Modified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations

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1 Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro- Differetial Equatios M. Al-Mazmumy Departmet of Mathematics, Faculty of Sciece-AL Faisaliah Campus Kig Abdulaziz Uiversity, Jeddah, Saudi Arabia S. O. Almuhalbedi Departmet of Mathematics; Faculty of Sciece Kig Abdulaziz Uiversity Jeddah, Saudi Arabia Copyright 27 M. Al-Mazmumy ad S. O. Almuhalbedi. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract I this paper, Modified Decompositio Method by Adomia ad Rach has bee implemeted to aalyze oliear Volterra itegro-differetial equatios. The suggested approach is much better tha the stadard versio of the Adomia decompositio method. Some examples are provided to illustrate the method. Mathematics Subject Classificatio: 45D5; 45E; 65M2 Keywords: Adomia decompositio method; Modified Decompositio Method by Adomia ad Rach; Volterra itegro-differetial equatios; oliear itegrodifferetial equatios. Itroductio I 99, Adomia ad Rach [] itroduced a modificatio to the classical Adomia method, based o Multiple decompositio. For computatioal coveiece purpose, they itroduced the decompositio of both the system iputs ad system operators together with the solutio process to achieve simple to itegrate series for liear ad oliear differetial equatios. I 99, Adomia ad Rach [2] defied a oliear Trasformatio Series that will be

2 58 M. Al-Mazmumy ad S. O. Almuhalbedi evaluated usig the Adomia polyomials i the decompositio method. Give a specific fuctio N(u(t)) with u give by a coverget series, the evaluatio of N(u(t)) ca be made by usig the Adomia polyomials used to represet the oliearities i differetial ad partial differetial equatios. For a aalytic fuctio N(u(t)),, it ca be writte N(u(t)) = = A ( u, u u 2,, u ) where the A, are give by A = = c(ν, ) N (ν) (u) u=u,. To determie c(ν, ), the sum of possible products of v compoets u i ca be formed, with i =,, 2,..., ad divide by the factorial of the umber of repetitios of the compoets. I [2] Adomia ad Rach proposed the coverget series N(u) with u = c x, ad they stated the followig theorem: = Theorem : N(u) = N( = c x ) = = A ( c, c c 2,, c ). I 992, Adomia ad Rach [3] geeralized the theorem o trasformatio of series preseted i Ref.[2] to fuctio of several variables. The result is valuable i derivig solutios of coupled ordiary differetial equatios with geeral oliear couplig terms. Usig the modified decompositio method of the authors, they stated the followig theorems: Theorem 2. If u = = a x, ad v = = b x, the N(u, v) = N( = A x ), A = A ( a, a,, a, b, b, b ) where the A are Adomia polyomials of fuctios of two variables N(u, v). This ca be exteded to -dimesios ad -dimesioal Adomia polyomials. I iitial-value problem defied by a system of liear homogeeous differetial equatios, the power series solutios yields simple recurrece relatios for the coefficiets. But such solutios are geerally ot adequate for oliear equatios, although they are applicable to some simple cases such as the Riccati equatio. The results from the Adomia decompositio method [] ad Theorem 2 were used by Adomia ad Rach o trasformatio of series, stated i Theorem, i order to exted the Maclauri method. Adomia solved the geeral operator equatios Fu = g by usig the decompositio u = = u, which is a special case of the expasio of N(u) = = A ( u, u u 2,, u ). The coefficiets A are derived by coveiet algorithms for N(u). Thus, N( = u ) = = A ( u, u u 2,, u ). If u is give by the series u = = c x, each compoet u of the power series ca be idetified. This will lead to A ( u,, u ) = x A ( c,, c ). The series = x A ( c,, c ) is coverget if the series = c x is coverget. This is simply a extesio to Taylor series: N ( c (x x ) ) = A ( c, c,, c )(x x ) = =

3 Modified decompositio method by Adomia ad Rach 59 Thus, the Adomia series is actually a geeralized form of Taylor series about a fuctio rather tha a poit. I a liear case, it ca be reduced to the well-kow Taylor series. Further, Maclauri series ca be made more useful by combiig it with Adomia polyomials ad decompositio techiques. I Refereces [2] ad [3], the authors showed that the exteded Maclauri series ca be used for coupled differetial equatios ad partial differetial equatios. They also showed that the Adomia series resultig from the decompositio method has superior covergece properties, i spite the improved power of the Maclauri series with the use of the Adomia polyomials ad decompositio techiques. It is also evidet that the power series solutio becomes complicated i collectig terms, while the decompositio series is always simple i this respect. Further work to Adomia ad Rach [4] showed the applicability of the method for the solutio of liear or oliear partial differetial equatios ad coupled equatios; ad they also showed that the covergece is faster usig Adomia decompositio series tha usig the exteded or improved power series. Usig the cocept of modified decompositio series, Adomia ad Rach [5] established that ihomogeeous oliear partial differetial equatios with variable coefficiets ad iputs are solvable ad the coefficiets were easily programmable. Also, they foud the solutios of oliear partial differetial equatios usig a exteded Maclauri series form of the decompositio method ad the Adomia polyomials [6]. Recetly, several other researchers have applied this modificatio for solvig a wide class of problems. Lazhar Bougoffa [7] studied the solvability of the predator ad prey system with variable coefficiets ad showed the compariso of the results with this modified decompositio method. I Ref. [8], M. Almazmumy, et al. used this modificatio i the solutio of the iitial value problems i ordiary differetial equatios. 2. Modified Decompositio Method by Adomia ad Rach We cosider the itegro-differetial equatio of the form u (x) = f(x) + b(x) a k(x, t). (Lu(t) + N(u(t))dt (2.) with iitial coditios u() = α, u () = β. x x Let L = d2 dx 2, so L (. ) = (. )dxdx, applyig L to both sides of (2.), ad usig iitial coditios, we obtai: u(x) = α + βx + L f(x) + L b(x) a k(x, t). (Lu(t) + Nu(t)) dt (2.2) We proceed the solutio u(x), the ihomogeeous term ad the oliear term i a series form u(x) = = a x (2.3) f(x) = = b x (2.4)

4 6 M. Al-Mazmumy ad S. O. Almuhalbedi N(u) = = A x (2.5) where the polyomials A, =,, are called Adomia polyomials, ad let the kerel be formed as k(x, t) = m= = c m, x m t = = c (x)t (2.6) where c (x) = m= c m, x m. Substitutig (2.3)- (2.6) i (2.2) gives = a x = α + βx + L = b x + +L b(x) c m, x m t ( = a t + = A t )dt (2.7) a = m= Thus = a x = α + βx + L = b x + +L b(x) [ c m, x m t ( = t (a + A ))] dt (2.8) a = m= We observe that ( = c m, t )( = a t ) = = t ( c m,v (a v ) ad Substitutig v= ) v= ) ( = c m, t )( = A t ) = = t ( c m,v (A v ) = a x = α + βx + L = b x + L b(x) [ x m t. (a v + A v )]dt a = m= v= c m,v So that = a x = α + βx + L b x + L [ = m= v= c m,v (a v + A v )dt = x m++ + = a x b = α + βx + = (+)(+2) x+2 + m= x +m+3 = c (+)(m++2)(m++3) v= m,v (a v + A v ) The = a x = α + βx + b 2 =2 ( ) x + m= x m+ 3 =3 c ( 2)(m+ )(m+) v= m,v (a v 3 + A v 3 ) So that a + a x + a 2 x 2 + =3 =3 a x = α + βx + 2 b x 2 + x m+ =3 ( ) b 2x + m= v= c m,v. (a v 3 + A v 3 ) ( 2)(m+ )(m+) 3

5 Modified decompositio method by Adomia ad Rach 6 Equatig the coefficiets of like powers of x i both sides a = α a = β a 2 = 2 b a = b ( ) c ( 2)( ) v=,v. (a v 3 + A v 3 ) a +m =, m (2.9) By determie the coefficiet ad substitutig i (2.3) we get the solutio. 3. Computatioal results Example 2. Cosider oliear Volterra itegro-differetial equatio u (x) = e x + 2 e2x x 2 u2 (t)dt, u() = With the exact solutio is u(x) = e x. Applyig L x (. ) = (. )dx i both sides give u(x) = L (e x + 2 e2x x 2 ) L u 2 (t)dt Busig modified decompositio method by Adomia ad Rach (MDAR), leadig to x A 2 =2 ( ) x = a x = + L b x L = A t dt = the a x = + a + a x + Where, =2 b x = = = b x a x = + b x + = L (e x + 2 e2x ) 2 Equatig the coefficiets =2 b x A 2 =2 ( ) x = e x + 4 e2x x = x + x2 + 2 x3 +

6 62 M. Al-Mazmumy ad S. O. Almuhalbedi b = b = 2 b 2 = 3 2 b 3 = We ow get the recursive relatio a = a = a = b A 2 ( ), 2 or a = a = a 2 = b 2 A 2 = 2 a 3 = b 2 3 A 6 = 3! a 4 = b 3 4 A 2 2 = 4! The exact solutio is give by u(x) = = a x = + x + 2 x2 + 3! x3 + 4! x4 + = e x Example 2.2 Cosider the system of oliear Volterra itegro-differetial equatio u x (x) = 2 3x 2 si(x) + 3 cos(x) + (u 2 (t) + v 2 (t))dt, u() = v (t) = 2 + si(x) + 2 cos(x) + si(2x) + 2 x (u2 (t) v 2 (t))dt, v() = 2 With the exact solutio (u(x), v(x)) = ( + si(x), + cos(x)). ApplyigL x (. ) = (. )dx i both sides u(x) = 2x 3 x 2 x2 + 2 cos(x) si(x) + L (u 2 (t) + v 2 (t))dt, v(x) = 2 + 2x cos(x) + 2 si(x) 4 cos(2x) + 5 x 4 + L (u 2 (t) v 2 (t))dt Usig the method (MDAR) we get

7 Modified decompositio method by Adomia ad Rach 63 = a x = + L b x + L ( A t + = = = B t )dt g x = 2 + L R x + L x = = ( A t = = B t )dt So that a + a x + g + g x + Where = Ad b x R = =2 =2 a x = + b x + g x = 2 + R x + x b =2 R =2 = ( 2x 3 2 x2 + 2 cos(x) 2 + 3si (x)) = x 5 2 x2 2 x3 + 2 x4 + 4 x5 + x = ( 2x cos(x) + 2 si(x) cos(2x) + 5 ) 4 4 =x 2 3 x x4 + 6 x5 + Equatig coefficiets ( ) x x + =2 (A 2 _ + B 2 ) ( ) x x + =2 (A 2 B 2 ) b = b = 5 b 2 = 3 2 b 3 = 3, R = R = 2 R 2 = R 3 = 5 6 b 4 = 8. R 4 = 2 We ow get the recursive relatio a = a = b a = b + ( ) (A 2 + B 2 ), 2 ad g = g = R g = R + ( ) (A 2 B 2 ), 2 or

8 64 M. Al-Mazmumy ad S. O. Almuhalbedi a = a = b = a 2 = 2 b + 2 (A + b ) = a 3 = 3 b (A + b ) = 6 a 4 = 4 b (A 2 + b 2 ) = a 5 = b (A b 3 ) = 2 ad g = 2 g = R = g 2 = 2 R + 2 (A b ) = 2 g 3 = 3 R (A b ) = g 4 = 4 R (A 2 b 2 ) = 24 g 5 = 5 R (A 3 b 3 ) = The the series solutio ca be writte as u(x) = = a x = a + a x + a 2 x = + x 3! x3 + 5! x v(x) = = g x = g + g x + gx 2 + = + ( 2! x2 + 4! x4 +.. ) Or u(x) = + si(x) v(x) = + cos(x) Example 2.3 Cosider oliear Volterra itegro-differetial equatio u (x) = xe x2 x x + xte u2 (t) dt, u() = With the exact solutio u(x) = x. Applyig L x (. ) = (. )dx i both sides give, u(x) = L ( 2 x + xe x2 2 Usig the method (MDAR) we get ) + L x xte u2 (t) dt x = a x = L = b x + L m= = c m, x m t = A t dt

9 Modified decompositio method by Adomia ad Rach 65 the = = x a x = L b x + L m= x m t c m,v A v dt So that b = x m++2 v= = a x = = + x+ + m= = c (+)(m++2) v= m,v A v a + a x + Where b =2 a x = b x + =2 x m+ b x x + =2 c,v A v 2 ( ) m= =2 c m,v A v 2 ( )(m+) 2 v= = x = L ( xe x2 x + ) = x x e x2 = x 8 x x6 96 x x + Equatig coefficiets b = b = b 2 = b 3 = 2 b 4 = 2 v= + b 5 = 4 m= Sice = c m, x m t = xt, we have c, = c, = c m,=, m, Therefore a = a = b a = b + 2 c ( ) v=,v. A v 2, 2 a +m =, 2, m Or a = a = b = a 2 = b + c,a = 2 2 a 3 = b 2 + c,a +c, A = 3 6 a 4 = b 3 + c,a 2 +c, A +c,2 A = 4 2 8

10 66 M. Al-Mazmumy ad S. O. Almuhalbedi The the series solutios ca be writte as u(x) = a x = x 8 x x6 96 x8 + = 48 x +.. The results produced by the preset method with oly few compoets (m=) are i a very good agreemet with the best of the results of the methods listed i Table. The preset method solutio compared with exact solutio i Figure (). x Exact (MDAR) Error....e e e e e e-3 Table (): compariso betwee exact solutio ad approximate solutio usig method Figure : compariso betwee exact solutio ad approximate solutio usig method (MDAR) Example 2.4 cosider the system of oliear Volterra itegro-differetial equatio u (x) = x + 2 x2 2 x4 x + ((x t)u 2 (t) + v 2 (t))dt, u() = v (t) = x 3 2 x2 2 x4 x + u 2 (t) + (x t)v 2 (t))dt, v() = With the exact solutio are ( + x, x). Applyig L x (. ) = (. )dx

11 Modified decompositio method by Adomia ad Rach 67 u(x) = + x 2 x2 + 6 x3 6 x5 + L x ((x t)u2 (t) + v2 (t))dt, v(x) = x 2 x2 2 x3 6 x5 + L x (u 2 (t) + (x t)v 2 (t))dt, Usig the method (MDAR) we get = a x = + L b x + L x (= t ( m= x m = v= c m,v A v + B ))dt g x = + L R x + L x = = ( t (A m= x m = c m,v B v ))dt therefore a + a x + ad g + g x + =2 =2 a x = + b x + = x + m= + x ( =2 b v= 2 c ( )(+m) m,va v 2 x m + B ( ) 2) v= R g x = + R x + = x + + =2 x ( A ( ) 2 m= 2 c ( )(+m) m,vb v 2 x m ) v= Where = b x 6 x5, equatig coefficiets = x 2 x2 + 6 x3 6 x5, ad = R x = x 2 x2 2 x3 b = b = b 2 = 2 b 3 = b 4 = 2 b =, 5, R = R = R 2 = 3 2 R 3 = R 4 = 2 R =, 5 =2 m= c, = c, = c m, =, m, 2 Sice c m, x m t = x t, we have we ow get the recursive relatio a = a = b a = b + ( 2 c ( ) v=,v A v 2 + B 2 ), 2 a +m =, 2, m

12 68 M. Al-Mazmumy ad S. O. Almuhalbedi ad g = g = R g = R + (A ( ) 2 2 v= c,v B v 2 ), 2 g +m =, 2, m Or a = a = b = a 2 = 2 b + 2 (c,a + B ) = a 3 = 3 b (c,a + c, A + B ) = a 4 = 4 b (c,a 2 + c, A + c,2 A + B 2 ) = 2 a 5 = b (c 2,A 3 + c, A 2 + c,2 A + c,3 A + B 3 ) = 5 ad g = g = R = g 2 = 2 R + 2 (A c, B ) = g 3 = 3 R (A c, B c, B ) = g 4 = 4 R (A 2 c, B 2 c, B c,2 B ) = 2 g 5 = 5 R (A 3 c, B 3 c, B 2 c,2 B ) = the the series solutio ca be writte as u(x) = = a x = a + a x + a 2 x = + x 3 x3 v(x) = = g x = g + g x + gx 2 + = x 3 2 x4 + 2 x4 5 x5 + The results produced by the preset method with oly few compoets (m=5) are i a very good agreemet with the best of the results of the methods listed i Table (2-a) ad (2-b). The (MDAR) solutio compared with exact solutio i Figure (2-a) ad Figure (2-b). x Exact (MDAR) Error....e e e e e e-2 Table (2-a): compariso betwee exact solutio u(x) ad Approximate Solutio usig method (MDAR)

13 Modified decompositio method by Adomia ad Rach 69 Figure (2-a): compariso betwee exact solutio u(x) ad Approximate Solutio usig method(mdar) x Exact M Error....e e e e e e-3 Table (2-b): compariso betwee exact solutio v(x) ad Approximate. Solutio usig method (MDAR) Figure (2-b): compariso betwee exact solutio v(x) ad Approximate. Solutio usig method (MDAR) 4. Coclusio I this study, the modificatio of Adomia ad Rach was applied to Volterra itegro-differetial equatios. We obtaied accurate approximatio aalytical solutio for Volterra itegro-differetial equatios ad the system of Volterra itegro-differetial equatios. Refereces [] R. Rach, G. Adomia, Multiple Decompositios for Computatioal Coveiece, Appl. Math. Lett., 3 (99), o. 3,

14 7 M. Al-Mazmumy ad S. O. Almuhalbedi [2] R. Rach, G. Adomia, Trasformatio of Series, Appl. Math. Lett., 4 (99), o. 4, 69-7l. [3] R. Rach, G. Adomia, Noliear Trasformatio of Series- PART II, Computers Math. Appl., 23 (992), o., [4] R. Rach, G. Adomia ad R. E. Meyers, A modified Decompositio, Computers Math. Appl., 23 (992), [5] G. Adomia ad R. Rach, Ihomogeeous oliear partial differetial equatios with variable coefficiets, Appl. Math. Lett., 5 (992), [6] G. Adomia ad R. Rach, Modified Decompositio Solutio of oliear partial differetial equatios, Appl. Math. Lett., 5 (992), [7] Lazhar Bougoffa, Solvability of the predator ad prey system with variable coefficiets ad compariso of the results with modified decompositio, Applied Mathematics ad Computatio, 82 (26), o, [8] M. Almazmumy, F. A. Hedi, H. O. Bakodah, H. Alzumi, Recet Modificatios of Adomia Decompositio Method for Iitial Value Problem i Ordiary Differetial Equatios, America Joural of Computatioal Mathematics, 2 (22), Received: Jauary 2, 27; Published: February 28, 27