Modification Adomian Decomposition Method for solving Seventh OrderIntegro-Differential Equations

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1 IOSR Journl of Mthemtics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 5 Ver. V (Sep-Oct. 24), PP Modifiction Adomin Decomposition Method for solving Seventh OrderIntegro-Differentil Equtions Smher M. Yssien Deprtment of MthemticsCollege of Eduction for Sciences pure Bghdd University,Irq Abstrct: In this pper, method bsed on modified domin decomposition method for solving Seventh order integro-differentil equtions (MADM). The distinctive feture of the method is tht it cn be used to find the nlytic solution without trnsformtion of boundry vlue problems. To test the efficiency of the method presented two emples re solved by proposed method. Keyword: Adomin decomposition method; boundry-vlue problems; integro-differentil eqution I. Introduction An nlyticl method clled the Adomin decomposition method (ADM) proposed by Adomin [] ims to solve frontier nonliner physicl problems. It hs been pplied to wide clss of deterministic nd stochstic problems, liner nd nonliner, in physics, biology nd chemicl rections etc. For nonliner models, the method hs shown relible results in supplying nlyticl pproimtions tht converge rpidly [2].The Adomin decomposition method (ADM) [3,4] hs been efficiently used to solve liner nd nonliner problems such s differentil equtions nd integrl equtions.the methodprovides the solution s n infinite series in which ech term cn be esily determined. The rpid convergence of the series obtined by this method is horoughly discussed bycherruult et l. [5]. Recently, Wzwz [6] proposed relible modified techniqueof ADM tht ccelertes the rpid convergence of decomposition series solution. The modified decomposition needs only slight vrition from the stndrd decompositionmethod. Although the modified decomposition method my provide the ect solution by using two itertions only nd sometimes without using the so-clled Adomin polynomils, its effectiveness is bsed on the ssumption tht the function fcn be divided into two prts, nd thus the success of themodifiedmethod depends on the proper choice of f nd f2.the ADM[7,8,9] is well-known systemtic method for solving liner nd nonliner equtions, including ordinry differentil equtions,prtil differentil equtions, integrl equtions nd integro-differentil equtions. The method permits us to solve both nonlinerinitil vlue problems nd boundry vlue problems. The method is well known, nd severl dvnced progresses re conducted inthis regrd. II. Description of the Modifiction Adomin decomposition method Since the beginning of the 98s, theadomin decompositionmethod hs been pplied to wide clss of integrl equtions.to illustrte the procedure, consider the following Volterrintegrl equtions of the second kind given by L(y ()) =()+λ K(, t) (R (y (t)) + N(y (t))) dt, λ =, () where the kernel K(, t) nd the function f() re given relvlued functions, λis prmeter, R(y()) nd N(y()) re liner nd nonliner opertors of y()[],the differentil opertor L(y ())is the highest order derivtive in the eqution, respectively, Then, we ssume tht L is invertible byusing the given conditions nd pplying the inverse opertor L - to both sides of (),we get the following eqution: y () =ψ +L - f()+l - (λ K(, t) (R (y (t)) + N(y (t))) dt),λ =, (2) where the functionψ is rising from integrting the source term from pplying the given conditions which re prescribed. And so on the Adomin decomposition method dmits the decomposition of y into n infinite series of components [] y()= n= yn()(3) Moreover, the Adomin decomposition method identifies the nonlinertermn(y()) by the decomposition series 72 Pge

2 Modifiction Adomin Decomposition Method for solving Seventh Order Integro-Differentil Equtions N(y)= n= An()(4) WhereAnis the so-clled Adomin polynomils, whichcn be evluted by the following formul [2, 3]: An= d n n! dλ N n n i= λ i y i, n =,, 2,....(5) Substituting (3) nd (4) into both sides of (2) gives n= yn()= ψ + L - f()+ L - (λ K, t [R n= y n t + n= A n (t)]dt), (6) The vrious components y n of the solution y cn be esily determined by using the recursive reltion Y =ψ + L - f(), Y k+= L - (λ K, t [R n= y k t + n= A k (t)]dt),fork. (7) Consequently, the first few components cn be written s Y = + L - f(), Y = L - ( K, t [R n= y t + n= A (t)]dt), (8) Y 2= L - (λ K, t [R n= y t + n= A (t)]dt), where the Adomin polynomil cn be evluted by (4),Hving determined the components Y n, n, the solution y in series form follows immeditely. As stted before, the series my be summed to provide the solution in closed form. We cn pply modifiction by ssuming tht the function F cn be written s F= ψ + L - f(), (9) The components Y n re determined by using the following reltion: Y = F,() Y k+= L - ( K, t [R n= y k t + n= A k (t)]dt), for k () From the bove equtions, we observe tht the component Y is identified by thefunction F. the modified Adomin decomposition method will minimize the volume of clcultions, we split the function F into two prts, F nd F. Let the function be s follows: F = F + F (2) Under this ssumption, we hve slight vrition for components Y nd Y, where F ssigned to Y nd F is combined with the other terms in () to ssign Y. The modified recursive lgorithm is s follows: y = F + L (λ K, t y K+ = L ( λ K, t y = F, R R n= y t + n= A t dt), n= y k t + n= A k t dt), (3) for k. However, the nonliner term F(y) cn be epressed by infinite series of the so-clled Adomin polynomils A n given in the form F(y) = n= A n (y,y, y 2,, y n ) (4) 73 Pge

3 Modifiction Adomin Decomposition Method for solving Seventh Order Integro-Differentil Equtions There re severl rules tht re needed to follow for Adomin polynomils of nonliner opertor F(y): A = F(y ), A 2 = y 2 F'(y ) + 2! y2 F''(y ), nd so on; see [], then substituting (5) into (4) gives A = y F'(y ), (5) F(y) = A + A + A 2 + (6) To illustrte the pplicbility nd effectiveness of the method, we presented two numericl emples. III. Appliction for Liner nd Nonlinerintegro-differentil equtions Emple[4]:- Consider the following linerintegro-differentil eqution: y (vii) () = 2-8e + y t dt, (7) withthe boundry conditions: y() =, y'() =, y''() = -, y'''() = -2, (8) y() =, y'() = -e, y''() = -2e. theectsolution is y() = (-)e. Eqution (8) cn be rewritten in opertor form s follows: Ly() = 2-8e + y t dt, (9) Operting with sevenfold integrl opertor L on (9) nd using the boundry conditions t =, we obtin the following eqution: Then, determine the constnts y 4 = A, y 5 = B, y 6 = C. y() = 2! 2 2 3! 3 + A 4! 4 + B 5! 5 + C 6! 6 +L - (2-8e )+L - ( y t dt) (2) Substituting the decomposition series (3) for y() into (2) yields n= yn()= 2! 2 2 3! 3 + A 4! 4 + B 5! 5 + C 6! 6 +L - (2-8e ) +L - ( y t dt) (2) Then, we split the terms into two prts which re ssigned to y () nd y () tht re notincluded under L in (2). We cn obtin the following recursive reltion: t y () =( 2 2 ), y ()= 2 3! 3 + A 4! 4 + B 5! 5 + C 6! 6 +L - (2-8e )+L - ( y t dt) (22) To determine the constnts A,Bnd C, we use the boundry conditions in (8) t = on the two-term pproimnt φ 2, where t t φ 2= k= yk,(23) 74 Pge

4 Modifiction Adomin Decomposition Method for solving Seventh Order Integro-Differentil Equtions The coefficients A,BndC, were obtined by using Mtlb with boundry conditions t = in (8) given A= , B = , C = (24) Then we get the series solution s follows: Y()=8 8e ( 2 3) + 9. Emple 2 [4] :- Consider the followingnonliner integro-differentil eqution: withthe boundry conditions: y (vii) () = + e y 2 t dt, (25) y() =, y'() =, y''() =, y'''() =, (26) y() = e, y'() = e, y''() = e. the ect solution is y() =e. Eqution (25) cn be rewritten in opertor form s follows: Ly() =+ e y 2 t dt,, (27) Operting with sevenfold integrl opertor L on (27) nd using the boundry conditions t =, we obtin the following eqution: y()=++ 2! 2 + 3! 3 + A 4! 4 + B 5! 5+ C 6! 6 +L - ( e y 2 t dt) (28) Then, determine the constnts y 4 = A, y 5 = B, y 6 = C. Substituting the decomposition series (3) for y()nd the series of polynomils (4) into (28) yields n= yn()=++ 2! 2 + 3! 3 + A 4! 4 + B 5! 5+ C 6! 6 +L - ( e n= A n ()dt) (29) Then, we split the terms into two prts which re ssigned to y () nd y () tht re notincluded under L in (29). We cn obtin the following recursive reltion: y () =, y () =+ 2! 2 + 3! 3 + A 4! 4 + B 5! 5+ C 6! 6 +L - ( e n= A ()dt) (3) y k+= - L - (y k ), for k. To determine the constnts A,Bnd C, we use the boundry conditions in (26) t = on the two-term pproimnt φ 2, where φ 2= k= yk,(3) The coefficients A,Bnd C, re obtined by using Mtlb which gives : A= , B= , C= Then we get the series solution s follows: Y()=3 2 7e - e Pge

5 y-is Modifiction Adomin Decomposition Method for solving Seventh Order Integro-Differentil Equtions The pproimte solutions of two numericl emples obtined with the help of MADM, in Tble - 2 respectively. From the numericl results, it is cler tht the MADM is more efficient nd ccurte. The grphicl comprison of ect nd pproimte solutions is shown in Figure -2 respectively. IV. Conclusions The objective of this pper is to present simple method to solve liner nd nonliner seventh order integro-differentil equtions without discretiztion, trnsformtion, lineriztion.modified domin decomposition methodgve good greements nd relible for results. Also this method is useful for finding n ccurte pproimtion of the ect solution. Tble : Comprison of numericl results for Emple Ect solution MADM Error MADM E E E E E E E E E E-5 Tble 2: Comprison of numericl results for Emple 2 Ect solution MADM Error MADM E E E E E E E E The solution t Y2 Approimte series Ect is Figure : Comprison between the ect nd MADM solution of emple Pge

6 y-is Modifiction Adomin Decomposition Method for solving Seventh Order Integro-Differentil Equtions Approimte series Ect The solution t Y is Figure 2: Comprison between the ect nd MADM solution of emple References []. G. Adomin, Solving Frontier Problems of Physics: The Decompositions Method, Kluwer, Boston,(994). [2]. W. Chen nd Z. Lu, An lgorithm for Adomin decomposition method.appl Mth Comput, 59 (24) [3]. G. Adomin, A review of the decomposition method nd some recent results for nonlinerequtions, Mthemticl nd Computer Modelling, vol. 3, no. 7, pp. 7 43, 99. [4]. G. Adomin, Solving Frontier Problems of Physics: The Decomposition Method, vol. 6 of Fundmentl Theories of Physics, Kluwer Acdemic, Dordrecht,TheNetherlnds, 994. [5]. Y. Cherruult, G. Sccomndi, nd B. Some, New results for convergence of Adomin s method pplied to integrl equtions, Mthemticl nd ComputerModelling, vol. 6, no. 2, pp , 992. [6]. A.-M.Wzwz, A relible modifiction of Adomin decomposition method, AppliedMthemtics nd Computtion, vol. 2, no., pp , 999. [7]. w. Abdul-Mjid, R. Rndolph nd D. Jun-Sheng, A study on the systems of thevolterr integrl forms of the Lne Emden equtions by the Adomindecomposition method, MOS subject clssifiction: 34A34; 35C5; 37C, Published online 2 Mrch 23 in Wiley Online Librry. [8]. Wzwz AM. Prtil Differentil Equtions nd SolitryWves Theory. HEP nd Springer: Beijing nd Berlin, 29. [9]. Rch R. A new definition of the Adominpolynomils.Kybernetes 28; 37: []. B. Roberto, A New Modifiction of Adomin Decomposition Method for Volterr Integrl Equtions of the Second Kind, Deprtment of Mthemtics, Fculty of Science, Ningbo University, Ningbo, Zhejing 352, Chin, Volume 23,Article ID 7955, 7 pges []. H.Wleed, Solving nth-order Integro-Differentil Equtions Using thecombined Lplce Trnsform-Adomin Decomposition Method, Deprtment ofmthemtics, College of Computer Science nd Mthemtics, University of Mosul, Mosul, Irq,Applied Mthemtics, 23, 4, [2]. A.-M. Wzwz, A new lgorithm for clculting Adomin polynomils for nonliner opertors, AppliedMthemtics nd Computtion, vol., no., pp. 33 5, 2. [3]. E. Bbolin nd Sh. Jvdi, New method for clculting Adomin polynomils, AppliedMthemtics nd Computtion, vol. 53, no., pp , 24. [4]. N. S. Jfr nd G. A. Fhimeh, Vritionl itertion method for solving Seventh order integro-differentil equtions, Deprtmn of Applied Mthemtics, School of Mthemticl Sciences, Ferdowsi University of Mshhd, Irn.-2 Mrch Pge