A Numerical Method for Solving Nonlinear Integral Equations

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1 Interntionl Mthemticl Forum, 4, 29, no. 17, A Numericl Method for Solving Nonliner Integrl Equtions F. Awwdeh nd A. Adwi Deprtment of Mthemtics, Hshemite University, Jordn wwdeh@hu.edu.jo, dwi@hu.edu.jo S. Al-Shr Deprtment of Mthemtics, Tfil Technicl University, Jordn sshr@ttu.edu.jo Abstrct In this pper, n itertive scheme bsed on the homotopy nlysis method HAM hs been used to solve nonliner integrl equtions. To check the numericl method, it is pplied to solve different test problems with known exct solutions nd the numericl solutions obtined confirm the vlidity of the numericl method nd suggest tht it is n interesting nd vible lterntive to existing numericl methods for solving the problem under considertion. Convergence is lso observed. Mthemtics Subject Clssifiction: 65H2, 65H5, 45G5, 45G1 Keywords: Homotopy Anlysis Method; Nonliner Integrl Equtions; Numericl Method 1 Introduction Integrl equtions ply n importnt role in mny brnches of liner nd nonliner functionl nlysis nd their pplictions in the theory of elsticity, engineering, mthemticl physics, potentil theory, electrosttics nd rditive het trnsfer problems. Therefore, mny different methods re used to obtin the solution of the liner nd nonliner integrl equtions. Some different vlid methods for solving integrl eqution hve been developed in the lst yers [1 8]. In [2], Brunner et l., introduced clss of methods depending on some prmeters to obtin the numericl solution of Abel integrl eqution of the second kind. The liner multistep methods were

2 86 F. Awwdeh, A. Adwi nd S. Al-Shr pplied in [7], to obtin the numericl solution of singulr nonliner Volterr integrl eqution. Also, in [8], Kilbs nd Sigo used n symptotic method to obtin numericlly the solution of nonliner Abel Volterr integrl eqution. In [9], Orsi used Product Nyström method, s numericl method, to obtin the solution of nonliner Volterr integrl eqution, when its kernel tkes logrithmic nd Crlemn forms. Vritionl itertion method [13], Homotopy perturbtion method [5-6] nd Adomin decomposition method [1] re effective nd convenient for solving integrl equtions. The homotopy nlysis method HAM [9-12] is generl nlytic pproch to get series solutions of vrious types of nonliner equtions, including lgebric equtions, ordinry differentil equtions, prtil differentil equtions, differentil-difference eqution. More importntly, different from ll perturbtion nd trditionl non-perturbtion methods, the HAM provides us simple wy to ensure the convergence of solution series, nd therefore, the HAM is vlid even for strongly nonliner problems. The HAM is bsed on homotopy, fundmentl concept in topology nd differentil geometry. Briefly speking, by mens of the HAM, one constructs continuous mpping of n initil guess pproximtion to the exct solution of considered equtions. An uxiliry liner opertor is chosen to construct such kind of continuous mpping, nd n uxiliry prmeter is used to ensure the convergence of solution series. The method enjoys gret freedom in choosing initil pproximtions nd uxiliry liner opertors. By mens of this kind of freedom, complicted nonliner problem cn be trnsferred into n infinite number of simpler, liner sub-problems, s shown by Lio nd Tn [12]. Until recently, the ppliction of the homotopy nlysis method in nonliner problems hs been devoted by scientists nd engineers, becuse this method is to continuously deform simple problem esy to solve into the difficult problem under study. In this pper, we use the HAM for nonliner integrl equtions such tht yx =gx+ Kx, tfytdt, where the upper limit my be either vrible or fixed, the kernel of the integrlkx, t nd gx re known functions, fy is known function of y nd yx is unknown function tht will be determined. Some exmples re tested, nd the obtined results suggest tht newly improvement technique introduces promising tool nd powerful improvement for solving integrl equtions.

3 Nonliner integrl equtions 87 2 Description of the Method Consider N[y] =yx gx Kx, tfytdt =, 1 where N is n opertor, yx is unknown function nd x the independent vrible. Let y x denote n initil guess of the exct solution yx, h n uxiliry prmeter, Hx n uxiliry function, nd L n uxiliry liner opertor with the property L[rx] = when rx =. Then using q [, 1] s n embedding prmeter, we construct such homotopy 1 ql[φx; q y x] qhhxn[φx; q] = Ĥ[φx; q; y x,hx,h,q]. 2 It should be emphsized tht we hve gret freedom to choose the initil guess y x, the uxiliry liner opertor L, the non-zero uxiliry prmeter h, nd the uxiliry function Hx. Enforcing the homotopy 2 to be zero, i.e., Ĥ[φx; q; y x,hx,h,q]= we hve the so-clled zero-order deformtion eqution 1 ql[φx; q y x] = qhhxn[φx; q]. 3 When q =, the zero-order deformtion eqution 3 becomes φx;=y x, 4 nd when q = 1, since h nd Hx, the zero-order deformtion eqution 3 is equivlent to φx;1=yx. 5 Thus, ccording to 4 nd 5, s the embedding prmeter q increses from to 1, φx; q vries continuously from the initil pproximtion y x to the exct solution yx. Such kind of continuous vrition is clled deformtion in homotopy. By Tylor s theorem, φx; q cn be expnded in power series of q s follows φx; q =y x+ y m xq m 6

4 88 F. Awwdeh, A. Adwi nd S. Al-Shr where y m x = 1 m φx; q m! q m q=. 7 If the initil guess y x, the uxiliry liner prmeter L, the nonzero uxiliry prmeter h, nd the uxiliry function Hx re properly chosen so tht the power series 6 of φx; q converges t q =1.Then, we hve under these ssumptions the solution series For brevity, define the vector yx =φx;1=y x+ y m x. 8 y n x ={y x,y 1 x,y 2 x,...,y n x}. 9 According to the definition 7, the governing eqution of y m x cn be derived from the zero-order deformtion eqution 3. Differentiting the zeroorder deformtion eqution 3 m times with respective to q nd then dividing by m! nd finlly setting q =, we hve the so-clled mth-order deformtion eqution where nd L[y m x χ m y m 1 x] = hhxr m y m 1 x, 1 y m =, R m y m 1 x = 1 m 1 N[φx; q] m 1! q m 1 q= 11 χ m = {, m 1 1, m > 1. Note tht the high-order deformtion eqution 1 is governing by the liner opertor L, nd the term R m y m 1 x cn be expressed simply by 11 for ny nonliner opertor N. According to the definition 11, the right-hnd side of eqution 1 is only dependent upon y m 1 x. Therefore y m x cn be esily gined, especilly by mens of computtionl softwre such s MATLAB. The solution yx given by the bove pproch is dependent of L, h, Hx, nd y x. Thus, unlike ll previous nlytic techniques, the convergence region nd rte of solution series given by the bove pproch might not be uniquely determined.

5 Nonliner integrl equtions 89 Here, we rigorous definitions nd then give some properties of the homotopyderivtive. These properties re useful to deduce the high-order deformtion equtions nd provide us with simple nd convenient wy to pply the HAM to nonliner problems. Let φ be function of the homotopy-prmeter q, then D m φ = 1 m φx; q m! q m q= is clled the mth-order homotopy-derivtive of φ, where m is n integer. According to the Leibnitz s rule for derivtives nd using the induction, one cn show the following properties of the homotopy-derivtive. Theorem 1 For homotopy-series φx; q = y m xq m, ψx; q = v k xq k, k= where m, l, n nd k m re positive integers, then 1 For p, positive integer, it holds m r 1 r 2 D m φ p = y r2 r 3... y m r1 y r1 r 2 r 1 = r 2 = r 3 = r pij 3 r pij 2 y rp 3 r p 2 r pij 2= r pij 1= y rp 2 r p 1 y rp 1. 2 D m fφ+ gψ =fd m φ+gd m ψ. 3 D m φ =y m. 4 D m q k φ=d m k φ. 5 D m φψ = m D i φd m i ψ = m D i ψd m i φ. i= i= 6 D m φ n ψ l = m D i φ n D m i ψ l = m D i ψ l D m i φ n. i= 7 it holds the recurrence formuls D e φ = e y, D m e φ = m 1 k= 8 it holds the recurrence formuls D sin φ = siny, D m sin φ = m 1 k= i= 1 k D k e φ D m k φ, for m 1. m 1 k m D k cos φd m k φ, for m 1.

6 81 F. Awwdeh, A. Adwi nd S. Al-Shr 9 it holds the recurrence formuls D cos φ = cosy, D m cos φ = m 1 k= 1 k m 3 Computtionl procedure D k sin φd m k φ, for m 1. In this section we will use the HAM pproch to consider nonliner integrl equtions of the type: yx =gx+ Kx, t[yt] p dt, 12 where the upper limit my be either vrible or fixed, p is positive integer, the kernel kx, t nd gx re known functions, wheres y is to be determined. Let N[y] =yx gx Kx, t[yt] p dt, The corresponding mth-order deformtion eqution 1 reds L[y m x χ m y m 1 x] = hhxr m 1 y m 1 x, 13 y m =, where nd R m 1 y m 1 x = y m 1 1 χ m g Kx, tr m 1 φ p dt R m φ p = m r 1 r 2 y m r1 y r1 r 2 r 1 = r 2 = r 3 = y r2 r 3... r p 3 r p 2 y rp 3 r p 2 r p 2 = r p 1 = y rp 2 r p 1 y rp 1. To obtin simple itertion formul for y m x, choose Ly = y s n uxiliry liner opertor, s zero-order pproximtion to the desired function yx, the solution y x =gx, is tken, the nonzero uxiliry prmeter h

7 Nonliner integrl equtions 811 nd the uxiliry function Hx, cn be tken s h = 1 nd Hx = 1. This is substituted into 13 to obtin y x = gx, y m x = Kx, tr m 1 φ p dt, m =1, 2, 3,.... The corresponding homotopy-series solution is given by yx = y m x 14 4 Convergence Anlysis Theorem 2 The Integrl eqution yx =gx+ Kx, tfytdt, 15 with the kernel Kx, t stisfies Kx, t <M for ll x, t [, b] [, b], gx is given continuous function defined on [, b] nd fy is Lipschitz continuous with fy fz L y z, hs unique solution whenever <α<1, where, α = LMb. Proof. Consider the spce C[, b] of ll continuous functions defined on the intervl [, b] with metric d given by dx, y = mx xt yt. Obviously, t [,b] 15 cn be written y = Ty where Tyx =gx+ Kx, tfytdt. Let y 1 nd y 2 be two different solutions to 15 then Ty 1 x Ty 2 x = Kx, t[fy 1 fy 2 ]dt Kx, t fy 1 fy 2 dt LMdy 1,y 2 dt αdy 1,y 2.

8 812 F. Awwdeh, A. Adwi nd S. Al-Shr Since <α<1, T becomes contrction nd the Bnch s fixed point theorem completes the proof. As the function fx =x p is Lipschitz continuous, the integrl eqution 12 hs unique solution. Theorem 3 Let Sx = y n x. Then for k 2, where k is n integer, n= R m φ k =S k x. Proof. The proof by induction on the k. From Theorem 1, for k = 2, we hve m R m φ 2 = y m j y j. j= Thus m R m φ 2 = y m j y j = = j= y m j y j j= m=j m y j y m j j= m=j = S 2. Put φ k+1 = φ k φ 1, with the help of Theorem 1, we obtin m R m φ k+1 = y m j R j φ k This ends the proof. = = j= y m j R j φ k j= m=j R j φ k j= = S k S = S k+1. m m=j y m j Theorem 4 As long s the series 14 converges, it must be the exct solution of the integrl eqution 12.

9 Nonliner integrl equtions 813 Proof. If the series 14 converges, we cn write Sx = y m x, nd it holds tht We cn verify tht lim m y mx =. 16 n [y m x χ m y m 1 x] = y 1 +y 2 y 1 + +y n y n 1 = y n x, which gives us, ccording to 16, [y m x χ m y m 1 x] = lim y n x =. 17 n Furthermore, using 17 nd the definition of the liner opertor L, we hve L[y m x χ m y m 1 x] = L[ [y m x χ m y m 1 x]] =. In this line, we cn obtin tht L[y m x χ m y m 1 x] = hhx R m 1 y m 1 x =, which gives, since h nd Hx, tht R m 1 y m 1 x =. 18 Substituting R m 1 y m 1 x into the bove expression, recll Theorem 3, nd

10 814 F. Awwdeh, A. Adwi nd S. Al-Shr simplifying it, we hve R m 1 y m 1 x = [y m 1 1 χ m g = = y m x gx y m x gx Kx, t Kx, t[ Kx, tr m 1 φ p dt] R m 1 φ p dt y m t] p dt = Sx gx Kx, t[st] p dt 19 From 18 nd 19, we hve Sx =gx+ Kx, t[st] p dt, nd so, Sx must be the exct solution of Eq Numericl Results nd Discussion The HAM provides n nlyticl solution in terms of n infinite power series. However, there is prcticl need to evlute this solution. The consequent series trunction, nd the prcticl procedure conducted to ccomplish this tsk, together trnsforms the nlyticl results into n exct solution, which is evluted to finite degree of ccurcy. In order to investigte the ccurcy of the HAM solution with finite number of terms, two exmples were solved. To show the efficiency of the present method for our problem in comprison with the exct solution we report bsolute error which is defined by Ey m HAM = y exct y m HAM where yham m = m y i x. MATLAB 7 is used to crry out the computtions. i= Exmple 1. Consider the nonliner Fredholm integrl eqution yx = lnx ln 21 x ln 2 + x 2x x t y 2 t dt.

11 Nonliner integrl equtions 815 For which the exct solution is yx = lnx +1. We begin with y x = lnx ln 21 x ln 2 + x 2x 5. Its itertion formultion 4 reds 1 y m x = [x t m 1 j= y j ty m j 1 t]dt, m =1, 2,.... Some numericl results of these solutions re presented in Tble 1. Exmple 2. The presented HAM itertive scheme is pplied for solving the nonliner integrl eqution yx = sinπx cosπx sinπt y 3 t dt. The exct solution to this eqution is yx = sinπx cosπx. The formuls corresponding to this problem re y x = sinπx 1 y m x = 1 [cosπx sinπt 5 m 1 i= y m i 1 i j= y j y i j ]dt, m =1, 2,.... Tble 2. shows bsolute errors of numericl results clculted ccording the presented method. Tble1. Numericl results of Exmple 1 x i y exct yham 15 y exct y HAM E E E E E E E E E E E 1

12 816 F. Awwdeh, A. Adwi nd S. Al-Shr Tble2. Numericl results of Exmple 2 x i y exct y 15 HAM y exct y Conclusion HAM E E E E E E E E E E 15 The proposed method is powerful procedure for solving integrl equtions. The exmples nlyzed illustrte the bility nd relibility of the method presented in this pper nd revels tht this one is very simple nd effective. The obtined solutions, in comprison with exct solutions dmit remrkble ccurcy. Results indicte tht the convergence rte is very fst, nd lower pproximtions cn chieve high ccurcy. References [1] S. Abbsbndy, Numericl solution of integrl eqution: Homotopy perturbtion method nd Adomin s decomposition method, Appl. Mth. Comput [2] H. Brunner, M.R. Crisci, E. Russo, A. Recchio, A fmily of methods for Abel integrl equtions of the second kind, J. Comput. Appl. Mth [3] N. Bildik, A. Konurlp, The use of vritionl itertion method, differentil trnsform method nd Adomin decomposition method for solving different types of nonliner prtil differentil equtions, Interntionl Journl of Nonliner Sciences nd Numericl Simultion [4] Driusz Bugjewski, On BV-Solutions of some nonliner integrl equtions, Integrl Equtions nd Opertor Theory

13 Nonliner integrl equtions 817 [5] M. El-Shhed, Appliction of He s homotopy perturbtion method to Volterr s integro-differentil eqution, Interntionl Journl of Nonliner Sciences nd Numericl Simultion [6] A. Golbbi, B. Kermti, Modified homotopy perturbtion method for solving Fredholm integrl equtions, Chos Solitons & Frctls 26, doi:1.116/j.chos [7] J.P. Kuthen, A survey of singulr perturbed Volterr equtions, Appl. Num. Mth [8] A.A. Kilbs, M. Sigo, On solution of nonliner Abel Volterr integrl eqution, J. Mth. Anl. Appl [9] S.J. Lio, Beyond Perturbtion: Introduction to the Homotopy Anlysis Method, Chpmn & Hll/CRC Press, Boc Rton, 23. [1] S.J. Lio, On the homotopy nlysis method for nonliner problems, Appl. Mth. Comput [11] S.J. Lio, Notes on the homotopy nlysis method: some definitions nd theorems, Communictions in Nonliner Science nd Numericl Simultion 28, doi: 1.116/j.cnsns [12] S.J. Lio nd Y. Tn, A generl pproch to obtin series solutions of nonliner differentil equtions. Studies in Applied Mthemtics, [13] X. Ln, Vritionl itertion method for solving integrl equtions, Computers nd Mthemtics with Applictions Received: August, 28