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The modiﬁed decomposition method for analytic treatment of diﬀerential equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA

Abstract The modiﬁed decomposition method is applied for analytic treatment of nonlinear diﬀerential equations. The modiﬁed method accelerates the rapid convergence of the series solution, dramatically reduces the size of work, and provides the exact solution by using few iterations only without any need to the so-called Adomian polynomials. Numerical illustrations that include nonlinear PDEs, nonlinear Klein–Gordon equations, and nonlinear Lane–Emden equation, are investigated to show the pertinent features of the technique. Ó 2005 Elsevier Inc. All rights reserved. Keywords: The modiﬁed decomposition method; Nonlinear partial diﬀerential equations; Nonlinear Klein–Gordon equation; Lane–Emden equation

1. Introduction Nonlinear phenomena, that appear in many areas of scientiﬁc ﬁelds such as solid state physics, plasma physics, ﬂuid dynamics, mathematical biology and

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.02.048

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chemical kinetics, can be modelled by partial diﬀerential equations. A broad class of analytical solutions methods and numerical solutions methods were used in to handle these problems. The Adomian decomposition method [1–3] has been proved to be eﬀective and reliable for handling diﬀerential equations, linear or nonlinear. Although the convergence provided by Adomian decomposition method is rapid [4], the modiﬁed decomposition method developed by Wazwaz in [5] accelerates this rapid convergence of the series solution. This new technique provides the exact solution by using two iterations only without any need to the so-called Adomian polynomials if used properly. The modiﬁed decomposition method has been used recently by many mathematicians because it facilitates the computational work and minimizes it. This method can eﬀectively improve the speed of convergence [6–14]. In this work we aim to introduce analytical treatment for nonlinear diﬀerential equations by using the modiﬁed decomposition method. Although the modiﬁed form introduces a slight change in the formulation of Adomian recursive relation, but it provides a qualitative improvement over standard Adomian method [6]. While this slight variation is rather simple, it does demonstrate the reliability, eﬃciency, and power of the method. In this paper, only a brief discussion of the modiﬁed decomposition analysis will be emphasized, because the complete details of the method are found in [5–14] and in many related works.

2. The modiﬁed decomposition method In this section, we will brieﬂy discuss the use of the modiﬁed decomposition method for nonlinear diﬀerential equations and for Lane–Emden equation. 2.1. Diﬀerential equations Without loss of generality, we consider the diﬀerential equation Lu þ Ru þ Nu ¼ g;

ð1Þ

with prescribed conditions, where u is the unknown function, L is the highest order derivative which is assumed to be easily invertible, R is a linear diﬀerential operator of less order than L, Nu represents the nonlinear terms, and g is the source term. Applying the inverse operator L1 to both sides of (1), and using the given conditions we obtain u ¼ f L1 ðRuÞ L1 ðNuÞ;

ð2Þ

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167

where the function f(x) represents the terms arising from integrating the source term g and from using the given initial or boundary conditions, all are assumed to be prescribed. The standard Adomian method deﬁnes the solution u(x) by the series 1 X uðxÞ ¼ un ðxÞ; ð3Þ n¼0

where the components u0, u1, u2, . . ., are usually determined recursively by using the relation u0 ðxÞ ¼ f ðxÞ; ð4Þ ukþ1 ðxÞ ¼ L1 ðRuk Þ L1 ðNuk Þ; k P 0. The decomposition method suggests that the zeroth component u0(x) is usually identiﬁed by the function f(x) deﬁned above. The components u0, u1, u2, . . . are determined recursively, therefore the solution u(x) in a series form deﬁned by (3) is readily determined. It is obvious that Adomian method changes the differential equations to obtaining an easily computable components [1–3]. The closed form for the solution u(x) if exists can be immediately obtained because of the rapid convergence presented by the method. The modiﬁed decomposition method [5] suggests that the function f(x) deﬁned above in (2) be decomposed into two parts, namely f0(x) and f1(x) such that f ðxÞ ¼ f0 ðxÞ þ f1 ðxÞ.

ð5Þ

The proper choice of the parts f0 and f1 depends mainly on trial basis. In view of this decomposition of f(x), a slight variation only on the components u0(x) and u1(x) should be introduced. The proposed variation is that only the part f0(x) be assigned to the zeroth component u0, whereas the remaining part f1(x) be combined with the other terms given in (4) to deﬁne u1(x). In view of this assumption, we formulate the following recursive relation for the modiﬁed decomposition method 8 > < u0 ðxÞ ¼ f0 ðxÞ; u1 ðxÞ ¼ f1 ðxÞ L1 ðRu0 Þ L1 ðNu0 ðxÞÞ; ð6Þ > : ukþ2 ðxÞ ¼ L1 ðRukþ1 ðxÞÞ L1 ðNukþ1 ðxÞÞ; k P 0. Comparing the recursive relation (4) of the standard Adomian method with the recursive relation (6) of the modiﬁed decomposition method leads to concluding that in (4) the zeroth component was deﬁned by the function f(x) = f1(x) + f2(x), whereas in (6), the zeroth component u0(x) is deﬁned only by f0(x) a part of f(x). The remaining part f1(x) of f(x) is added to the deﬁnition of the component u1(x) in (4).

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In our earlier works [6–14], we have shown that only two iterations are sufﬁcient to determine the exact solution for most of the examined cases. If more than two iterations are needed, then N(u(x, t)) should be represented by Adomian polynomials that can be computed for all types of nonlinearity. The choice of f0(x) for u0(x) to contain minimal number of terms has a strong inﬂuence on facilitating the recurrence relation, and, as a consequence, accelerates the rapid convergence of the solution. As stated before, the success of the modiﬁed method depends mainly on the proper choice of the parts f0(x) and f1(x). We have been unable to establish any criterion to judge what forms of f0(x) and f1(x) can be used to yield the acceleration demanded. It appears that trials are the only criteria that can be applied so far. Numerical applications used in [6– 14] showed that the modiﬁed algorithm compares favorably with standard decomposition method. 2.2. Lane–Emden equation The modiﬁed decomposition strategy presented before will be applied to Lane–Emden equation characterized by singular behavior. Many problems in the literature of the diﬀusion of heat perpendicular to the surfaces of parallel planes are modeled by the heat equation xr ðxr ux Þx þ af ðxÞgðuÞ ¼ hðxÞ 0 < x 6 L; r > 0; or equivalently r uxx þ ux þ af ðxÞgðuÞ ¼ hðxÞ; x

0 6 x 6 L; r > 0;

ð7Þ

ð8Þ

where u(x) is the temperature. For the steady-state case, and for r = 2, h(x) = 0, Eq. (8) is the Emden–Fowler equation [1–3] given by 2 u00 þ u0 þ af ðxÞgðuÞ ¼ 0; x

yð0Þ ¼ y 0 ; y 0 ð0Þ ¼ 0;

ð9Þ

where f(x) and g(u) are some given functions of x and u respectively. For f(x) = 1 and g(u) = un, Eq. (9) is the standard Lane–Emden equation that was used to model the thermal behavior of a spherical cloud of gas [15,16] acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics. For other special forms of g(u), the well-known Lane–Emden equation was used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, and theory of thermionic currents [15,16]. A substantial amount of work has been done on this type of problems for various structures of g(u) in [2,3,7–10]. The singularity behavior that occurs at the point x = 0 is the main diﬃculty in the analysis of Eqs. (8) and (9). The motivation for the analysis presented in

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this paper comes actually from the aim of extending our previous works in [7– 10]. A reliable framework depends mainly on Adomian decomposition method was introduced in [1–3,7–10] to handle this type of models. The introduced algorithm was mainly used for Lane–Emden equation and for the Emden– Fowler equation where there is no time dependence for u. To successfully use the modiﬁed decomposition method, we propose a framework to overcome the diﬃculty of the singular point at x = 0. A new choice for the diﬀerential operator L will be proposed. The diﬀerential operator L is deﬁned in terms of the two derivatives, u0 þ xr u0 . Following [7–10], we ﬁrst rewrite (9) in the form Lu ¼ af ðxÞgðuÞ;

ð10Þ

where the diﬀerential operator L employs the ﬁrst two derivatives in the form d d xr L ¼ xr ; ð11Þ dx dx in order to overcome the singularity behavior at x = 0. In view of (11), the best encountered deﬁnition of L1 is an inverse twofold deﬁnite integration operator deﬁned by Z x Z x 1 r L ðÞ ¼ x xr ðÞdx dx; r > 0. ð12Þ 0

0

1

Applying L of (12) to the ﬁrst two terms u00 þ xr u0 of Eq. (9) we ﬁnd Z x Z x r r L1 u00 þ u0 ¼ xr xr u00 þ u0 dx dx; x x Z0 x 0 Z x Z x ¼ xr xr u0 rxr1 u0 dx þ rxr1 u0 dx dx; 0 0 Z0 x u0 dx; ¼ 0

¼ uðxÞ uð0Þ;

ð13Þ

where integration by parts is used to integrate xru00 . Notice that only u(0) is sufﬁcient to carry out the analysis and the other condition can be used to conﬁrm that the obtained solution satisﬁes this given condition. In view of the presented analysis, the modiﬁed decomposition method can be employed in a parallel manner to the implementation on the partial diﬀerential equations. In order to evaluate the performance of the modiﬁed decomposition method, six illustrative examples, selected from nonlinear PDEs, nonlinear Klein– Gordon equation, and the nonlinear Lane–Emden diﬀerential equations, will be used in this work to show the pertinent feature of this method and to show that it eﬀectively reduces the size of work.

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3. Nonlinear partial diﬀerential equations To achieve our goal, we start the analysis by considering the following nonlinear partial diﬀerential equations. Example 1. We ﬁrst consider the nonlinear PDE utt þ u2x þ u u2 ¼ tex ;

ð14Þ

with initial conditions ut ðx; 0Þ ¼ ex .

uðx; 0Þ ¼ 0;

ð15Þ

In an operator form, Eq. (14) becomes Lu ¼ tex u2x u þ u2 ;

ð16Þ

where L is a second order differential operator that is assumed invertible, and L1 is given by Z tZ t ðÞds ds. ð17Þ L1 ðÞ ¼ 0

1

Applying L

0

to both sides of (16) and using the initial conditions we ﬁnd

1 uðx; tÞ ¼ tex þ t3 ex L1 ðu2x þ u u2 Þ. 6

ð18Þ

Using the decomposition series (3) for u(x, t) into both sides of (18) and applying the modiﬁed recursive relation (6) we obtain u0 ðx; tÞ ¼ tex ; 1 u1 ðx; tÞ ¼ t3 ex L1 ðu20x þ u0 u20 Þ. 6

ð19Þ

This gives u0 ðx; tÞ ¼ tex ; 1 u1 ðx; tÞ ¼ t3 ex L1 ðtex Þ; 6 ¼ 0;

ð20Þ

and consequently, the remaining components vanish. This in turn gives the solution in a closed form by uðx; tÞ ¼ tex .

ð21Þ

Example 2. We next consider the nonlinear PDE uxx þ uux ¼ x þ ln t;

t > 0;

ð22Þ

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171

with boundary conditions uð0; tÞ ¼ ln t;

ux ð0; tÞ ¼ 1.

ð23Þ

In an operator form, Eq. (22) becomes Lu ¼ x þ ln t uux ;

ð24Þ 2

where L is second order differential operator oxo 2 that is assumed invertible, and L1 is given by 1

L ðÞ ¼

Z 0

x

Z

x

ðÞds ds.

ð25Þ

0

Applying L1 to both sides of (24) and using the initial conditions we ﬁnd 1 1 uðx; tÞ ¼ x þ ln t þ x3 þ x2 ln t L1 ðuux Þ. 6 2

ð26Þ

Using the decomposition series (3) for u(x, t) into both sides of (26) and recalling that the modiﬁed decomposition method (6) admits the use of the recurrence relation u0 ðx; tÞ ¼ x þ ln t; 1 1 u1 ðx; tÞ ¼ x3 þ x2 ln t L1 ðu0 u0x Þ. 6 2

ð27Þ

This gives u0 ðx; tÞ ¼ x þ ln t; 1 1 u1 ðx; tÞ ¼ x3 þ x2 ln t L1 ðx þ ln tÞ; 6 2 ¼ 0;

ð28Þ

and consequently, the other components uk = 0, k P 2. The exact solution uðx; tÞ ¼ x þ ln t;

ð29Þ

is readily obtained. It is obvious from the previous two examples that by proper choice of f0 and f1 we can signiﬁcantly reduce the size of the work. The exact solutions were obtained by determining two components u0 and u1 only without using the Adomian polynomials for the nonlinear terms. It is well known that any nonlinear function g(u) should be represented by an inﬁnite series of polynomials 1 X gðuðx; tÞÞ ¼ An ðy 0 ; y 1 ; . . . ; un Þ; ð30Þ n¼0

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where Adomian polynomials An are deﬁned by "

n X 1 dn i y i ðkÞ An ¼ ng n! dk i¼0

!# .

ð31Þ

k¼0

Adomian polynomials An can be constructed for various classes of nonlinearity according to speciﬁc algorithms presented in [1–3,7–10] and the references therein.

4. Nonlinear Klein–Gordon equation We next aim to study the following nonlinear Klein–Gordon equations. Example 1. We ﬁrst consider the nonlinear Klein–Gordon equation utt uxx þ u2 ¼ 6xtðx2 t2 Þ þ x6 t6 ;

ð32Þ

with initial conditions uðx; 0Þ ¼ 0;

ut ðx; 0Þ ¼ 0.

ð33Þ

In an operator form, Eq. (32) becomes Lu ¼ 6xtðx2 t2 Þ þ x6 t6 þ uxx u2 ; where L is a second order differential operator, and L1 is given by Z tZ t 1 ðÞds ds. L ðÞ ¼ 0

ð34Þ

ð35Þ

0

Applying L1 to both sides of (34) and using the initial conditions we ﬁnd uðx; tÞ ¼ x3 t3

3 5 1 xt þ x6 t8 þ L1 ðuxx u2 Þ. 10 56

ð36Þ

Proceeding as before and by properly decomposing f(x, t) we ﬁnd u0 ðx; tÞ ¼ x3 t3 ; 3 1 u1 ðx; tÞ ¼ xt5 þ x6 t8 þ L1 ðu0xx u20 Þ; 10 56

ð37Þ

so that u0 ðx; tÞ ¼ x3 t3 ; 3 1 u1 ðx; tÞ ¼ xt5 þ x6 t8 þ L1 ð6xt3 x6 t6 Þ; 10 56 ¼ 0;

ð38Þ

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173

and as a result, uk = 0, k P 2. This in turn gives the solution in a closed form by uðx; tÞ ¼ x3 t3 .

ð39Þ

Example 2. We next consider the nonlinear Klein–Gordon equation utt uxx þ u2 ¼ x cos t þ x2 cos2 t;

ð40Þ

with initial conditions uðx; 0Þ ¼ x;

ut ðx; 0Þ ¼ 0.

ð41Þ

1

to both sides of (40) and using the initial conditions we ﬁnd 1 1 1 uðx; tÞ ¼ x cos t þ x2 t2 cos2 t þ ð42Þ þ L1 ðuxx u2 Þ. 4 4 4

Applying L

Proceeding as before and by properly decomposing f(x, t) we ﬁnd u0 ðx; tÞ ¼ x cos t; 1 1 2 1 2 2 t cos t þ u1 ðx; tÞ ¼ x þ L1 ðu0xx u20 Þ; 4 4 4

ð43Þ

so that u0 ðx; tÞ ¼ x cos t; 1 1 2 1 2 2 t cos t þ þ L1 ðx2 cos2 tÞ; u1 ðx; tÞ ¼ x 4 4 4

ð44Þ

¼ 0; and as a result, Uk = 0, k P 2. This in turn gives the solution in a closed form by uðx; tÞ ¼ x cos t. ð45Þ This conﬁrms the conclusion made above in that the work can be signiﬁcantly reduced by the proper choice of f0 and f1. The exact solutions were obtained by determining two components u0 and u1 only without using the Adomian polynomials for the nonlinear terms.

5. Lane–Emden equation We close our analysis by considering the following nonlinear Lane–Emden ordinary diﬀerential equations. Example 1. We ﬁrst start by considering the inhomogeneous Lane–Emden equation

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2 uxx þ ux þ u3 ¼ 6 þ x6 ; x

ð46Þ

with initial conditions ux ð0Þ ¼ 0.

uð0Þ ¼ 0;

ð47Þ

Using the approach presented above for the Lane–Emden equation, and set Eq. (46) in an operator form by Lu ¼ 6 þ x6 u3 ;

ð48Þ 1

where L is a differential operator for the ﬁrst two terms, r = 2, and L by Z x Z x 1 2 L ðÞ ¼ x x2 ðÞdx dx. 0

is given

ð49Þ

0

Applying L1 to both sides of (48), noting that r = 2, and using the initial condition we ﬁnd uðxÞ ¼ x2 þ

1 8 x L1 ðu3 Þ. 72

ð50Þ

Using the decomposition series (3) for u(x) gives 1 X n¼0

0 !3 1 1 X 1 8 1 @ un ðxÞ ¼ x þ x L un ðxÞ A. 72 n¼0 2

ð51Þ

The modiﬁed decomposition method admits the use of the recurrence relation u0 ðxÞ ¼ x2 ; 1 u1 ðxÞ ¼ x8 L1 ðu30 ðxÞÞ; 72 ¼ 0;

ð52Þ

and as a result uk(x) = 0, k P 2. The exact solution is therefore given by uðxÞ ¼ x2 .

ð53Þ

Example 2. We ﬁnally close our analysis by studying the inhomogeneous Lane–Emden equation 4 uxx þ ux þ u2 ¼ 4 þ 18x þ 4x3 þ x6 ; x

ð54Þ

with initial conditions uð0Þ ¼ 2;

ux ð0Þ ¼ 0.

ð55Þ

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Notice that r = 4, and therefor L1 is given by Z x Z x 1 4 x x4 ðÞdx dx; r > 0. L ðÞ ¼ 0

175

ð56Þ

0

Applying L1 to both sides of (54), and using the initial condition we ﬁnd 2 1 1 uðxÞ ¼ 2 þ x3 þ x2 þ x5 þ x8 L1 ðu2 Þ. 5 10 88

ð57Þ

The modiﬁed decomposition method admits the use of the recurrence relation u0 ðxÞ ¼ 2 þ x3 ; 2 1 1 2 u1 ðxÞ ¼ x2 þ x5 þ x8 L1 ðð2 þ x3 Þ Þ; 5 10 88 ¼ 0;

ð58Þ

and as a result Uk(x) = 0, k P 2. The exact solution uðxÞ ¼ 2 þ x3 ;

ð59Þ

follows immediately.

6. Concluding remarks In this work, we carefully employed the reliable modiﬁed decomposition method, that accelerates the rapid convergence of the decomposition series solution. The success of the modiﬁed decomposition method depends mainly on the proper choice of f0(x) and f1(x) of f(x). We have been unable to establish any criterion to judge what forms of f0(x) and f1(x) can be used to yield the acceleration demanded. It appears that trials are the only criteria that can be applied so far. The method can signiﬁcantly reduce the volume of computational work. In comparisons with the standard Adomian method, the modiﬁed algorithm gives better performance in many cases. Moreover, the Adomian polynomials, that may cause diﬃcult integrations and proliferation [17] of terms in Adomian scheme, were not used here and this gives the modiﬁed decomposition method an advantage over the standard Adomian method.

References [1] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, 1994. [2] G. Adomian, R. Rach, N.T. Shawagfeh, On the analytic solution of Lane–Emden equation, Found. Phys. Lett. 8 (2) (1995) 161–181.

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[3] N.T. Shawagfeh, Nonperturbative approximate solution for Lane–Emden equation, J. Math. Phys. 34 (9) (1993) 4364–4369. [4] Y. Cherruault, Convergence of AdomianÕs method, Kybernetes 18 (1989) 31–38. [5] A.M. Wazwaz, A reliable modiﬁcation of AdomianÕs decomposition method, Appl. Math. Comput. 102 (1999) 77–86. [6] A.M. Wazwaz, The modiﬁed decomposition method and the PadeÕ approximants for solving Thomas–Fermi equation, Appl. Math. Comput. 105 (1999) 11–19. [7] A.M. Wazwaz, A new method for solving diﬀerential equations of the Lane–Emden type, Appl. Math. Comput. 118 (2/3) (2001) 287–310. [8] A.M. Wazwaz, A new method for solving singular initial value problems in the second order ordinary diﬀerential equations, Appl. Math. Comput. 128 (2002) 47–57. [9] A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Emden–Fowler equation, Appl. Math. Comput. 161 (2005) 543–560. [10] A.M. Wazwaz, Partial Diﬀerential Equations: Methods and Applications, Balkema, The Netherlands, 2002. [11] A.M. Wazwaz, A First Course in Integral Equations, WSPC, New Jersey, 1997. [12] A.M. Wazwaz, A reliable treatment for mixed Volterra–Fredholm integral equations, Appl. Math. Comput. 127 (2002) 405–414. [13] A.M. Wazwaz, The existence of noise terms for systems of inhomogeneous diﬀerential and integral equations, Appl. Math. Comput. 146 (1) (2003) 81–92. [14] A.M. Wazwaz, The numerical solution of special fourth-order boundary value problems by the modiﬁed decomposition method, Int. J. Comput. Math. 79 (3) (2002) 345–356. [15] S. Chandrasekhar, Introduction to the Study of Stellar Structure, Dover Publications, New York, 1967. [16] O.U. Richardson, The Emission of Electricity from Hot Bodies, London, 1921. [17] K. Maleknejad, M. Hadizadeh, A new computational method for Volterra–Fredholm integral equations, J. Comput. Math. Appl. 37 (1999) 1–8 (1994) 339–348.

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