He s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations

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1 nternational Journal of Theoretical an Alie Mathematics 2017; 3(6): htt:// oi: /j.ijtam SSN: (Print); SSN: (Online) He s Homotoy Perturbation Metho for solving Linear an Non-Linear Freholm ntegro-differential Equations Bijan Krishna Saha *, A. M. Mohiuin, Sushanta Parua Deartment of Mathematics, The University of Barisal, Barisal, Banglaesh aress: bijanumath@gmail.com (B. K. Saha), bijan203_math@yahoo.com (B. K. Saha) * Corresoning author To cite this article: Bijan Krishna Saha, A. M. Mohiuin, Sushanta Parua. He s Homotoy Perturbation Metho for solving Linear an Non-Linear Freholm ntegro-differential Equations. nternational Journal of Theoretical an Alie Mathematics. Vol. 3, No. 6, 2017, oi: /j.ijtam Receive: Aril 29, 2017; Accete: October 8, 2017; Publishe: November 10, 2017 Abstract: n this aer, linear an non-linear Freholm ntegro-differential Equations with initial conitions are resente. Aiming to fin out an analytic an aroximate solutions to linear an non-linear Freholm ntegro-differential Equations, this aer resents a comarative stuy of He s Homotoy erturbation metho with other traitional methos namely the Variational iteration metho (VM), the Aomian ecomosition metho (ADM), the Series solution metho (SSM) an the Direct comutation metho (DCM). Comarison of the alie methos of analytic solutions reveals that He s Homotoy erturbation metho is tremenously owerful an effective mathematical tool. Keywors: Homotoy Perturbation, Variational teration, Aomian Decomosition, Series Solution, Direct Comutation Metho 1. ntrouction Many researchers an scientists stuie the integroifferential equations through their work in science alications like heat transformer, neutron iffusion, an biological secies coexisting together with increasing an ecreasing rates of generating an iffusion rocess in general. These kins of equations can also be foun in hysics, biology an engineering alications, as well as in moels ealing with avance integral equations such as [1-2], [22-23]. A new erturbation metho calle Homotoy erturbation metho (HPM) was roose in [8-19] by He in 1997, an a systematical escrition was given in 2000 which is in fact, a couling of the traitional erturbation metho an Homotoy in toology. This new metho was further eveloe an imrove by He an alie to various linear an non-linear roblems. =+,+, 0=,0 1 (1) Where is the n-th erivative of the unknown function that will be etermine,, is the kernel of the integral equation, is a known analytic function, an are linear an nonlinear functions of resectively. For =0 the equation (1) turn out to be a classical Freholm integro- ifferential equation. The Freholm integroifferential equations (1) arise from the mathematical moeling of the satiotemoral eveloment of an eiemic moel in aition to various hysical an biological moels an also from many other scientific henomena. Nonlinear henomena, which aear in many alications in scientific fiels, such as flui ynamics, soli state hysics, lasma hysics, mathematical biology an chemical kinetics, can be moele by artial ifferential equations an by integral equations as well. This aer shows a comarative stuy between He's Homotoy erturbation metho [8-19] an four traitional methos for analytic treatments of linear an nonlinear integro-ifferential equations. He's Homotoy erturbation metho, well-aresse in [8-19] has a constructive attraction that rovies the exact solution by comuting only a few iterations, mostly two iterations, of the solution series. n aition, He's technique may give the exact

2 175 Bijan Krishna Saha et al.: He s Homotoy Perturbation Metho for solving Linear an Non-Linear Freholm ntegro-differential Equations solution for linear an nonlinear equations without any nee for the so-calle He's olynomials. Also this aer, will only focus on a brief iscussion of He's Homotoy erturbation metho because the etails of the metho are foun in [8-19], an in many relate works. For the sake of self-sufficiency of the article, the variational iteration metho [7, 18-21], the Aomian ecomosition metho [5-6], the irect comutation metho (DCM) [3] an the series solution metho [4] are remine an emloye for the comarison goal. 2. Materials an Methos Homotoy erturbation metho, Variational iteration metho, Aomian ecomosition metho, series solution metho, an Direct comutation metho have been alie to analyze the behavior of the solution of freholm integroifferential equations. Finally a comarative stuy has been mae among these methos Basic ea of He s Homotoy Perturbation Metho The Homotoy erturbation metho (HPM) roose by Shijun Liao in 1992 is base on the concet of the Homotoy, a funamental concet in toology an ifferential geometry. Consier the nonlinear ifferential equation, +=, Ω (2) With bounary conitions, "#, $% &=0, ' Where, L: A linear oerator, N: A nonlinear oerator, f(r): A known Analytic function, B: A bounary oerator, ': The bounary of the omain Ω, By He s Homotohy erturbation technique (He, 1999) Define a Homotoy ),*:Ω -0,1. This satisfies 0),*=1 *-).+ *-)+).=0, (3) Or, 0),*=) + * + *-).=0, (4) Where, Ω, * -0,1. is an embeing arameter an is an initial aroximation, which satisfies the bounary conitions. Clearly $ 0),0=) =0, 0),1=)+) =0 As changes from 0 to 1, Then v(r, ) changes from to. This is calle a eformation an), )+) are sai to be Homotoy in toology. Accoring to the Homotoy erturbation metho, the embeing arameter can be use as a small arameter an assume that the solution of equation (3) an (4) can be exresse as a ower series, that is ) =) +*) +* 1 ) 1 + (5) For =1, the aroximate solution of equation (1) therefore, can be exresse as ) =lim 6 )=) +) +) 1 + (6) The series in equation (6) is convergent in most cases an the convergence rate of the series eens on the nonlinear oerator [Biazar an Ghazvini, 2009; He, 1999]. Moreover, the following jugments are mae by He (1999, 2006) 1. The secon orer erivative of ) with resect to ) must be small as the arameter may be reasonably large, that is, * # $9 &7 Must be smaller than one, so that, the series $: converges 2.2. Variational teration Metho We consier the general n-th orer integro-ifferential equations of the tye [24] ; = +;+,; < With initial conitions =?,@<< (7) ;@=,; 8, (8) Where D,E=0,1,, 1, are real constants, m an n are integer an F<. n equation (7) the function f, g an k are given an y is the solution to be etermine. Assume that the equation (7) has the unique solution. Here, change the roblem to a system of orinary integro ifferential equations an aly the variational iteration to solve it, so that the Lagrange multilier can be effectively ientifie. Using the following notation ;=;, GH =; G 1, GJ H =; G K,, G LMN H =; J G, (9) LMN Rewrite the integro-ifferential equation (7) as the system of orinary integro-ifferential equations With initial conitions GH N =; G, GH J =; G K, GH O =; G P,, GH L =? ; = G,; <Q ; =,; 8 To illustrate the basic concets of the variational iteration metho, consier the following ifferential equation -.+-.=?, Where L is a linear oerator, N is a nonlinear oerator an (10) g(x) is given continuous function. The basic character of the metho is to construct a correction functional for the system, which reas Q = + S- + T?., (11) Where S is a general Lagrange multilier which can be ientifie otimally via variational theory, it is useful to summarize the Lagrange multiliers as

3 nternational Journal of Theoretical an Alie Mathematics 2017; 3(6): C +U, C U=0,S= 1, CC +U, C U, CC U=0,S=U CCC +U, C U, CC U, CCC U=0,S= 1 2! U 1,.. +U, C U, CC U,, U=0,S= 1 8! U 8, is the n-th aroximate solution, an T enotes a restricte variation i.e, XT =0. Accoring to the variational iteration metho, to solve the system (7), construct the following correction functional ; Q =; + S,-; C ;T Q., j=1, 2,, n-1, ; Q =; +Z S,[; C?+;T = +Z,\ ;T <Q \\], Where the suerscrit (k) is the number of iterations stes. Calculating variation with resect to ; ^=1,2,3,, resectively, an noting that X; =0, We have X; Q =X; +XZ S,`; C ;T Q a =X; +S,X; c Z es, X; e =#1 S,&X; +Z f es, gx; e =0,^=1,2,., 1, X; Q =X; +XZ S,[; C?+;T c +Z,\;T <Q \\ ] =1+S,X; +Z f es, gx; e =0 For arbitrary X;,^=1,2,.,, the following stationary conitions are obtaine: = An the natural bounary conition $h N,c $c = $h J,c $c = = $h L,c $c =0, (12) 1+S,=0,^=1,2,,. The Lagrange multiliers therefore, can be ientifie as S,= 1,^=1,2,,. An the following iteration formula can be obtaine as ; Q =; Z -; C ; Q.,where ^=1,2,, 1, ; Q =; -; C?+; Beginning with ; =n,; 1 =n,; K = n 1, ; =n 8, by the iteration formula (13), we can obtain the numerical solution of equation (7) The Aomian Decomosition Metho The rincial of the ADM when alie to a general nonlinear equation in the following form ++=? (14) nvers oerator L, with 8. =., equation (14) can be written as, = 8? 8 8 (15) = +,\; <Q \\.. (13) = (16) The nonlinear oerator Nu=q is ecomose as = r (17) Where r are Aomian olynomial which are efine as, r =! Consequently, it can be written as The ecomosition metho reresents the solution of equation (15) as the following infinite series =u+ 8, = 8 8 r, 1 = 8 8 r,, = r 8 G L Gs Lq- D td; D. s =1,2,3 (18) Substituting equation (16) an (17) into equation (15) we have, = =0 8 8 r (19)

4 177 Bijan Krishna Saha et al.: He s Homotoy Perturbation Metho for solving Linear an Non-Linear Freholm ntegro-differential Equations 2.4. The Series Solution Metho Assuming that u(x) is an analytic function, it can be reresente by a series given by (20) are constants that will be etermine recursively. The first few coefficients can be etermine by using the rescribe initial conitions where we may =0,@ = C 0,@ 1 = 1 2! CC 0,. Substituting (20) into both sies of (7), an assuming that the kernel v, is searable as v,=? h), obtain ya y 2.5. The Direct Comutation Metho x y =+? (21) Assume a stanar form to the freholm integro-ifferential equation given by =+ v,, 0=,0 1 (22) Where inicates the n-th erivative of u(x) with resect to x an are constants that efine the initial conitions. This yiel, =+? v,, 0=,0 1 (23) t can easily observe that the efinite integral in the integro-ifferential equation (18) involves an integran that comletely eens on the variable t, an therefore, it seems reasonable to set that efinite integral in the right sie of (23) to a constant α, that is we set = h. (24) With α efine in (24), the equation (23) can be written by =+?. (25) t remains to etermine the constant n to evaluate the exact solution. To fin α, we shoul erive a form for by using (25), followe by substituting this form in (24). To achieve this we integrate both sies of (25) n times from 0 to x, an by using the given initial conitions 0 =,0 1 Obtain an exression for given by =*;, (26) Where *; n is the result erive from integrating (25) an by using the given initial conitions. Substituting (26) into the right han sie of (24), integrating an solving the resulting equation lea to a comlete etermination of α. The exact solution of (22) follows immeiately uon substituting the resulting value of α into (26). 3. Results an Discussions 3.1. Solving the Following Linear Freholm ntegro-differential Equation by Different Methos Examle: Consier the Freholm ntegro-differential equation CCC J = \++Z CC, 0=0, C 0=1, CC 0=0, (27) Using He s Homotoy Perturbation Metho A Homotoy can be reaily constructe as follows 0),*=) CCC + \ + J) CC, ƒ=0 (28) Substituting ) =) +*) +* 1 ) 1 + into (28) an rearranging the resultant equation base on ower of -terms, one has * : ) CCC + \ =0, (29) * :) CCC J) CC * 1 :) CCC 1 J) CC an so on With the following conitions =0, (30) =0 (31) ) 0=0,) C 0=1,

5 nternational Journal of Theoretical an Alie Mathematics 2017; 3(6): ) 0=0, G Gc ) 0=0,=1,2,3, (32) With the effective initial aroximation for ) from the conitions (32) an solution of (29), (30), (31) can be written as follows ) = E, ) = ) 1 =0, an so on. n the same manner, the rest of comonents were obtaine using the Mathemtica ackage =lim 6 =) +) +) 1 + (33) Therefore, substituting the values of ),),) ) K in (33) yiels = E Using Variational teration Metho n the view of variations iteration metho, construct a correction functional for this equation (27) is given by, Q = +S # CCC U+ \U U JUT CC &U(34) Where S is a Lagrange multilier, therefore SU= 1! U 1. (35) Now, the following variational iteration formula can be obtaine Q = 1! U 1 CCC U+ \U U JU CC U (36) We can use the initial conition to select =0+ C 0+ J 1! CC 0= Using this selection into the correction functional gives the following successive Aroximations From (36) we have, =, = + P 4! + E, 1 = K P 1152 K P ˆ P E, an so on. The variation amits the use of =lim gives the exact solution = E Using Aomian Decomosition Metho From (27) we have, CCC = \++ J CC, 0=0, C 0=1, CC 0=0, (37) Alying the three-fol integral oerator 8 efine by, J J 8.=Z Z Z. To both sies of (37), that is integrating both sies of (37) thrice from 0, an using the given initial conition we 1 obtain= E+ + J CC P! P! Using = an the recurrence relation we obtaine, = E+ P 4!, = P 4! + K P 1152, Similarly, 1 = O Ž1 + 1 ˆP, an so on. This gives the solution in the series form, = E+ P P P 4! P 4! + K 1152 K ˆP Ž = E J an this converges to the exact solution = E Using Series Solution Metho Equation (27) gives, CCC = \++ J CC, 0=0, C 0=1, CC 0=0, (38) Substituting by the series (39) Using (39) in both sies of equation (38) we = CCC Differentiating the left sies twice an by evaluating the integral at the right sies J CC 1 2@ 8K =#1 J + + &++ J K 2@ @ K +12@ P 1 +20@ Ž K + (40) Using the initial conition an equating the coefficients of the owers of x in both sies of (40) gives the recurrence relation,

6 179 Bijan Krishna Saha et al.: He s Homotoy Perturbation Metho for solving Linear an Non-Linear Freholm ntegro-differential =0,@ =1,@ 1 = K = 1 3!,@ P =0,@ Ž = 1 5!, Substituting this results into, gives the series solution, = E This gives aroximate solution Using the Direct Comutation Metho We first set n = J CC So that given equation (27) can be written as (41) CCC = \++n (42) ntegrating three times both sies of equation (42) we have = E+ + + N J K (43) Alying initial conitions in (43) we have = E+# + &P (44) Now we have n = P 48+ K K n which gives n = 1 By utting this value of n in (44) leas to the exact solution = E Solving the Following Non-Linear Freholm ntegro-differential Equation by Different Methos Examle: C = \ 1, 0=0, n (45) Using He s Homotoy Perturbation Metho A Homotoy can be reaily constructe as follows 0),*=) C \+*# P ) 1 &=0 (46) Substituting ) =) +*) +* 1 ) 1 + into (46) an rearranging the resultant equation base on ower of -terms, one has * : ) C \ =0, (47) * :) C + P * 1 :) 1 C ) 1 ) 1 an so on. With the following conition ) 0=0, =0, (48) =0, (49) ) 0=0, G Gc ) 0=0,=1,2,3, (50) With the effective initial aroximation for ) from the conitions (50) an solution of (47), (48), (49) can be written as follows ) = E,) =) 1 = =0, n the same manner, the rest of comonents were obtaine using the mathematica ackage =lim 6 =) +) +) 1 + (51) Therefore, substituting the values of ),),) 1 r ) K in (51) yiels = E Using Variation teration Metho The correction functional for this equation (45) is, Q = Z 1 24 Z UU, C U \U+ 48 U Where, S = 1 for first-orer integro-ifferential equation. Using the initial conition to select = 0= 0. This selection into the correction functional gives the following successive aroximations =0, = E , 1 = E , K = E , an so on. This gives the exact solution by = E Using Aomian Decomosition Metho From (45), C = \ 1, 0=0, (52) Alying the integral oerator 8 efine by, 8.=. To both sies of (52), that is integrating both sies of (52) from0, an using the given initial conition, which gives = E J šˆ +8 # 1 & Using = an the recurrence relation = E = Similarly, 1 = , K an.. so on. This gives the solution in the series form,

7 nternational Journal of Theoretical an Alie Mathematics 2017; 3(6): = E Thus = E Using series Solution Metho From (45), C = \ Substituting by the series 1, 0=0 (53) Using (54) in both sies of equation (53), = \ C Differentiating the left sies an by evaluating the integral at the right sies 8 =#1 J @ +@ 1 1 +@ K K +@ P P +@ Ž Ž + (55) Using the initial conition an equating the coefficients of the owers of in both sies of (55) gives the recurrence =0,@ =1,@ 1 = K = K!,@ P = 0,@ Ž =, Substituting this results into, = gives the series solution, ux=sinx x Using the Direct Comutation Metho First set, n = 1 So that given equation (45) can be written as C = \ + P ntegrating both sies of equation (57), (56) (57) = E J šˆ + J (58) Alying initial conitions in (58) = E # 96 n 48 &1 C = \ 4. Conclusion 1, 0=0. This aer shows He s Homotoy erturbation metho of solving linear an non-linear Freholm ntegro-differential Equation an conucte a comarative stuy between He s Homotoy erturbation metho an the traitional methos that is Variational iteration metho, Aomian ecomosition metho, Series solution metho an Direct comutation metho. The main avantage of the He s Homotoy erturbation metho are the fact that it rovies its user with an analytical aroximation, in many cases an exact solution in raily convergent sequence with elegantly comute terms. Also this metho hanles linear an non-linear equations in a straightforwar manner. The four traitional methos suffer from the teious work of calculation. Other traitional methos, that is usually use to solve integro-ifferential equations analytically an numerically, were not examine in this work, ue to the huge size of calculations neee by these methos. Generally seaking, He's Homotoy erturbation metho is convenient an more efficient comare to other techniques. Thus from (56) an (58), n = P 48+ K K n (59) Equation (59) gives, n = By utting this value of n in (59) leas to the aroximate solution = E X Comarison HPM VM ADM SSM DCM Figure 1. A Comarison of He s Homotoy Perturbation Metho an Traitional Methos for Solving Non-Linear Frehlom ntegro-differential Equation. References [1] A. J. Jerri, ntrouction to ntegral Equations with Alications, Marcel Dekker, New ork, [2] A. M. Golberg, Solution Methos for ntegral Equations: Theory an alications, Plenum Press, New ork, [3] A. M. Wazwaz, A First Course in ntegral Equations, Worl Scientific, [4] F. G. Tricomi, ntegral Equations, Dover, [5] G. Aomian, Solving Frontier Problems of Physics: The Decomosition Metho, Kluwer, [6] G. Aomian, A review of the ecomosition metho an some recent results for nonlinear equation, Math. Comut. Moel 13 (1992) 17_43. [7] J. Biazar, H. Ghazvini, He's variational iteration metho for solving linear an non-linear systems of orinary ifferential equations, Al. Math. Comut.191 (2007) 287_297.

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