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||k1 <1 , T , (1)( , 1997)1. 2{ei (x)}ni=1 , n Xn , (x)n(x)=ni=1ciei (x) (4) ci v X =C [ a , b] , X nPn v=nj=1v(xj)ej(x) (5) xj (1)Pn(x)=PnbaF (x, y , (y))dy +Pn f(x)n(x)=baPnF(x, y, (y))dy +Pn f(x)(4)(5) ,ni=1ciei(x)=ba ni=1F(xi , y , n(y))ei(x)dy +ni=1f(xi)ei(x), i=1, 2, , nni=1{ci -baF(xi , y, n(y))dy – f(xi)}ei(x)=0 ei(x) ,ci -baF(xi , y, n(y))dy – f(xi)=0 (6) i=1, 2, , n (6) , ,(6) ( , 2000) (6)In(x)=Nj=1jx(tj)baF(xi , y, n(y))dy =Nj=1jF(xi , yj , n(yj))=Nj=1jF(xi , yj , ni=1ciei(yj))(6)ci -Nj=1jF(xi , yj , Ni=1ciei(yj)) – f(xi)=0 (7) i=1, , n (6)(7) (7)F I(c1 , c2 , , cn)=0; i=1, 2, , n2 , Matlab ( , 2002), (x)=10 2xt14+2(t)dt+x2-x2arctan12(8)(8) (x)=x2 ,( n):e1(x)=0 , x2x1n(x2 -x), x1xx2ei(x)=0, 0xxi – 11n(x -xi -1), xi -1xxi1n(xi+1 – x), xixxi+10, xi+1x1en(x)=1n(x – xn -1), xn – 1xxn0, xxn – 1(6)(N =3):10F(x)dx118[ 5F (x1) +8F(x2)+5F (x3)] , x1 =0.113 , x2 =0. 5 , x3 =0. 8878 n =100 n =200 , 1 2 1(8)(n=100, N =3)F ig. 1Compar ison o f two so lu tions o fequa tion(8) (n =100, N =3) 2(8)(n=200, N =3)F ig. 2Compar ison o f two so lu tions o fequa tion(8)(n=200, N =3) 2 , n=200 , . 1994.[ J] . , 4:418 431.. 1995. Fredholm [ J] . , 17(2):20 22. , ,. 2000. [M ] . :.295 3: Fredholm ,. 2003. – [ J] . , 26(2):115 117. , ,. 2002. Lap lace Cau chy [ J] . , 25(4):356 360.. 1997. Numerical S olu tions for Non linear F redholm In tegra l E-quation s of the Second k ind and Their Superconvergence[ J] . Jour-na l of ShangH aiUn iversi ty, 1(2):98 104.Numerical Solutions of Nonlinear Fredhoml IntegralEquations of the Second K indLIU Tang-we i , Y ING Zheng-wei, WU Zh i-qiang(Faculty ofM athematics and Informa tion Science, EastChina Institu te o f Technology, Fuzhou, JX 344000 , China)Abstract:The co llocationme thod is used to so lve non linear Fredho lm integ ral equa tions o f the Second kind in ge-ophy sical prob lems. W e transfo rm non linear F redho lm In tegra l Equa tions into the nonlinear algebraic equations,then so lve the algebraic equaitons. Numerical experiments are g iven w ith Gauss numerica l integ ra l formulas. Pro-gramm ing is based on the symbo l func tion o fM atlab softw are. This pape r so lve s the prob lem of hard-programm ingof the nonlinear equa tions. Numerical experimen ts show the efficiency o f the method. The article is va luable forsolv ing non linear in teg ra l equa tionsand nonlinear algebraic equa tions.KeyW ords:non linear Fredholm integ ral equations; the co llocation method ;numerica l so lu tions;Matlab soft-ware.296 2005