theorem 1. for α = β =1, lso let d = π 2, so step size for Sinc function is h = π. (2 1 2 ow let E( = mx f(t j P num (f(t j j where t j s re colloction nodes, nd ssume Cond(A = A A 1 where A is coefficient mtrix in lgeric system of equtions. Then umericl results re shown for different vlues of in Tle 1. E( Cond(A TABLE I UMERICAL RESULTS FOR EXAMPLE 1 Exmple 2. In this exmple, we hve 1 0 (sin(tt 3 + s 2 f(tdt = s 2 nd the exct solution is f(t =t(1 t. This is cler tht f(t stisfies ssumption (1 nd (2 is Theorem 1. Also, let e(t = f(t P num (f(t, t [, ] s solute error function. Error function hs een drwn for different vlues of nd results re shown in Figures 1, Fig. 1. Error Function for = COCLUSIO Properties of the Sinc-colloction method re utilized to reduce the computtion of this prolem to some lgeric eqution nd then get the numericl results with high ccurcy nd little computtionl efforts. Our method is shown to e of good convergence, esy to progrm. So, we expect our method cn e extended to the nonliner Fredholm nd Volterr type equtions. This is left for our next pper. REFERECES [1] M. Muhmmd, A. urmuhmmd, M. Mori, M. Sugihr, umericl solution of integrl equtions y mens of the Sinc colloction method sed on the doule exponentil trnsformtion, Journl of Computtionl nd Applied Mthemtics 177 ( [2] K. Mleknejd, K. ouri, M. osrti Shln, Convergence of pproximte solution of nonliner Fredholm-Hmmerstein integrl equtions, Communictions in onliner Science nd umericl Simultion, (In Press. [3] M. Rsty, M. Hdizdeh, A product integrtion pproch sed on new orthogonl polynomils for nonliner wekly singulr integrl equtions, Act Appl. Mth. (In Press. [4] K. Mleknejd, M. osrti, The Method of Moments for Solution of Second Kind Fredholm Integrl Equtions Bsed on B-Spline Wvelets Interntionl Journl of Computer Mthemtics, (In Press. [5] Adel Mohsen nd Mohmed El-Gmel, A Sinc-Colloction method for the liner Fredholm integro-differentil equtions, Z. Angew. Mth. Phys. 58 ( [6] K. Mleknejd, K. ouri, R. Mollpoursl, Existence of solutions for some nonliner integrl equtions, Communictions in onliner Science nd umericl Simultion, 14 (2009, [7] W. Volk, The iterted Glerkin methods for liner integro-differentil equtions, J. Comp. Appl. Mth. 21 (1988, [8] K. Mleknejd, F. Mirzee, Using rtionlized Hr wvelet for solving liner integrl equtions, Appl. Mth. Comp., 160 (2005, [9] A. Avudinygm, C. Vni, Wvelet Glerkin method for integrodifferentil equtions, Appl. umer. Mth. 32 (2000, [10] F. Stenger, umericl Methods Bsed on Sinc nd Anlytic Functions, Springer, ew York, [11] M. Sugihr, T. Mtsuo, Recent developments of the Sinc numericl methods, J. Comput. Appl. Mth ( [12] M.Sugihr, er optimlity of the Sinc pproximtion, Mth. Comput. 72 ( [13] X. Shng, D. Hn, umericl solution of Fredholm integrl equtions of the first kind y using liner Legendre multi-wvelets, Applied Mthemtics nd Computtion, 191, (2007, [14] E. Bolin, Z. Msouri, Direct method to solve Volterr integrl eqution of the first kind using opertionl mtrix with lock-pulse functions, Journl of Computtionl nd Applied Mthemtics, 220, (2008, [15] E. Bolin, T. Lotfi, M. Pripour, Wvelet moment method for solving Fredholm integrl equtions of the first kind, Applied Mthemtics nd Computtion, 186, (2007, [16] C.T.H. Bker, The numericl tretment of integrl equtions, Clrendon Press, Oxford, [17] Kendll E. Atkinson, The umericl Solution of Integrl Equtions of the Second Kind, Cmridge University Press, [18] R. Kress, Liner Integrl Eqution, Springer-Verlg, ew York, Interntionl Scholrly nd Scientific Reserch & Innovtion 4( scholr.wset.org/ /12240

5 World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences [19] A. Kroui, Wvelets: Properties nd pproximte solution of second kind integrl eqution, Computers & Mthemtics with Applictions, 46 (2003, Interntionl Science Index, Mthemticl nd Computtionl Sciences wset.org/puliction/12240 Khosrow Mleknejd received the M.S. degree in pplied mthemtics from Tehrn University, Irn, in 1972 nd the PhD degree in numericl nlysis from university of Wles, Aerystwyth, UK in In Septemer 1976, he joined the fculty of the Bsic Science, Deprtment of Applied Mthemtics t Irn University of Science & Technology; he is currently professor since During , he lso served s vice-chir for grdute students. He ws visiting professor t university of Cliforni t Los Angeles (UCLA in His reserch interests re in numericl nlysis in solving Ill-posed prolems nd solving Fredholm nd Volterr integrl equtions. He hs uthored or couthored more thn 160 reserch ppers on these topics. He is n editor-in-chief of Interntionl Journl of Mthemticl Sciences nd memer of editoril ord of some journls. He is memer of the AMS. His pper ws selected s the est pper in 34th Annul Irnin Mthemtics Conference, 30 Aug-2 Sep, Shhrood University, Irn, Rez Mollpoursl received the B.Sc, M.Sc nd Ph.D degrees in pplied mthemtics from Irn University of Science & Technology, Irn, in 2003, 2005 nd 2009 respectively. His reserch interests re studying on theoreticl nd numericl solution of some integrl nd dely differentil equtions. He hs pulished some reserch rticles in numericl nd nlyticl points of view of the integrl nd differentil equtions in some journls nd interntionl conferences. Prvin Tori received the B.Sc degree in pplied mthemtics from Isfhn University of Technology, Irn in 2001 nd M.Sc degree in pplied mthemtics (optiml control from Shhid Chmrn University of Ahvz, Irn in Her reserch interests re studying on numericl solution of some integrl equtions nd control theory. She hs pulished some reserch rticles in numericl solution of integrl equtions in some interntionl conferences. Mhdiyeh Alizdeh received the B.Sc degree in pplied mthemtics from Islmic Azd University of Kermn, Irn in 2003 nd M.Sc degree in pplied mthemtics (umericl nlysis from Islmic Azd University of Krj, Irn in Her reserch interests re studying on numericl solution of some integrl nd differentil equtions. She hs pulished some reserch rticles in numericl solution of integrl equtions in some journls nd interntionl conferences. Interntionl Scholrly nd Scientific Reserch & Innovtion 4( scholr.wset.org/ /12240