New implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations

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1 014 (014) 1-7 Avilble online t Volume 014, Yer 014 Article ID cn-0005, 7 Pges doi: /014/cn-0005 Reserch Article ew implementtion of reproducing kernel Hilbert spce method for solving clss of functionl integrl equtions Shhnm Jvdi 1, Esmil Bbolin 1, Eslm Mordi 1 (1) Deprtment of Mthemtics, Fculty of Mthemticl Sciences nd Computer, Khrzmi University, Tehrn, Irn Copyright 014 c Shhnm Jvdi, Esmil Bbolin nd Eslm Mordi. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct In this pper, we pply the new implementtion of reproducing kernel Hilbert spce method to give the pproximte solution to some functionl integrl equtions of the second kind. To show its effectiveness nd convenience, some exmples re given. Keywords: Fredholm integrl eqution, Volterr integrl eqution, Reproducing kernel Hilbert spce method. 1 Introduction Integrl equtions (IEs) hve n importnt role in the fields of science nd engineering [1,, 3]. Some boundry vlue problems rising in electromgnetic theory led to the problem of solving functionl IEs [4]. Functionl IEs rise in solid stte physics, plsm physics, quntum mechnics, strophysics, fluid dynmics, cell kinetics, chemicl kinetics, the theory of gses, mthemticl economics, hereditry phenomen in biology. Some nlyticl nd numericl methods hve been developed for obtining pproximte solutions to IEs. For instnce we cn mention the following works. Bbolin et l. [5] pplied numericl method for solving clss of functionl nd two dimensionl integrl equtions, Abbsbndy [6] used Hes homotopy perturbtion method for solving functionl integrl equtions, Rshed [7] used Lgrnge interpoltion nd Chebyshev interpoltion for obtining numericl solution of functionl differentil, integrl nd integro-differentil equtions. In 1986, Cui Minggen [8] introduced the reproducing kernel spce W 1 [,b] nd its reproducing kernel. This technique hs successfully been treted singulr liner two-point BVP [9, 10], singulr nonliner two-point periodic BVP [11, 1], nonliner system of BVP [13], fifth-order BVP [14], singulr intergl equtions [15, 16], nd nonliner prtil differentil equtions [17] in recent yers. This pper investigtes the pproximte solution of the following Fredholm nd Volterr functionl integrl equtions using new implementtion of reproducing kernel Hilbert spce method (RKHSM) nd y(x) + p(x)y(h(x)) + λ k(x,t)y(t)dt = f (x), x b, (1.1) x y(x) + p(x)y(h(x)) + λ k(x,t)y(t)dt = f (x), x b, (1.) Corresponding uthor. Emil ddress: eslm.mordi@gmil.com, Tel:

2 Pge of 7 where p(x), h(x), f (x) re nlyticl known functions defined on the intervl [,b], k(x,t) is given continuous function on the region [,b] [,b], unknown function y(x) is continuous on the intervl [,b] nd λ is given constnt. Obviously, if we put p(x) = 0 in Eqs. (1.1) or (1.), then Fredholm or Volterr integrl equtions of the second kind would derived. As we known, Grm - Schmidt orthogonliztion process is numericlly unstble nd in ddition it my tke lot of time to produce numericl pproximtion. Here, insted of using orthogonliztion process, we successfully mke use of the bsic functions which re obtined by RKHSM. The structure of this pper is orgnized s follows. In the following section, we introduce the reproducing kernel spce W 1 [,b]. Section 3 is devoted to solve Eqs. (1.1) nd (1.) by new implementtion of RKHSM. Some numericl exmples re presented in Section 4. We end the pper with few conclusions. Reproducing kernel spce W 1 [,b] Definition.1. [18] For nonempty set X, let (H,.,. H ) be Hilbert spce of rel-vlued functions on some set X. A function k : X X R is sid to be the reproducing kernel of H if nd only if 1. k(x,.) H, x X,. φ(.),k(x,.) H = φ(x), φ H, x X (reproducing property). Also, Hilbert spce of functions (H,.,. H ) tht possesses reproducing kernel k is Reproducing kernel Hilbert spce (RKHS); we denote it by (H,.,. H,k). In the following we often denote by k x the function k(x,.) : t k(x,t). Definition.. [8, 18] W 1 [,b] = {y(x) y(x) is n bsolute continuous rel vlued functions on the intervl [,b] nd y (x) L [,b]}. The inner product nd the norm in the function spce W 1 [,b] re defined s follows. u,v W 1 = u()v() + u (ζ )v (ζ )dζ, u W 1 = u,u W 1. Let s ssume tht function R x (t) W 1 [,b] stisfies the following generlized differentil equtions { = δ(t x), R x () R x() = 0, R x (t) R x (b) = 0, where δ is Dirc delt function. Therefore, the following theorem holds. Theorem.1. Under the ssumptions of Eq. (.3), Hilbert spce W 1 [,b] is RKHS with the reproducing kernel function R x (t), nmely for ny y(t) W 1[,b] nd ech fixed x [,b], there exists R x(t) W 1 [,b], t [,b, such tht y(t),r x (t) W 1 = y(x). Proof. Applying integrtion by prts six times, since R x (t) W 1 [,b], we hve y(t),r x (t) W 1 = y()r x () + + y() [ R x () R x() y (t)( R x(t) ] )dt = y(b) R x(b) y(t)( R x (t) )dt. (.3) So, Eq. (.3) imply tht y(t),r x (t) W 1 = y(t)δ(t x)dt = y(x).

3 Pge 3 of 7 While x t, function R x (t) is the solution of the following constnt liner homogeneous differentil eqution with orders, with the boundry conditions: R x (t) = 0, (.4) R x () R x() = 0, R x (b) = 0. (.5) We know tht Eq. (.4) hs chrcteristic eqution λ = 0, nd the eigenvlue λ = 0 is root whose multiplicity is. Hence, the generl solution of Eq. (.3) is R x (t) = { α1 (x) + α (x)t, t x, β 1 (x) + β (x)t, t > x. (.6) ow, we re redy to clculte the coefficients α i (x) nd β i (x), i = 1,. Since we hve R x (x ) = R x (x + ), R x (t) R x (x ) = δ(t x), R x(x + ) = 1. (.7) Then, using Eqs. (.5) nd (.7), the unknown coefficients of Eq. (.6) re uniquely obtined. Therefore { 1 +t, t x, R x (t) = 1 + x, t > x. Theorem.. [16] Let {x i } i=1 be dense subset of intervl [,b], then {R x i (t)} i=1 is bsis of W 1 [,b]. 3 The reproducing kernel method In this section, we shll give the exct nd pproximte solution of Eqs. (1.1) nd (1.) in reproducing kernel spce W 1 [,b]. We ssume tht Eqs. (1.1) nd (1.) hve unique solution. To del with the system, we consider Eqs. (1.1) nd (1.) s Ly(x) = f (x), x b, (3.8) where Ly(x) = y(x) + p(x)y(h(x)) + λ k(x,t)y(t)dt or Ly(x) = y(x) + p(x)y(h(x)) + λ x k(x,t)y(t)dt, it is cler tht L is the bounded liner opertor of W 1[,b] W 1 [,b]. We shll give the representtion of nlyticl solution of Eq. (3.8) in the spce W 1[,b]. Set ψ i(x) = L t R x (t) t=xi, where the subscript t in the opertor L indictes tht the opertor L pplies to the function of t. Theorem 3.1. Let {x i } i=1 be dense subset of intervl [,b], then {ψ i(x)} i=1 is complete system of W 1 [,b]. Proof. For ech fixed y(x) W 1 [,b], let y(x),ψ i(x) W 1 = 0, i = 1,,..., which mens tht y(x),l t R x (t) t=xi W 1 = L y(x),r x (t) W 1 (x i ) = Ly(x i ) = 0, i = 1,,... Since {x i } i=1 is dense on [,b], hence, Ly(x) = 0. Due to the existence of L 1, then y 0. Therefore, {ψ i (x)} i=1 is the complete system of W 1 [,b].

4 Pge 4 of 7 Usully, normlized orthogonl system is constructed from {ψ i (x)} i=1 by using the Grm-Schmidt lgorithm, nd then the pproximte solution will be obtined by clculting truncted series bsed on these functions. However, Grm-Schmidt lgorithm hs some drwbcks such s numericl instbility nd high volume of computtions. Here, to fix these flws, we stte the following Theorem in which the following nottions re used. Ψ = [ψ i j ] i, j=1,...,, Ψ = [ ψ i j ] i, j=1,...,,b = [β i j ] i, j=1,...,, where β i j = 0, i < j nd β ii > 0, 1 i, ψ i j = Lψ j,ψ i, ψ i j = L ψ j, ψ i, i, j = 1,...,. And where ˆf i = f,ψ i, i = 1,...,. â = (â 1,â,...,â ) T,ā = (ā 1,ā,...,ā ) T, ˆF = ( ˆf 1, ˆf,..., ˆf 3 ) T, Theorem 3.. Suppose tht {ψ i (x)} i=1 is linerly independent set in W 1[,b] nd { ψ i(x)} i=1 be normlized orthogonl system in W 1[,b], such tht ψ i(x) = i k=1 β ikψ k (x). If y(x) = i=1 āiψ i (x) y (x) = i=1 āiψ i (x) = i=1 âi ψ i (x) then Ψā = F. Proof. Suppose tht y(x) W 1 [,b] then y(x) = i=1 āiψ i (x) = i=1 âi ψ i (x). ow, by truncting -term of the two series, becuse of y (x) = i=1 āiψ i (x) = i=1 âi ψ i (x) nd since ψ i (x) = i k=1 β ikψ k (x) so i=1 ā i ψ i (x) = i=1 â i ψ i (x) = i â i i=1 k=1 β ik ψ k (x) = ( k=1 i=k Due to the liner independence of {ψ i (x)} i=1, ā k = i=k âiβ ik, k = 1,..., therefore Eq. (3.8), imply Ly (x) = f (x). For i = 1,..., we hve Ly, ψ i = f, ψ i j=1 â i β ik )ψ k (x). ā = B T â. (3.9) â j L ψ j, ψ i = f, ψ i i j â j β ik j=1 k=1 l=1 i j â j j=1 k=1 l=1 j=1 â j (BΨB T ) i j = (BΨB T )â = B F. β jl Lψ l,ψ k = β ik Lψ l,ψ k β T i k=1 i k=1 i l j = k=1 β ik f,ψ k, β ik f,ψ k β ik f,ψ k Eq. (3.9), imply BΨā = B F, hence Ψā = F.

5 Pge 5 of 7 Tble 1: Absolute Error y(x) y (x) for Exmple 4.1 h(x) = x h(x) = sin(x) h(x) = x = 0 = 50 = 0 = 50 x= E E E E-4 x= E E E E-4 x= E E E E-4 x= e E E E-6 x= e E E E-4 x= E E E E-4 x= E E E E-5 x= E E E E-5 x= E E E E-4 x= E E E E-4 It is necessry to mention tht here we solve the system Ψā = F which obtined without using the Grm-Schmidt lgorithm. Tble : Absolute Error y(x) y (x) for Exmple 4. h(x) = sin(x) h(x) = x = 50 = 100 = 50 = 100 x= E E E E-7 x= E E E E-5 x=0.3.77e E E E-5 x= E E E E-6 x= E E E E-5 x= E E E E-5 x= e e E E-5 x= E E E E-6 x= E E E E-5 x= E-3.165E E E-5 4 umericl exmples To illustrte the effectiveness nd the ccurcy of the proposed method, some numericl exmples re considered in this section. The exmples re computed using Mple 16. The numericl results in Tbles 1, nd 3 show tht the pproximte solution converge to the exct solution. Exmple 4.1. [5] Tking p(x) = e x, k(x,t) = e x t, f (x) = 3e x + e h(x) x in Eq. (1.1), the exct solution is y(x) = e x. The numericl results re given in Tble 1 with = 1,b = 1,x i = + (i 1) 1,i = 1,..., for = 0,50. Here we consider three different options for h(x) s x, sin(x) nd x. Exmple 4.. [6, 7] Tking f (x) = x[x + h (x)e x ] + (xt t 3 )dt, p(x) = xe x, k(x,t) = x t in Eq. (1.1), the exct solution y(x) = x. The numericl results re given in following Tble with = 0,b = 1.1,x i = (i 1)b 1,i = 1,..., for = 50,100. Here we consider two different options for h(x) s sin(x) nd x.

6 Pge 6 of 7 Exmple 4.3. [5] Tking p(x) = x, k(x,t) = x + 1, h(x) = x, f (x) = 3 4 x3 + 3 x + 3x + 3 in Eq. (1.), the exct solution is y(x) = x +1. The numericl results re given in Tble 3 with = 1,b = 1,x i = + (i 1) 1,i = 1,..., for = 50,100. Tble 3: Absolute Error y(x) y (x) for Exmple 4.3 (h(x) = x ) ode = 50 = 100 ode = 50 = E E E E E E E E E E E E E E E E E E E E-5 5 Conclusion In this pper, we introduced the new implementtion of reproducing kernel Hilbert spce method to obtin the pproximte solution to some of the Fredholm nd Volterr integrl equtions of the second kind. The relibility of the method nd reduction of the mount of computtion gives this method wider pplicbility. The obtined numericl results confirm tht the method is rpidly convergent nd show tht the pproximte solution converge to the exct solution. Acknowledgements The uthors re extending their hertfelt thnks to the reviewers for their vluble suggestions for the improvement of the rticle. References [1] L. M. Delves, J. L. Mohmed, Computtionl Methods for Integrl Equtions, Cmbridge University Press, Cmbridge, (1985). [] P. Schivne, C. Constnd, A. Mioduchowski, Integrl Methods in Science nd Engineering, Birkhuser, Boston, (00). [3] A. Kyselk, Properties of systems of integro-differentil equtions in the sttistics of polymer chins, Polym. Sci. USSR, 19 (11) (1977) [4] F. Bloom, Asymptotic bounds for solutions to system of dmped integro-differentil equtions of electromgnetic theory, J. Mth. Anl. Appl, 73 (1980) [5] E. Bbolin, S. Abbsbndy, F. Ftthzdeh, A numericl method for solving clss of functionl nd two dimensionl integrl equtions, Appl. Mth. Comput, 198 (008) [6] S. Abbsbndy, Appliction of Hes homotopy perturbtion method to functionl integrl equtions, Chos Solitons Frctls, 31 (007)

7 Pge 7 of 7 [7] M. T. Rshed, umericl solution of functionl differentil, integrl nd integro-differentil equtions, Appl. Mth. Comput, 156 (004) [8] Cui Minggen, Deng Zhongxing, On the best opertor of interpltion, J. Mth. umeric. Sinic, 8 () (1986) [9] M. Cui, F. Geng, Solving singulr two-point boundry vlue problem in reproducing kernel spce, J. Comput. Appl. Mth, 05 (007) [10] Wenyn Wng, Minggen Cui, Bo Hn, A new method for solving clss of singulr two-point boundry vlue problems, Appl. Mth. Comput, 06 (008) [11] F. Geng, M. Cui, Solving singulr nonliner two-point boundry vlue problems in the reproducing kernel spce, J. of the Koren Mth. Society, 45 (3) (008), [1] F. Geng, M. Cui, Solving singulr nonliner second-order periodic boundry vlue problems in the reproducing kernel spce, Appl. Mth. Comput, 19 (007) [13] F. Geng, M. Cui, Solving nonliner system of second order boundry vlue problems, J. Mth. Anl. Appl, 37 (007) [14] Gh. Akrm, H. Rehmn, Solution of fifth order boundry vlue problems in reproducing kernel spce, Middle- Est Journl of Scientific Reserch, 10() (011) [15] H. Du, J. Shen, Reproducing kernel method of solving singulr integrl eqution with cosecnt kernel, J. Mth. Anl. Appl, 348 (008) [16] Zhong Chen, YongFng Zhou, A new method for solving Hilbert type singulr integrl equtions, Appl. Mth. Comput, 18 (011) [17] Mrym Mohmmdi, Rez Mokhtri, Solving the generlized regulrized long wve eqution on the bsis of reproducing kernel spce, J. Comput. Appl. Mth, 35 (011) [18] M. Cui, Y. Lin, onliner umericl Anlysis in Reproducing Kernel Hilbert Spce, ov Science Publisher, ew York, (009).