An improvement to the homotopy perturbation method for solving integro-differential equations

Transcription

1 Avilble online t Int J Industril Mthemtics (ISSN ) Vol 4, No 4, Yer 212 Article ID IJIM-241, 12 pges Reserch Article An improvement to the homotopy perturbtion method for solving integro-differentil equtions E Bbolin, A R Vhidi b, Z Azimzdeh () Deprtment of Mthemtics, Science nd Reserch Brnch, Islmic Azd University, Tehrn, Irn (b) Deprtment of Mthemtics, Shhr-e-Rey Brnch, Islmic Azd University, Tehrn, Irn Abstrct In this pper, new form of the homotopy perturbtion method (NHPM) hs been dopted for solving integro-differentil equtions In the present study, firstly the NHPM is used to the integro-differentil eqution, which yields the Mclurin series of the exct solution By pplying the Lplce trnsformtion to the truncted Mclurin series nd then the Pde pproximtion to the solution derived from the Lplce trnsformtion, we obtin mostly the exct solution of this kind of equtions Illustrtive exmples re given to represent the efficiency nd the ccurcy of the proposed method Keywords: Homotopy perturbtion method (HPM); Integro-differentil equtions; Pde pproximnt 1 Introduction Since mny physicl problems re modeled by integro-differentil equtions, the numericl solutions of such integro-differentil equtions hve been highly studied by mny uthors [8,23] In recent yers, numerous works hve been focusing on the development of more dvnced nd efficient methods for integro-differentil equtions such s WveletGlerkin method [4] nd Tu method [22] nd semi nlyticl-numericl techniques such s Tylor polynomils [28,29], Adomins decomposition method [6], nd the HPM [1,3] Perturbtion techniques re widely used in science nd engineering to hndle liner nd nonliner problems [26] The HPM ws first proposed by He in [18] nd further developed nd improved by him [17,19] This method is bsed on the use of trditionl perturbtion method nd homotopy technique In this method, the solution is considered s the summtion of n infinite series which converges rpidly to the exct solutions The pplictions of the HPM in nonliner problems hve been demonstrted by mny reserchers, cf [2,5,7,9,1,12,27,3,31] Corresponding uthor Emil ddress: ebbolin@yhoocom 353

2 354 E Bbolin, et l /IJIM Vol 4, No 4 (212) Severl uthors hve proposed vriety of the modified homotopy perturbtion methods E Yusufoğlu in [32] proposed the improved homotopy perturbtion method for solving Fredholm type integro-differentil equtions M Jvidi in [25] proposed the modified homotopy perturbtion method (MHPM) to solve nonliner Fredholm integrl equtions The result revels tht the MHPM is very effective nd convenient In nother study, he pplied the MHPM for solving the system of liner Fredholm integrl equtions [24] A Golbbi in [15,16] used the MHPM for solving Fredholm integrl equtions nd nonliner Fredholm integrl equtions of the first kind In nother work, he introduced new itertive methods for nonliner equtions by the MHPM [14] In this pper, we introduce the NHPM to solve integro-differentil equtions Bsiclly, the NHPM is the sme s the HPM In the HPM, in order to find the solution, the series components re to be clculted nd the sum of the prtil series of these components is considered s n pproximtion of the solution However, in the NHPM, the method is designed in such wy tht only one term of the series is clculted In this work, firstly the NHPM is pplied to solve integro-differentil equtions By this method, we get truncted series solution tht often coincides with the Mclurin expnsion of the true solution By pplying the Lplce trnsformtion to the truncted Mclurin series nd then the Pdé pproximtion to the solution derived from the Lplce trnsformtion, in most cses, we obtin the exct solution of this kind of equtions 2 Description of HPM To illustrte the HPM, consider the following integro-differentil eqution [11] A(u(x)) = y(r(x)), r(x) Ω, (21) with boundry conditions B(u(x), u(x) ) =, r(x) Γ, (22) n where A is generl integrl opertor, B is boundry opertor, y(r(x)) is known nlytic function nd Γ is the boundry of the domin Ω The opertor A cn be generlly divided into two prts L nd R, where L is n identity opertor nd R = A L Therefore, Eq (21) cn be rewritten s follows L(u(x)) + R(u(x)) = y(r(x)) (23) He in [2] constructed homotopy which stisfies H(v(x), p) = (1 p)[l(v(x)) L(u (x))] + p[a(v(x)) y(r(x))] =, (24) where r(x) Ω, p [, 1] tht is clled the homotopy prmeter, nd u is n initil pproximtion of Eq (21) Hence, it is obvious tht H(v(x), ) = L(v(x)) L(u (x)) =, H(v(x), 1) = A(v(x)) y(r(x)) =, (25) nd the chnging process of p from to 1, is just tht of H(v(x), p) chnging from L(v(x)) L(u (x)) to A(v(x)) y(r(x)) While in topology, this is clled deformtion, L(v(x)) L(u (x)) nd A(v(x)) y(r(x)) re clled homotopic Applying the perturbtion technique

3 E Bbolin, et l / IJIM Vol 4, No 4 (212) [26], due to the fct tht p 1 cn be considered s smll prmeter, we cn ssume tht the solution of (24) cn be expressed s series in p, s follows: v(x) = v (x) + pv 1 (x) + p 2 v 2 (x) + (26) when p 1, Eq (24) corresponds to Eq (23) nd becomes the pproximte solution of Eq (23), i e, u(x) = lim p 1 v(x) = v (x) + v 1 (x) + v 2 (x) + (27) The series (27) is convergent in most cses, nd the convergence rte depends on A(u(x)) y(r(x)) [21] Note tht in the HPM in order to obtin n pproximte solution, the components v i (x) for i =, 1, must be clculted Specilly for i 3, it needs lrge nd sometimes complicted computtions nd, in the cse of nonlinerity, the use of He s polynomils [13] To obvite this problem, the NHPM is introduced, in which v (x) is clculted in such wy tht v i (x) = for i 1 So, the number of computtions decreses in comprison with tht in the HPM The NHPM for liner integro-differentil equtions will be discussed in detil in the following section 3 NHPM for integro-differentil equtions In order to illuminte the solution procedure of the NHPM, we consider the Fredholm integro-differentil eqution u (x) = g(x) + b k(x, t, u(t))dt, x b (38) where, b re constnts, g(x), k(x, t, u) re known functions nd u(x) is solution to be determined By considering the convex homotopy defined in Eq (24), we hve H(v(x), p) = L(v(x)) L(u (x)) + pl(u (x)) + p[r(v(x)) y(r(x))] =, (39) which cn be written in the following form L(v(x)) = L(u (x)) pl(u (x)) pr(v(x)) + py(r(x)) (31) where L(v(x)) = v (x), L(u (x)) = u (x), R(v(x)) = b k(x, t, v(t))dt nd y(r(x)) = g(x) An equivlent expression (31) is v (x) = u (x) pu (x) + pg(x) p b k(x, t, u(t))dt (311) Suppose tht the initil pproximtion of Eq (38) hs the form u (x) = n F n (x), (312) n=

4 356 E Bbolin, et l /IJIM Vol 4, No 4 (212) where, 1, 2, re unknown coefficients nd F (x), F 1 (x), F 2 (x), re specific functions depending on the problem Denote d dx by G, we hve G 1 s integrtion from to x By pplying G 1 to both sides of Eq (311), we hve v(x) = T (x) + (u (x) ) p(u (x) ) + p p b g(x) k(x, t, v(t))dtdx, (313) where T incorportes the constnts of integrtion nd stisfies GT = According to the HPM, we ssume tht the solution of Eq (313) cn be represented s series in p v(x) = p n v n (x) n= By substituting Eqs (314) nd (312) into the Eq (313), we obtin p n v n (x) = T (x) + ( n F n (x) ) p( n F n (x) ) + p n= p b n= k(x, t, n= g(x)dx (314) p n v n (t))dtdx (315) n= Compring coefficients of terms with identicl powers of p leds to p : v (x) = T (x) + n F n (x), n= p 1 : v 1 (x) = ( n F n (x) ) + p 2 : v 2 (x) = p j : v j (x) = n= b b g(x)dx b k(x, t, p n v n (t))dtdx, k(x, t, v 1 (t))dtdx, (316) k(x, t, v j 1 (t))dtdx, Now, we solve these equtions in such wy tht v 1 (x) =, then Eq (316) results in v 2 (x) = v 3 (x) = = Therefore, the exct solution my be obtined s the following u(x) = v (x) = T (x) + n F n (x) (317) n= Similrly, the NHPM re pplied for the Volterr integro-differentil equtions Note tht, the dvntge of this method in comprison with the HPM is tht only one term of the series is clculted To show the cpbility of the method, we pply the NHPM to some exmples in the next section n=

5 E Bbolin, et l / IJIM Vol 4, No 4 (212) Exmples In this section, we present four exmples to illustrte the efficiency of the NHPM for integro-differentil equtions It is indicted tht through the NHPM, in most cses, the exct solution to these types of equtions cn be figured out Exmple 41 Consider the Fredholm liner integro-differentil eqution [8] u (x) = xe x + e x x + 1 xu(t)dt, (418) with the initil condition u() = nd the exct solution u(x) = xe x In order to pply the NHPM to Eq (418), consider the convex homotopy (31), where L(v(x)) = v (x), L(u (x)) = u (x) R(v(x)) = 1 xv(t)dt nd y(r(x)) = xex +e x x Therefore, considering Eq (31) one gets v (x) = u (x) pu (x) + p(xe x + e x x) p Denoting becomes 1 xv(t)dt (419) d dx by G, we hve G 1 s integrtion Using the opertor G, Eq (419) Gv(x) = u (x) pu (x) + p(xe x + e x x) p 1 xv(t)dt (42) Applying the inverse opertor G 1 to both sides of Eq (42) nd using the initil conditions, we obtin v(x) = p u (x)dx p 1 u (x)dx + p (xe x + e x x)dx xv(t)dtdx, (421) By replcing u (x) = n= nf n (x), where F n (x) = x n in the bove eqution, we obtin v(x) = ( n x n ) p( n x n ) + p p n= 1 n= (xe x + e x x)dx xv(t)dtdx (422) Substituting v(t) = n= pn v n (t) into Eq (422), considering the Mclurin series of the excittion term e x 1 + x + x2 2 + x3 6 + x x5 12, (423) nd equting the terms with the identicl powers of p give p : v (x) = n x n, n=

6 358 E Bbolin, et l /IJIM Vol 4, No 4 (212) p 1 : v 1 (x) = ( n x n ) + p 2 : v 2 (x) = n= 1 1 p n+1 : v n+1 (x) = (1 + x x x x x5 )dx xv (t)dtdx, (424) xv 1 (t)dtdx, 1 xv n (t)dtdx, Solving the bove equtions for v 1 (t) leds to the result v 1 (t) = (1 1 )x ( )x 2 + ( 1 2 3)x )x 2 + ( 9 2 3)x 3 + ( )x 4 + (425) Eliminting v 1 (t) lets the coefficients n for n =, 1, 2, tke the following vlues =, 1 = 1, 2 = 1, 3 = 1 2, 4 = 1 6, 5 = 1 24, 6 = 1 12, (426) By substituting the bove vlues into v (t), we obtin u(x) = v (x) = x + x 2 + x3 2 + x4 6 + x (427) which is the prtil sum of the Tylor series of the exct solution t x = In order to obtin more ccurte solution, we use the truncted series of (427) Consider five terms in u(x) s φ(x) = x + x 2 + x3 2 + x4 6 + x5 24 (428) By pplying the Lplce trnsformtion to both sides of Eq (428), we hve L[φ(x)] = 5 s s s s s 2 (429) If s = 1 x, then L[φ(x)] = x + 2x 3 + 3x 4 + 4x 5 + 5x 6 (43) All of the [ L M ] Pde pproximnt of Eq (43) with L 2, M 2 nd L + M 1 yield [ L M ] = x 2 1 2x + x 2 Replcing x = 1 s, we obtin [ L M ] in terms of s s (431) [ L M ] = 1 (1 + 1 s 2 ) (432) s 2 2 s By using the inverse Lplce trnsformtion to Eq (432), we obtin the exct solution xe x

7 E Bbolin, et l / IJIM Vol 4, No 4 (212) Exmple 42 Consider the Fredholm nonliner integro-differentil eqution [8] u (x) = x + xu 2 (t)dt, (433) with the initil condition u() = nd the exct solution u(x) = x In order to pply the NHPM to Eq (433), consider the convex homotopy (31), where L(v(x)) = v (x), L(u (x)) = u (x), R(v(x)) = 1 xv2 (t)dt nd y(r(x)) = 1 1 3x Therefore, considering Eq (31) one gets v (x) = u (x) pu (x) + p( x) p xv 2 (t)dt (434) Denoting becomes d dx by G, we hve G 1 s integrtion Using the opertor G, Eq (434) Gv(x) = u (x) pu (x) + p( x) p xv 2 (t)dt (435) Applying the inverse opertor G 1 to both sides of Eq (435) nd using the initil conditions, we obtin v(x) = p u (x)dx p x 1 u (x)dx + p (1 1 3 x)dx xv 2 (t)dtdx, (436) By replcing u (x) = n= nf n (x), where F n (x) = x n, in the bove eqution, we obtin 1 v(x) = ( n x n ) p( n x n ) + p(x x2 6 ) p xv 2 (t)dtdx (437) n= n= Substituting v(t) = n= pn v n (t) into Eq (437) nd equting the terms with the identicl powers of p give p : v (x) = ( n x n ), n= p 1 : v 1 (x) = ( n x n ) + (x x2 6 ) p 2 : v 2 (x) = n= x 1 xv 2 (t)v 1 (t)dtdx, 1 Solving the bove equtions for v 1 (t) leds to the result xv 2 (t)dtdx (438) v 1 (x) = x x2 6 x x2 2 1 x x x (439) Eliminting v 1 (x) lets the coefficients n for n =, 1, 2, tke the following vlues =, 1 = 1, 2 =, 3 =, 4 =, (44) By substituting the bove vlues into v (x), we obtin u(x) = v (x) = x which is the exct solution Eq (433) (441)

8 36 E Bbolin, et l /IJIM Vol 4, No 4 (212) Exmple 43 Consider the Volterr nonliner integro-differentil eqution u (x) = 1 + u(t)u (t)dt, (442) with the initil condition u() = nd the exct solution u(x) = 2 T n[ x 2 ] The Tylor series of the exct solution t x = is u(x) = x x x x7 + (443) In order to pply the NHPM to Eq (442), consider the convex homotopy (31), where L(v(x)) = v (x), L(u (x)) = u (x), R(v(x)) = u(t)u (t)dt nd y(r(x)) = 1 Therefore, considering Eq (31) one gets v (x) = u (x) pu (x) + p p Denoting becomes v(t)v (t)dt (444) d dx by G, we hve G 1 s integrtion Using the opertor G, Eq (444) Gv(x) = u (x) pu (x) + p p v(t)v (t)dt (445) Applying the inverse opertor G 1 to both sides of Eq (445) nd using the initil conditions, we obtin v(x) = u (x)dx p u (x)dx + px p v(t)v (t)dtdx, (446) By replcing u (x) = n= nf n (x), where F n (x) = x n, in the bove eqution, we obtin v(x) = ( n x n ) p( n x n ) + px p n= n= v(t)v (t)dtdx (447) Substituting v(t) = n= pn v n (t) into Eq (447)nd equting the terms with the identicl powers of p give p : v (x) = n x n, n= p 1 : v 1 (x) = ( n x n ) + x p 2 : v 2 (x) = n= p n+1 : v n+1 (x) = v 1 (t)v 1(t)dtdx, v n (t)v n(t)dtdx, v (t)v (t)dtdx (448)

9 E Bbolin, et l / IJIM Vol 4, No 4 (212) Solving the bove equtions for v 1 (x) leds to the result v 1 (x) = (1 1 )x 2 x 2 + ( )x 3 + ( )x 4 + (449) 4 Eliminting v 1 (x) lets the coefficients n for n =, 1, 2, tke the following vlues =, 1 = 1, 2 =, 3 = 1 6, 4 =, 5 = 1 3, 6 =, 7 = , (45) By substituting the bove vlues into v (t), we obtin u(x) = v (x) = x x x x7 + (451) which is the Tylor series of the exct solution t x = 5 Conclusion In this work, we pplied n improvement of the HPM to the integro-differentil equtions The NHPM yielded the Mclurin series of the exct solution By pplying the Lplce trnsformtion to the truncted Mclurin series nd then the Pdé pproximtion to the solution derived from the Lplce trnsformtion, we obtined the exct solution of some of these equtions Note tht in the bove process, the obtined solution did not depend on the number of the selected terms in the truncted Mclurin series The nlyzed exmples illustrted the bility nd the relibility of the NHPM nd reveled tht the presented improvement of the HPM ws simple but very effective fctor for clculting the exct solution of this kind of equtions References [1] S Abbsbndy, Homotopy perturbtion method for qudrtic Riccti differentil eqution nd comprison with Adomin s decomposition method, Appl Mth Comput 172 (26) [2] S Abbsbndy, Iterted He s homotopy perturbtion method for qudrtic Riccti differentil eqution, Appl Mth Comput 175 (26) [3] S Abbsbndy, Numericl solutions of the integrl equtions: Homotopy perturbtion method nd Adomin s decomposition method, Appl Mth Comput 173 (26) [4] A Avudinygm, C Vni, Wvelet-Glerkin method for integro-differentil equtions, Appl Numer Mth 32 (2) [5] E Bbolin, N Dstni, Numericl solutions of two-dimensionl liner nd nonliner Volterr integrl equtions: homotopy perturbtion method nd differentil trnsform method, Int J Industril Mthemtics 3 (211)

10 362 E Bbolin, et l /IJIM Vol 4, No 4 (212) [6] E Bbolin, A Dvri, Numericl implementtion of Adomin decomposition method for liner Volterr integrl equtions of the second kind, Appl Mth Comput 165 (25) [7] J Bizr, H Ghzvini, Numericl solution for specil non-liner Fredholm integrl eqution by HPM, Appl Mth Comput 195 (28) [8] P Drni, A Ebdin, A method for the numericl solution of the integro-differentil equtions, Appl Mth Comput 188 (27) [9] R Ezzti, M Aqhmohmdi, Solving the non-liner system of third-order boundry vlue problems by using He s homotopy perturbtion method, Int J Industril Mthemtics 1 (29) [1] M Ghnbri, Numericl solution of fuzzy initil vlue problems under generlized differentibility by HPM, Int J Industril Mthemtics 1 (29) [11] M Ghsemi, M Tvssoli Kjni, E Bbolin, Appliction of Hes homotopy perturbtion method to nonliner integro-differentil equtions, Appl Mth Comput 188 (27) [12] M Ghsemi, M Tvssoli kjni, E Bbolin, Numericl solutions of the nonliner integro-differentil equtions: Wvelet-Glerkin method nd homotopy perturbtion method, Appl Mth Comput 188 (27) [13] A Ghorbni, Beyond Adomin polynomils: He polynomils, Chos Solitons nd Frctls 39 (29) [14] A Golbbi, M Jvidi, New itertive methods for nonliner equtions by modified HPM, Appl Mth Comput 191 (27) [15] A Golbbi, B Kermti, Modified homotopy perturbtion method for solving Fredholm integrl equtions, Chos Solitons nd Frctls 37 (28) [16] A Golbbi, B Kermti, Solution of non-liner Fredholm integrl equtions of the first kind using modified homotopy perturbtion method, Chos Solitons nd Frctls 39 (29) [17] JH He, A coupling method of homotopy technique nd perturbtion technique for nonliner problems, Int J NonLiner Mech 35 (2) [18] JH He, Approximte nlyticl solution of Blsius eqution, Commun Nonliner Sci Numer Simul 3 (1998) [19] JH He, A simple perturbtion pproch to Blsius eqution, Appl Mth Comput 14 (23) [2] JH He, Bookkeeping prmeter in perturbtion methods, Int J NonLiner Mech 2 (3) (21) [21] JH He, Homotopy perturbtion technique, Comput Meth Appl Mech Engin 178 (1999)

11 E Bbolin, et l / IJIM Vol 4, No 4 (212) [22] SM Hosseini, S Shhmord, Tu numericl solution of Fredholm integro-differentil equtions with rbitrry polynomil bses, Appl Mth Model 27 (23) [23] Z Jckiewicz, M Rhmn, BD Welfert, Numericl solution of Fredholm integrodifferentil eqution modelling Θ-neurl networks, Appl Mth Comput 195 (28) [24] M Jvidi, Modified homotopy perturbtion method for solving system of liner Fredholm integrl equtions, Mth Comput Model 5 (29) [25] M Jvidi, A Golbbi, Modified homotopy perturbtion method for solving non-liner Fredholm integrl equtions, Chos Solitons nd Frctls 4 (29) [26] A Nyfeh, Perturbtion Methods, John Wiley nd Sons, New York, 2 [27] Sh Sdigh Behzdi, A Yildirim, A method to estimte the solution of wekly singulr non-liner integro-differentil equtions by pplying the homotopy methods, Int J Industril Mthemtics 4 (212) [28] M Sezer, A method for the pproximte solution of the second order liner differentil equtions in terms of Tylor polynomils, Int J Mth Educ Sci Technol 6 (1996) [29] M Sezer, Tylor polynomil solution of Volterr integrl equtions, Int J Mth Educ Sci Technol 5 (1994) [3] A Yildirim, Homotopy perturbtion method for the mixed Volterr-Fredholm integrl equtions, Chos Solitons nd Frctls 42 (29) [31] E Yusufoglu, A homotopy perturbtion lgorithm to solve system of Fredholm- Volterr type integrl equtions, Mth Comput Model 47 (28) [32] E Yusufoğlu, Improved homotopy perturbtion method for solving Fredholm type integro-differentil equtions, Chos Solitons nd Frctls 41 (29) 28-37