Adomian Decomposition Method Using Integrating Factor

Transcription

1 Commun. Theor. Phys. 6 (213) Vol. 6, No. 2, August 15, 213 Adomian Decomposition Method Using Integrating Factor Yinwei Lin, 1, Tzon-Tzer Lu, 2, and Cha o-kuang Chen 1, 1 Department of Mechanical Engineering, National Cheng Kung University, Tainan 711, Taiwan, China 2 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 8424, Taiwan, China (Received November 3, 212; revised manuscript received May 1, 213) Abstract This paper proposes a new Adomian decomposition method by using integrating factor. Nonlinear models are solved by this method to get more reliable and efficient numerical results. It can also solve ordinary differential equations where the traditional one fails. Besides, the complete error analysis for this method is presented. PACS numbers: 2.6.-x, 2.3.Hq, 2.3.Mv Key words: Adomian decomposition, integrating factor, nonlinearity, analytic solution, Taylor series, error analysis 1 Introduction Since Adomian in early 198 s developed the concept of the Adomian decomposition method (ADM), [1 ADM as well as its various modified versions has been extensively used to solve many nonlinear problems, including ordinary differential equations (ODE), [2 5 partial differential equations, [6 7 differential-algebraic equations, [8 9 differential-difference equations, [1 11 and integro-differential equations. [12 16 Numerical solutions can also be obtained. [17 24 Furthermore, Lai et al. [25 applied ADM to solve some eigenvalue problems. It is well known now in literature that this algorithm gets the rapidly convergent solution. The purpose of this paper is to introduce a new reliable modification of ADM, i.e. the ADM with integrating factor. Our new method gives better approximation of the solution than the traditional one. For those problems that the standard ADM fails, our new scheme may still converge. Furthermore, we have a detail error analysis of this method. 2 Modified Technique We first give a brief outline of the standard ADM. For this reason, the differential equation Lu(x) + Ru(x) + N(u(x)) = g(x), (1) is considered, where L is the highest order derivative and easily or trivially invertible linear operator, R is the remaining part of the linear operator and N is the nonlinear operator. Assume this ODE has a unique solution. Applying the inverse operator L 1 of L on both sides of (1), we obtain u = L 1 g + ϕ(x) L 1 Ru L 1 N(u). (2) Here ϕ(x) satisfies Lϕ =, which is normally found by the initial conditions. According to the Adomian decomposition method, u(x) is defined by the series u = u i (3) i= and the nonlinear terms are by the series N(u) = A n (u, u 1,...,u n ). (4) n= The components of A i are called the Adomian polynomials, e.g. given by Wazwaz [26 A = N(u ), A 1 = u 1 N (u ), A 2 = u 2 N (u ) + 1 2! u2 1N (u ), A 3 = u 3 N (u ) + u 1 u 2 N (u ) + 1 3! u3 1N (u ), (5) The other polynomials can be produced by the general form A n = 1 [ d n ( n! dλ n N λ i u i )λ=. i= We can find A depends only on u, A 1 depends only on u and u 1 and so on. From (2), we have the iteration of standard ADM u = L 1 g + ϕ(x), u 1 = L 1 Ru L 1 A, u 2 = L 1 Ru 1 L 1 A 1, u i+1 = L 1 Ru i L 1 A i, (6) panguapig@yahoo.com.tw ttlu@math.nsysu.edu.tw Corresponding author, chen@mail.ncku.edu.tw c 213 Chinese Physical Society and IOP Publishing Ltd

2 16 Communications in Theoretical Physics Vol. 6 for i = 1, 2,... So the solution u(x) is decomposed into an infinite series (3) and the series is supposed to be convergent. Now, we present our new ADM using integrating factor. Consider the first order ordinary differential equation on I = [, B y (x) + L(x)y(x) = c(x)n(y(x)) + r(x), y() = a, (7) where L(x) is the coefficient of linear term and N the nonlinear term. In the sequel we assume (7) has a unique solution. Assume L(t) is continuous. Multiplying the integrating factor exp [ L(x)dx on both side of (7), we have d dx ( e L(x)dx y ) = e L(x)dx (y + Ly) = e L(x)dx (cn + r). Then we integrate it from to x. The function ϕ in (2) satisfies dϕ/dx =, i.e. ϕ is a constant. In fact it is the constant of integration, which can be found by the initial value. So ϕ = y() = a. Finally we have the solution [ y(x) = a exp L(x)dx [ + exp L(x)dx e L(s)ds [c(t)n(y(t)) + r(t)dt. (8) Let g(x) = L(t)dt, then (8) becomes y(x) = e g(x){ } a + e g(t) [c(t)n(y(t)) + r(t)dt. (9) The standard ADM for (9) can obtain its true solution y(x) = y i (x), (1) where i= y (x) = e g(x)[ a + e g(t) r(t)dt, y i (x) = e g(x) e g(t) c(t)a i 1 (t)dt (11) for i = 1, 2,... In general itis time-consuming to compute the integrating factor e L = e g(x). By Taylor series expansion e z 1 = zj, e z ( 1) j = z j, (12) j= j= j= we use m+1 finite terms of the following series instead in numerical computation e g(x) g j (x) =, e g(x) ( 1) j g j (x) =. (13) j= So we have truncated version of (9) ( 1) j g j (x) { ŷ(x) = a + j= [ m j= g j (t) where By ADM its solution ŷ (x) = ŷ i (x) = j= j= } [c(t)n(ŷ(t)) + r(t)dt. (14) ŷ(x) = ŷ i (x), (15) i= ( 1) j g j (x) [ a + ( 1) j g j (x) [ m j= j= g j (t) g j (t) r(t)dt, c(t)a i 1 (t)dt (16) for i = 1, 2,..., n. For numerical solution, only finite terms are possible and the partial sum S n = n i= ŷi(x) is the desired approximation. Beside (5), El-Kalla [27 had another choice for the Adomian polynomials A n, which are obtained by the following formula n 1 A n = N(S n ) A i. (17) i= We will use this set of A n in Secs. 3 and 4. 3 Convergence Analysis In this section we do the error analysis of our new method and state several theoretic theorems. The function space we consider is C(I), all continuous functions on bounded interval I. The norm we use is the sup-norm f(x) = max x I f(x). We have several assumptions for the following results. Suppose the nonlinear term N in (7) is Lipschitz continuous with Lipschitz constant K, i.e. N(y) N(z) K y z. (18) If L(t) is continuous, then so is g(x) = L(t)dt. For simplicity in later proofs, let ( 1) j g j (x) g j (x) p(x) =, q(x) =. j= j= Since a continuous function on finite interval [, B is also bounded, suppose P = p(x) = max x [,B p(x), Q = q(x) = max x [,B q(x), P = e g(x), Q = e g(x). Assume both c(x) and r(x) in (7) are continuous too, and C = c(x), R = r(x). Theorem 1 (Uniqueness of (14)) Assume L(y), c(x) and r(x) are continuous on [, B, the nonlinear term N is Lipschitz continuous (18). If α = PQBCK < 1, then (14) has at most one solution. Proof Let ŷ and ẑ be two solutions of (14), then ŷ(x) ẑ(x) = p(x) q(t)c(t)[n(ŷ(t)) N(ẑ(t))dt p(x) q(t) c(t) N(ŷ(t)) N(ẑ(t)) dt

3 No. 2 Communications in Theoretical Physics 161 PQCK PQCK ŷ(t) ẑ(t) dt ŷ(t) ẑ(t) dt PQCK ŷ ẑ B, x [, B. (19) The above inequality does not depend on x in the interval [, B, so we have Since α < 1, ŷ ẑ = max ŷ(x) ẑ(x) α ŷ ẑ. (1 α) ŷ ẑ ŷ ẑ = ŷ(x) = ẑ(x). The same technique can be applied to the original ODE (8), or equivalent (7), to get its uniqueness of solution by replacing m = in Theorem 1. Theorem 2 (Uniqueness of (7)) Under the same assumptions of Theorem 1 and α = P QBCK < 1, (7) has at most one solution. Theorem 3 (Convergence and Existence) Under the same assumptions of Theorem 1 and α < 1, the ADM solution (15) converges. So the perturbed ODE (14) has a unique solution. Proof We will prove that the sequence of partial sums {S n } is a Cauchy sequence in the Banach space (C(I), ). For k n and x I, k S k S n = max S k S n = max ŷ i (x) From (17), we have = maxp(x) = maxp(x) q(t)c(t) i=n+1 k i=n+1 i=n A i 1 dt k 1 q(t)c(t) A i dt. k 1 A i = N(S k 1 ) N(S n 1 ). i=n Similar to the inequality (19), S k S n = max p(x) q(t)c(t)[n(s n 1 ) N(S k 1 )dt So PQCKB S n 1 S k 1 = α S n 1 S k 1. For k = n + 1, we have S n+1 S n α S n S n 1 α 2 S n 1 S n 2 α n S 1 S = α n ŷ 1 (x). S k S n S k S k 1 + S k 1 S k S n+1 S n (α k 1 + α k α n ) ŷ 1 (x) = α n( 1 α k n ) ( α n ) ŷ 1 (x) ŷ 1 (x). 1 α 1 α Notice that ŷ 1 (x) = maxp(x) q(t)c(t)n(ŷ (t))dt PQCB max N(ŷ (t)) = α K N(ŷ (x)). Since g(x) and r(x) are continuous, so is ŷ (t) of (16). In view of the Lipschitz continuity of N, N(ŷ (x)) is also continuous and N(ŷ (x)) is finite. Thus by < α < 1, S k S n ( αn 1 α ) α K N(ŷ (x)), n. (2) So {S n } converges to some function ŷ(x) in C(I). By ADM (16) and (17), [ S n = p(x) a + q(t)r(t)dt + p(x) { = p(x) a + q(t)c(t) n A i 1 (t)dt i=1 } q(t)[r(t) + c(t)n(s n 1 (t))dt. Let n, then we can see the limit ŷ(x) of S n satisfies the ODE (14). This proves the existence of solution of (14). The uniqueness part is from Theorem 1. Theorem 4 (Truncation Error) Under the same assumptions of Theorem 3, the numerical solution S n of (15) has the error S n ŷ(x) j= αn+1 K(1 α) N(ŷ (x)). Proof This result follows directly from setting k in (2). Similarly using the ADM iteration (1) and (11), we can show the existence of solution of ODE (8), or its equivalence (7). Theorem 5 (Existence of (7)) Under the same assumptions of Theorem 2, (7) has a unique solution. Proof The partial sum of S n = n i= y i(x) of (1) is a Cauchy sequence in (C(I), ) as proved in Theorem 3. So { S n } converges to some function y(x) in C(I), which satisfies the ODE (8). From the right hand side of (8) we see this y(x) is differentiable, so it satisfies (7). This proves the existence of solution of (7). Its uniqueness is from Theorem 2. Notice that the Taylor series (12) of e z and e z converge uniformly in any finite interval. So in [, B, since g(x) is bounded, the series (13) of e g(x) and e g(x) also have uniform convergence. Hence give any ε > there is an positive integer m such that e g(t) g j (t) ε, e g(x) j= ( 1) j g j (x) ε. (21) Also recall that (7), (8) and (9) are equivalent. Theorem 6 (Perturbed Error) Under the same assumptions of Theorem 3 and (21), the solutions of (9) and (14)

4 162 Communications in Theoretical Physics Vol. 6 can be very close, i.e. ŷ(x) y(x) for some positive constant β. εβ 1 α Proof In view of (9) and (14) we have y(x) ŷ(x) a e g(x) p(x) (22) + e g(x) e g(t) c(t)n(y(t))dt p(x) p(x) for x [, B. For (22) q(t)c(t)n(ŷ(t))dt (23) + e g(x) e g(t) r(t)dt q(t)r(t)dt (24) a e g(x) p(x) a e g(x) p(x) a ε. The second part (23) e g(x) e g(x) + p(x) + p(x) e g(t) c(t)n(y(t))dt p(x) e g(t) c(t)n(y(t))dt p(x) e g(t) c(t)n(y(t))dt p(x) q(t)c(t)n(y(t))dt p(x) q(t)c(t)n(ŷ(t))dt e g(t) c(t)n(y(t))dt q(t)c(t)n(y(t))dt q(t)c(t)n(ŷt))dt e g(x) p(x) PC N(y(x)) B + P e g(t) q(x) C N(y(x)) B + PQCK y ŷ B ε PBC N(y) + εpbc N(y) + PQBCK y ŷ. The last term (24) e g(x) e g(t) r(t)dt p(x) e g(x) q(t)r(t)dt e g(t) r(t)dt e g(x) q(t)r(t)dt + e g(x) q(t)r(t)dt p(x) P e g(x) q(x) RB + e g(x) p(x) QRB ε PRB + εqrb. Since the above inequalities hold for all x [, B, (22)-(24) give q(t)r(t)dt y(x) ŷ(x) a ε + ε PBC N(y) + εpbc N(y) + α y ŷ + ε PRB + εqrb, (1 α) y ŷ a ε + ε(p + P)BC N(y) + ε( P + Q)RB, y ŷ ε 1 α [ a + (P + P)BC N(y) + ( P + Q)RB. Theorem 7 (Convergence) Under the same assumptions of Theorem 3, the numerical solution S n of (14) converges to the exact solution y(x) of (9) as m, n. Proof By Theorem 4 and 6 S n y(x) S n ŷ(x) + ŷ(x) y(x) αn+1 K(1 α) N(ŷ ) + εβ 1 α. In view of (21), ε can be as small as we wish if m is large enough. 4 Numerical Examples Now we show some numerical experiments of our new ADM with integrating factor and compare the results with the traditional one without integrating factor. All computations are coded under the computer algebra package Mathematica with 16 working decimal digits. Example 1 For the nonlinear differential equation y (x) + xy(x) = (x + 2x 3 )y 3 (x), y() = 1 3, (25) in [28, Exercise 5.8.3d, its exact solution is y(x) = (3 + 2x e x2 ) 1/2. Multiply the integrating factor e x2 /2 to both sides of (25), we have d 2 dx (ex /2 y) = e x2 /2 (y + xy) = e x2 /2 (x + 2x 3 )y 3. (26) Integrate (26) both sides and substituting the initial value y() = 1/3, then y(x) = 1 3 e x2 /2 + e x2 /2 By (16), we have ŷ (x) = 1 ( x 2 /2) j, 3 j= e t2 /2 (t+2t 3 )y 3 (t)dt. (27)

5 No. 2 Communications in Theoretical Physics 163 [ m ( x 2 /2) j ŷ i (x) = j= [ m (t 2 /2) j j= (t + 2t 3 )A i 1 (t)dt, i 1, (28) where the nonlinear terms by (17) are A = ŷ 3, A 1 = (ŷ + ŷ 1 ) 3 ŷ 3, A 2 = (ŷ + ŷ 1 + ŷ 2 ) 3 (ŷ + ŷ 1 ) 3, For numerical solution we consider the domain [, B = [,.5 and m = 1. Since P Q = max q(x) = q(.5) x [,B ( ) j = 1.13, j= K =.75 and C = max x + 2x 3 =.75, α = PQBCK = =.359 < 1. So our new method converges. The maximum errors of S n of the standard ADM for (25) and the present method in x.5 are listed as in Table 1. Table 1 The maximum errors of S n of the standard ADM for (25) and the present method in x.5. n Standard ADM Present method The assumptions of Theorem 3 are only the sufficient conditions to guarantee the convergence of our new ADM, but not necessary. The next two examples show that our method might also converge even though α > 1. Example 2 Consider the nonlinear ODE of model y (x) = y(x)/2 y 2 (x), y() = 1, (29) which exact solution is y(x) = (2 e x/2 ) 1. Multiply the integrating factor e x/2 to both sides of (29), we have d dx (ex/2 y) = e x/2( y + y ) = e x/2 ( y 2 ). (3) 2 Integrate (3) both sides and substituting the initial value y() = 1, then y(x) = e x/2 e x/2 e t/2 y 2 (t)dt. By (16), we have ( x/2) j ŷ (x) =, j= ŷ i (x) = i 1, ( x/2) j j= where according to (17) A = ŷ 2, A 1 = 2ŷ ŷ 1 + ŷ 2 1, (t/2) j A i 1 (t)dt, j= A 2 = 2ŷ ŷ 2 + 2ŷ 1 ŷ 2 + ŷ 2 2, (31) Numerically we consider m = 7 and the domain [, 1 with B = 1. Since P = 1, Q = max x [,1 q(x) = q(1) = 7 K = 6 and C = 1, j= (.5) j α = PQBCK = = 9.9 > , Although we do not have Theorem 3, our new method still converges and is better than the standard ADM applied to (29). The comparison of their maximum errors are shown as in Table 2. Table 2 The comparison of the maximum errors for Example 2, i.e. Eq. (29). n Standard ADM Present method There are examples that our new method converges but the standard one fails. The next example is one of such cases. Example 3 The nonlinear differential equation y (x) + 2 x y(x) = 2 x 2 y2 (x) + 3γ 2γ 2, y() = (32) has the exact solution y(x) = γx, where γ is a constant. Notice that x = is a singular point of this ODE. Multiply the integrating factor x 2 to both sides of (32), we have d dx (x2 y) = x 2 y + 2xy = 2y 2 + (3γ 2γ 2 )x 2. (33) Integrate (33) both sides and substituting the initial value y() =, then y(x) = 1 x 2 [2y 2 (t) + (3γ 2γ 2 )t 2 dt. Since p(x) = 1/x 2 is easy to compute, we do not need the truncated version as in (14). By (11) and (31), we have y (x) = 1 x 2 (3γ 2γ 2 )t 2 dt,

6 164 Communications in Theoretical Physics Vol. 6 y i (x) = 1 x 2 2A i 1 (t)dt, i 1. For numerical computation we consider γ =.1 and the domain x 1. Notice that p(x) is unbounded at x =, so P is not finite. Hence this ODE does not satisfy the assumptions of theorem 5. But we still have convergent results as listed in Table 3. Table 3 The comparison of the maximum errors for Example 3, i.e. Eq. (32). n Standard ADM Present method On the other hand the traditional ADM for (32) with (31) gives u = (3γ 2γ 2 )dx =.32x, u i+1 = 2 x u i + 2 x 2 A i, i 1. So we obtain u 1 = 2 x u + 2 x 2 u2 =.8448x, u 2 = xx, u 3 = x, u 4 = x, Actually, its solution u(x) = u + u 1 + u 2 + diverge as shown in Table 3, which exhibits the comparison of the maximum errors of both methods. 5 Conclusion This paper proposes a new modification of the ADM with integrating factor. The numerical results demonstrate that it is more reliable and efficient than the traditional ADM. Our new method can solve certain ordinary differential equations where the traditional one fails like Example 3. Our ADM also has faster convergent rate. Besides, we give complete theoretic error analysis for the convergence and errors of this new method. As a byproduct, we can show the existence and uniqueness of solution of the original and perturbed ODEs. The underlined theory for ODE with singular point, however, still remains open and is deserved further investigation. References [1 G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic publishers, Boston, MA (1994). [2 M.M. Rodrigues, Appl. Math. E-Notes 11 (211) 5. [3 Y.Q. Hasan and L.M. Zhu, J. Concr. Appl. Math. 7(4) (29) 37. [4 Y.Q. Hasan and L.M. Zhu, Commun. Nonlinear Sci. Numer. Simul. 14(5) (29) [5 M.M. Hosseini and H. Nasabzadeh, Appl. Math. Comput. 186(1) (27) 117. [6 A.H. Oda, Q. Al and H.F. Lafta, Int. J. Contemp. Math. Sci. 5(49 52) (21) 255. [7 N. Bildik and A. Konuralp, Int. J. Comput. Math. 83(12) (26) 973. [8 M.M. Hosseini, Appl. Math. Comput. 181(2) (26) [9 M.M. Hosseini, J. Comput. Appl. Math. 197(2) (26) 495. [1 M.A. Abdou, Int. J. Nonlinear Sci. 12(1) (211) 29. [11 Z. Wang, L. Zou, and Z. Zong, Appl. Math. Comput. 218(4) (211) [12 A.M. Wazwaz, Appl. Math. Comput. 216(4) (21) 134. [13 M. Matinfar, S.H. Nasseri, M. Ghanbari, and H. Abdollahi, Int. J. Appl. Math. 22(5) (29) 677. [14 A.R. Vahidi, E. Babolian, C.G. Asadi, and Z. Azimzadeh, Int. J. Math. Anal. 3(33-36) (29) [15 H.A. Zedan and E. Al-Aidrous, Int. J. Math. Comput. 4 (29) 9. [16 I. Hashim, J. Comput. Appl. Math. 193(2) (26) 658. [17 H. Zhu, H. Shu, and M. Ding, Comput. Math. Appl. 6(3) (21) 84. [18 P.Y. Tsai and C.K. Chen, J. Franklin Inst. 347(1) (21) 185. [19 J.C. Hsu, H.Y. Lai, and C.K. Chen, Journal of Sound and Vibration 325 (29) 451. [2 W.C. Tien and C.K. Chen, Chaos, Solitons & Fractals 39(5) (29) 293. [21 Y.T. Yang, S.K. Chien, and C.K. Chen, Energy Conversion and Management (49)(1) (28) 291. [22 H.Y. Lai, C.K. Chen, and J.C. Hsu, CMES Comput. Model. Eng. Sci. 34 (28) 87. [23 C.H. Chiu and C.K. Chen, J. Heat Transfer (125)(2) (23) 312. [24 C.H. Chiu and C.K. Chen, International Journal of Heat and Mass Transfer (45)(1) (22) 267. [25 H.Y. Lai, J.C. Hsu and C.K. Chen, Comput. Math. Appl. 56(12) (28) 324. [26 A.M. Wazwaz, Appl. Math. Comput. 111(1) (2) 53. [27 I.L. El-Kalla, Int. J. Differ. Equ. Appl. 1(2) (25) 225. [28 R.L. Burden and J.D. Faires, Numerical Analysis, 8th ed., Thomson-Books/Cole, Pacific Grove, CA 9395, USA (25). [29 M.J. Jang, C.L. Chen, and Y.C. Liy, Appl. Math. Comput. 115 (2) 145.