International Mathematical Forum, Vol. 7, 2012, no. 11, 515 - 521
Sumudu Decomposition Method for Nonlinear Equations Devendra Kumar1 , Jagdev Singh2 and Sushila Rathore3 1
Department of Mathematics Jagan Nath Gupta Institute of Engineering and Technology Jaipur-302022, Rajasthan, India
[email protected] 2
Department of Mathematics, Jagan Nath University Village-Rampura, Tehsil-Chaksu, Jaipur-303901, Rajasthan, India
[email protected] 3
Department of Physics, Jagan Nath University Village-Rampura, Tehsil-Chaksu, Jaipur-303901, Rajasthan, India
[email protected] Abstract In this paper, we propose a new method, namely sumudu decomposition method (SDM) for solving nonlinear equations. This method is a combination of the sumudu transform method and decomposition method. The nonlinear terms can be easily handled by the use of Adomian polynomials. The technique is described and illustrated with some examples. The results reveal that the proposed method is very efficient, simple and can be applied to other nonlinear problems.
Mathematics Subject Classification: 45A05, 45B05, 45D05, 45J05 Keywords: Sumudu decomposition method, Nonlinear partial differential equations, Adomian polynomials, Noise terms phenomena
1
Introduction
Non-linear phenomena, that appear in many areas of scientific fields such as solid state physics, plasma physics, fluid mechanics, population models and chemical kinetics, can be modeled by nonlinear differential equations. The importance of obtaining the exact or approximate solutions of nonlinear partial
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D. Kumar, J. Singh and S. Rathore
differential equations in physics and mathematics is still a significant problem that needs new methods to discover exact or approximate solutions. Various powerful mathematical methods such as Adomian decomposition method (ADM) [1], homotopy perturbation method (HPM) [2], variational iteration method (VIM) [3, 4], Laplace decomposition method (LDM) [5, 6] and homotopy perturbation sumudu transform method (HPSTM) [7] have been proposed to obtain exact and approximate analytic solutions. Inspired and motivated by the ongoing research in this area, we introduce a new method called sumudu decomposition method (SDM) for solving the nonlinear equations in the present paper. It is worth mentioning that the proposed method is an elegant combination of the sumudu transform method and decomposition method which was first introduced by Adomian [1, 8]. The proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form. This article considers the effectiveness of the sumudu decomposition method (SDM) in solving nonlinear partial differential equations. In early 90’s, Watugala [9] introduced a new integral transform, named the sumudu transform. The sumudu transform is defined over the set of functions A = {f(t) | ∃ M, τ1 , τ2 > 0, | f (t) | < M e|t|/τj , if t ∈ ( − 1)j × [0,∞)} by the following formula ¯f(u) = S [f(t)] =
0
∞
f (ut) e−t dt, u ∈ ( − τ1 , τ2 ).
(1)
For further detail and properties of this transform, see [10-12].
2
Sumudu decomposition method (SDM)
To illustrate the basic idea of this method, we consider a general nonlinear non-homogenous partial differential equation with the initial conditions of the form: L U(x, t) + RU(x, t) + N U(x, t) = g(x, t), (2) U(x, 0) = h(x),
Ut (x, 0) = f (x),
where L is the second order linear differential operator L = ∂ 2 /∂t2 , R is the linear differential operator of less order than L, N represents the general nonlinear differential operator and g(x, t)is the source term. Taking the sumudu transform on both sides of eq. (2), we get S [LU(x, t)] + S [R U(x, t)] + S [N U(x, t)] = S [g(x, t)].
(3)
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Sumudu decomposition method
Using the differentiation property of the sumudu transform and above initial conditions, we have S [U(x, t)] = u2 S [g(x, t)] + h(x) + uf (x) − u2 S [R U(x, t) + N U(x, t)] . (4) Now, applying the inverse sumudu transform on both sides of eq. (4), we get
U(x, t) = G(x, t) − S −1 u2 S [R U(x, t) + N U(x, t)] ,
(5)
where G(x, t) represents the term arising from the source term and the prescribed initial conditions. The second step in sumudu decomposition method is that we represent solution as an infinite series given below U(x, t) =
∞ n=0
Un (x, t)
(6)
and the nonlinear term can be decomposed as N U(x, t) =
∞ n=0
An ,
(7)
where An are Adomian polynomials [14] of U0 , U1 , U2 , ..., Un and it can be calculated by formula given below
∞ 1 dn An = N λi Ui n n! dλ i=0
, n = 0, 1, 2, ...
(8)
λ=0
Using eq. (6) and eq. (7) in eq. (5), we get ∞ n=0
Un (x, t) = G(x, t) − S
−1
2
u S R
∞ n=0
Un (x, t) +
∞ n=0
An
.
(9)
On comparing both sides of the eq. (9), we get U0 (x, t) = G(x, t),
(10)
U1 (x, t) = −S −1 u2 S [R U0 (x, t) + A0 ] ,
(11)
U2 (x, t) = −S −1 u2 S [R U1 (x, t) + A1 ] .
(12)
In general the recursive relation is given by
Un+1 (x, t) = −S −1 u2 S [R Un (x, t) + An ] , n ≥ 0.
(13)
Now first of all applying the sumudu transform of the right hand side of eq. (13) then applying the inverse sumudu transform, we get the values of U0 , U1 , U2 , ..., Un respectively.
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D. Kumar, J. Singh and S. Rathore
Applications
In order to elucidate the solution procedure of the sumudu decomposition method (SDM), we consider the nonlinear partial differential equations. Example 3.1. Consider a nonlinear partial differential equation [13]: Ut + UUx = x + xt2 ,
(14)
U(x, 0) = 0.
(15)
with initial condition By applying the aforesaid method subject to the initial condition, we have S[U(x, t)] = xu + 2xu3 − u S[UUx ].
(16)
The inverse of sumudu transform implies that U(x, t) = xt +
xt3 − S −1 [u S[UUx ]] . 3
(17)
Following the technique, if we assume an infinite series solution of the form (6), we obtain
∞
∞ xt3 − S −1 u S Un (x, t) = xt + An (U) 3 n=0 n=0
,
(18)
where An (U)are Adomian polynomials [14] that represent the nonlinear terms. The first few components of Adomian polynomials, are given by A0 (U) = U0 U0x , A1 (U) = U0 U1x + U1 U0x , .. .
(19)
The recursive relation is given below U0 (x, t) = xt +
xt3 , 3
(20)
U1 (x, t) = −S −1 [u S[A0(U)]] ,
(21)
Un+1 (x, t) = −S −1 [u S[An (U)]] , n ≥ 0.
(22)
The other components of the solutions can be easily found by using above recursive relation U1 (x, t) = −S −1 [u S[A0(U)]] = −S −1 [u S[U0 U0x ]]
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Sumudu decomposition method
=−
xt3 xt7 2xt5 − − , 3 63 15
(23)
U2 (x, t) = −S −1 [u S[A1(U)]] = −S −1 [u S[U0 U1x + U1 U0x ]] =
2xt5 22xt7 38xt9 2xt11 + + + , 15 315 2835 2079 .. .
(24)
It is important to recall here that the noise terms appear between the components U0 (x, t) and U1 (x, t), where the noise terms are those pairs of terms that are identical but carrying opposite signs. More precisely, the noise terms ± 13 xt3 between the components U0 (x, t) and U1 (x, t) can be cancelled and the remaining terms of U0 (x, t) still satisfy the equation. Therefore, the exact solution is given by U(x, t) =
∞
n=0
Un (x, t) = xt.
(25)
Example 3.2. Now, consider the following nonlinear partial differential equation [13]: Utt − Ut Uxx = −t + U, (26) with initial conditions U(x, 0) = sin x,
(27)
Ut (x, 0) = 1.
(28)
By applying the aforesaid method subject to the initial conditions, we have S[U(x, t)] = u + sin x − u3 + u2 S[U + Ut Uxx ].
(29)
The inverse of sumudu transform implies that U(x, t) = t + sin x −
t3 + S −1 u2 S[U + Ut Uxx ] . 6
(30)
Now, applying the same procedure as in previous example we arrive at recursive relation given below t3 (31) U0 (x, t) = t + sin x − , 6
U1 (x, t) = S −1 u2 S[U0 + B0 (U)] ,
Un+1 (x, t) = S −1 u2 S[Un + Bn (U)] , n ≥ 1,
(32) (33)
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where Bn (U) are Adomian polynomials [14] that represent the nonlinear terms in the above equation (33). The other components of the solutions can be easily found by using above recursive relation
U1 (x, t) = S −1 u2 S[U0 + B0 (U)]
= S −1 u2 S[U0 + U0t U0xx =
t5 t4 t3 − + sin x, 6 120 24 .. .
(34)
It is important to recall here that the noise terms appear between the components U0 (x, t) and U1 (x, t), where the noise terms are those pairs of terms that 3 are identical but carrying opposite signs. More precisely, the noise terms ± t6 between the components U0 (x, t) and U1 (x, t)can be cancelled and the remaining terms of U0 (x, t)still satisfy the equation. Therefore, the exact solution is given by U(x, t) =
∞
n=0
4
Un (x, t) = t + sin x.
(35)
Conclusions
The present paper develops the sumudu decomposition method (SDM) for solving nonlinear problems. We have also solved two nonlinear partial differential equations with initial conditions. The proposed technique has shown to computationally efficient in these examples that are important to researchers in the field of applied sciences. It is worth mentioning that the method is capable of reducing the volume of the computational work as compared to the classical methods while still maintaining the high accuracy of the numerical result; the size reduction amounts to an improvement of the performance of the approach. In conclusion, the SDM may be considered as a nice refinement in existing numerical techniques and might find the wide applications.
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[3] J.H. He, Variational iteration method—a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics, 34 (1999), 699–708. [4] J.H. He, G.C. Wu and F. Austin, The variational iteration method which should be followed, Nonlinear Science Letters A, 1 (2009), 1–30. [5] S.A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics, 1 (2001), 141–155. [6] Yasir Khan, An effective modification of the Laplace decomposition method for nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10 (2009), 1373–1376. [7] J. Singh, D. Kumar and Sushila, Homotopy Perturbation Sumudu Transform Method for Nonlinear Equations, Adv. Theor. Appl. Mechanics, 4(4) (2011), 165-175. [8] G. Adomian, Serrano SE, J. Appl. Math. Lett., 2 (1995), 161-164. [9] G.K. Watugala, Sumudu transform- a new integral transform to solve differential equations and control engineering problems, Math. Engg. Indust., 6(4) (1998), 319-329. [10] M.A. Asiru, Sumudu transform and the solution of integral equation of convolution type, International Journal of Mathematical Education in Science and Technology, 32(2001), 906-910. [11] F.B.M. Belgacem, A.A. Karaballi and S.L Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Mathematical problems in Engineering, 3 (2003), 103-118. [12] F.B.M. Belgacem and A.A. Karaballi, Sumudu transform fundamental properties investigations and applications, International J. Appl. Math. Stoch. Anal., (2005), 1-23. [13] A.M. Wazwaz, Partial differential equations methods and applications, Netherland Balkema Publisher, 2002. [14] A.M. Wazwaz, A new technique for calculating Adomian polynomials for nonlinear polynomials, Appl. Math. Compt., 111 (2002), 33-51. Received: September, 2011
