An iterative method for solving nonlinear functional equations

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1 J. Mth. Anl. Appl. 316 (26) An itertive method for solving nonliner functionl equtions Vrsh Dftrdr-Gejji, Hossein Jfri Deprtment of Mthemtics, University of Pune, Gneshkhind, Pune 4117, Indi Received 17 Februry 25 Avilble online 9 June 25 Submitted by M. Innelli Abstrct An itertive method for solving nonliner functionl equtions, viz. nonliner Volterr integrl equtions, lgebric equtions nd systems of ordinry differentil eqution, nonliner lgebric equtions nd frctionl differentil equtions hs been discussed. 25 Elsevier Inc. All rights reserved. Keywords: Nonliner Volterr integrl equtions; System of ordinry differentil equtions; Itertive method; Contrction; Bnch fixed point theorem; Frctionl differentil eqution 1. Introduction A vriety of problems in physics, chemistry nd biology hve their mthemticl setting s integrl equtions [1]. Therefore, developing methods to solve integrl equtions (especilly nonliner), is receiving incresing ttention in recent yers [1 8]. In the present work we describe n itertive method which cn be utilized to obtin solutions of nonliner functionl equtions. The method when combined with lgebric computing softwre (Mthemtic, e.g.) turns out to be powerful. * Corresponding uthor. E-mil ddresses: vsgejji@mth.unipune.ernet.in (V. Dftrdr-Gejji), jfri_h@mth.com (H. Jfri) X/$ see front mtter 25 Elsevier Inc. All rights reserved. doi:1.116/j.jm

2 754 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) The present pper hs been orgnized s follows. In Section 2 the itertive method is described nd in Section 3 existence of solutions for nonliner Volterr integrl equtions hs been proved using this method. Illustrtive exmples hve been presented in Section 4 followed by the conclusions in Section An itertive method Consider the following generl functionl eqution: y = N(y)+ f, where N is nonliner opertor from Bnch spce B B nd f is known function. We re looking for solution y of Eq. (1) hving the series form: y = y i. (2) i= The nonliner opertor N cn be decomposed s ( ) { ( i ) N y i = N(y ) + N y j N i= i=1 j= From Eqs. (2) nd (3), Eq. (1) is equivlent to { ( i ) y i = f + N(y ) + N y j N i= i=1 j= ( i 1 (1) y j )}. (3) j= ( i 1 y j )}. (4) We define the recurrence reltion: y = f, y 1 = N(y ), y m+1 = N(y + +y m ) N(y + +y m 1 ), m = 1, 2,... Then nd (y 1 + +y m+1 ) = N(y + +y m ), m = 1, 2,..., (6) y = f + y i. i=1 If N is contrction, i.e. N(x) N(y) K x y, <K<1, then y m+1 = N(y + +y m ) N(y + +y m 1 ) K y m K m y, m =, 1, 2,..., nd the series i= y i bsolutely nd uniformly converges to solution of Eq. (1) [5], which is unique, in view of the Bnch fixed point theorem [1]. j= (5) (7)

3 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) Nonliner Volterr integrl eqution Consider the Volterr integrl eqution y(x) = f(x)+ F ( x,t,y(t) ) dt, (8) where x α, t α, F is continuous function of its rguments nd stisfies Lipschitz condition, F(x,t,φ) F(x,t,ψ) <K φ ψ.let F(x,t,φ) <M. Define y (x) = f(x), y 1 (x) = F ( x,t,y (t) ) dt, y m+1 (x) = m = 1, 2,... F(x,t,y + +y m ) F(x,t,y + +y m 1 ) dt, (9) We prove i=1 y i (x) is uniformly convergent. y1 (x) x ( F x,t,y (t) ) dt M(x ) Mα, y2 (x) x ( F x,t,y (t) + y 1 (t) ) F ( x,t,y (t) ) x dt K y 1 (t) dt (x )2 MK M (Kα) 2, 2! K 2! y 3 (x) F ( x,t,y (t) + y 1 (t) + y 2 (t) ) F ( x,t,y (t) + y 1 (t) ) dt K y 2 (t) dt MK2 (x )3 3! M (Kα) 3, K 3!.. ym+1 (x) x F(x,t,y + +y m ) F(x,t,y + +y m 1 ) dt K y m 1 (t) dt MKm (x )m+1 (m + 1)! M K (Kα) m+1 (m + 1)!. (1)

4 756 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) Hence i= y i (x) is bsolutely nd uniformly convergent nd y(x) stisfies Eq. (8). If Eq. (8) does not possess unique solution, then this itertive method will give solution mong mny (possible) other solutions. 4. Illustrtive exmples (i) Consider the following nonliner differentil eqution with exct solution y = x 1 for x>: y = y 2, y(1) = 1. (11) The initil vlue problem in Eq. (11) is equivlent to the integrl eqution y(x) = 1 y 2 dt. 1 Following the lgorithm given in (9): (12) y = 1, y 1 = N(y ) = 1 y 2 dt = 1 x, y 2 = N(y + y 1 ) N(y ) = x + 2 x2 x3 3, y 3 = x x2 85x3 + 41x x x6 9 x7 63, In Fig. 1 we hve plotted 5 i= y i (x), which is lmost equl to the exct solution y = x 1. Fig. 1.

5 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) (ii) Consider the following liner Volterr integrl eqution: y(x) = Following (9) we get 1 + y(t) 1 + t dt. 1 y =, y 1 (x) = N(y ) = dt = log(1 + x), 1 + t y 2 (x) = N(y + y 1 ) N(y ) = log(1 + x) 1 + t dt = y m+1 (x) = N(y + +y m ) N(y + +y m 1 ) = m = 1, 2,... (log(1 + x))2, 2! Hence y(x) = y i (x) = exp(log(1 + x)) 1 = x. (iii) Consider the following nonliner Fredholm integrl eqution: y(x) = 7 8 x xt y 2 (t) dt. Then from (9), first few terms of y(x) re: y = 7 8 x =.875x, y 1(x) = N(y ) = 1 2 y 2 (x) = x The solution in series form is given by 1 (log(1 + x))m, m! xty 2 49 (t) dt = 512 x, t [( y (t) + y 1 (t) ) 2 y 2 (t) ] dt = x. y(x) =.875x x x x + x. This eqution hs two exct solutions, x nd 7x [13]. (iv) Consider the following nonliner Volterr integrl eqution: y(x) = e x 1 3 xe3x + 1 x 3 x + xy 3 (t) dt. We use here modified itertive scheme [14], in which we tke:

6 758 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) y = e x, y 1 (x) = 1 3 xe3x x + N(y ), y 1 (x) = 1 3 xe3x + 1 x 3 x + y 2 (x) = N(y + y 1 ) N(y ) =, y m+1 (x) =, m. xy 3 (t) dt =, Thus we find tht the solution is y(x) = e x, which is the exct solution. (v) Consider the following system of nonliner ordinry differentil equtions: nd y 1 (x) = 2y2 2, y 1() = 1, y 2 (x) = e x y 1, y 2 () = 1, y 3 (x) = y 2 + y 3, y 3 () =. (13) System (13) is equivlent to the following system of integrl equtions: y 1 (x) = 1 + y 2 (x) = 1 + y 3 (x) = Eqution (9) leds to 2y 2 2 dt, e t y 1 dt, (y 2 + y 3 )dt. (14) y 1 (x) = 1, y 11 (x) = N 1 (y 1,y 2,y 3 ) = y 2 (x) = 1, y 21 (x) = N 2 (y 1,y 2,y 3 ) = y 3 (x) =, y 31 (x) = N 3 (y 1,y 2,y 3 ) = 2y 2 2 dt, e t y 1 dt, (y 2 + y 3 )dt, y i,m (x) = N i (y 1 + +y 1,m,y 2 + +y 2,m,y 3 + +y 3,m ) N i (y 1 + +y 1,m 1,y 2 + +y 2,m 1,y 3 + +y 3,m 1 ), i = 1, 2, 3, m= 1, 2,... In Figs. 2 4, the pproximtion solutions y 1 = 5 m= y 1m,y 2 = 5 m= y 2m nd y 3 = 5m= y 3m hve been plotted, which coincide with the exct solution.

7 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) Fig. 2. Exct solution is e 2x. Fig. 3. Exct solution is e x. Fig. 4. Exct solution is xe x. Comment. Exmples (iii) nd (iv) hve been solved by Wzwz [13] nd exmple (v) hs been solved by Bizr et l. [3] using Adomin decomposition method (ADM) [1]. The method presented here is esier compred to ADM, nd gives the nswers to the sme ccurcy.

8 76 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) (vi) Consider the nonliner lgebric eqution x 6 5x 5 + 3x 4 + x 3 + 2x 2 8x.5 =. We rewrite this eqution s x = x6 5 8 x x x x2 = x + N(x), where x = 1 16.AsN (x )<1, we cn employ the lgorithm given in (5) to get one solution s follows: x = 1 16, x 1 = N(x ) = 1 8 x6 5 8 x x x x2, x m+1 = N(x + +x m ) N(x + +x m 1 ), m = 1, 2,... The first five terms re: x 1 = , x 2 = N(x + x 1 ) N(x ) = , x 3 = N(x + x 1 + x 2 ) N(x + x 1 ) = , x 4 = N(x + +x 3 ) N(x + x 1 + x 2 ) = , x 5 = N(x + +x 4 ) N(x + +x 3 ) = The sum of first five terms is x x + +x 5 = , which mtches with the pproximte solution given by Ouedrogo et l. [11]. Their nswer is x = The nswer given by Mthemtic softwre is x = [7]. (vii) Consider the following system of nonliner lgebric equtions: x1 2 1x 1 + x =, x 1 x2 2 + x 1 1x =. We rewrite this system s x 1 = x x2 2 = x 1 + N 1 (x 1,x 2 ), x 2 = x x 1x2 2 = x 2 + N 2 (x 1,x 2 ). We cn employ (5) to get solutions s follows: x 1 = 8 1, x 2 = 8 1, x 11 = N 1 (x 1,x 2 ) = 1 1 x x2 2, x 21 = N 2 (x 1,x 2 ) = 1 1 x x 1x2 2, x 1m+1 = N 1 (x 1 + +x 1m,x 2 + +x 2m ) N 1 (x 1 + +x 1m 1,x 2 + +x 2m 1 ), x 2m+1 = N 2 (x 1 + +x 1m,x 2 + +x 2m ) N 2 (x 1 + +x 1m 1,x 2 + +x 2m 1 ),

9 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) m = 1, 2,... The first five terms re: x 1 =.8, x 2 =.8, x 11 =.128, x 21 =.1312, x 12 = , x 22 = , x 13 = , x 23 = , x 14 = , x 24 = , x 15 = , x 25 = The sum of first five terms is x 1 x 1 + +x 15 = , x 2 x 2 + +x 25 = The exct solution is x = (1, 1) t. Revised ADM [8] (using five itertions) gives the nswer (.99778, ) t, wheres the stndrd ADM (using five itertions) gives the nswer ( , ) t [2]. (viii) We pply this method to solve system of nonliner frctionl differentil equtions. Consider the system of nonliner frctionl differentil equtions: D.5 y 1 = 2y2 2, y 1() =, D.4 y 2 = xy 1, y 2 () = 1, D.3 y 2 = y 2 y 3, y 3 () = 1, where D α denotes Cputo frctionl derivtive of order α [12]. We pply new itertive method for solving this system. For definitions of Riemnn Liouville frctionl integrl nd Cputo frctionl derivtive, we refer the reder to [6,12]. The bove system is equivlent to the following system of integrl equtions: y 1 = 2I.5 y2 2 (x) = 2 Ɣ(.5) y 2 2 (t) (x t) y 2 = y 2 () + I.4 xy 1 (x) = Ɣ(.4).5 dt, y 3 = y 3 () + I.3( y 2 (x)y 3 (x) ) = Ɣ(.3) ty 1 (t) dt, (x t).6 y 2 (t) y 3 (t) dt, (x t).7 where I α denotes Riemnn Liouville frctionl integrl of order α [12]. First few terms of the itertion re given below: y 1 =, y 11 = x.5, y 12 =, y 2 = 1, y 21 =, y 22 = x 1.9, y 3 = 1, y 31 = x.3, y 32 = x.6.

10 762 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) Fig. 5. Using new itertive method. Fig. 6. Using ADM. This exmple hs been solved using ADM s well [9]. In Figs. 5 nd 6, the pproximte solutions using the new itertive method nd the ADM hve been plotted, respectively. It should be noted tht the solution plotted in Fig. 5 hs been obtined by summing first five terms of the new itertive method, wheres the solution plotted in Fig. 6 corresponds to the sum of first 7 terms of ADM. 5. Conclusions An itertive method for solving functionl equtions hs been discussed. The proof of existence of solution for nonliner Volterr integrl equtions is presented. Illustrtive exmples deling with lgebric eqution, Volterr integrl equtions, systems of ordinry differentil eqution, nonliner lgebric equtions nd frctionl differentil equtions hve been given. The method proves to be simple in its principles nd convenient for computer lgorithms. Mthemtic hs been used for computtions in this pper.

11 V. Dftrdr-Gejji, H. Jfri / J. Mth. Anl. Appl. 316 (26) Acknowledgments Hossein Jfri thnks University Grnts Commission, New Delhi, Indi for the wrd of Junior Reserch Fellowship nd cknowledges Dr. S.K. Mirni, University of Mzndrn, Bbolsr, Irn for encourgement. References [1] G. Adomin, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Acd. Publ., [2] E. Bbolin, J. Bizr, A.R. Vhidi, Solution of system of nonliner equtions by Adomin decomposition method, J. Mth. Anl. Appl. 15 (24) [3] J. Bizr, E. Bbolin, R. Islm, Solution of the system of ordinry differentil equtions by Adomin decomposition method, Appl. Mth. Comput. 147 (24) [4] J. Bizr, E. Bbolin, R. Islm, Solution of the system of Volterr integrl equtions of the first kind by Adomin decomposition method, Appl. Mth. Comput. 139 (23) [5] Y. Cherruult, Convergence of Adomin s method, Kybernetes 8 (1988) [6] V. Dftrdr-Gejji, H. Jfri, Adomin decomposition: tool for solving system of frctionl differentil equtions, J. Mth. Anl. Appl. 31 (25) [7] S.M. El-Syed, The modified decomposition method for solving nonliner lgebric equtions, Appl. Mth. Comput. 132 (22) [8] H. Jfri, V. Dftrdr-Gejji, Revised Adomin decomposition method for solving system of nonliner equtions, submitted for publiction. [9] H. Jfri, V. Dftrdr-Gejji, Solving system of nonliner frctionl differentil equtions, submitted for publiction. [1] A.J. Jerri, Introduction to Integrl Equtions with Applictions, second ed., Wiley Interscience, [11] R.Z. Ouedrogo, Y. Cherruult, K. Abboui, Convergence of Adomin s decomposition method pplied to lgebric equtions, Kybernetes 29 (2) [12] I. Podlubny, Frctionl Differentil Equtions, Acdemic Press, [13] A. Wzwz, A First Course in Integrl Equtions, World Scientific, [14] A. Wzwz, A relible modifiction of Adomin decomposition method, Appl. Mth. Comput. 12 (1999)