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Variational iteration method for solving nonlinear boundary value problems
Applied Mathematics and Computation 183 (2006) 1351–1358 www.elsevier.com/locate/amc
Variational iteration method for solving nonlinear boundary value problems Shaher Momani
a,*
, Salah Abuasad a, Zaid Odibat
b
a
b
Department of Mathematics, Mutah University, P.O. Box 7, Mutah, Jordan Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balq’a Applied University, Salt, Jordan
Abstract In this paper, He’s variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems. The variational iteration method, which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. Numerical examples are given and comparisons are made with the Adomian decomposition method. The fact that this method solves nonlinear equations without using Adomian polynomials can be considered as an advantage of this method over Adomian decomposition method. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Variational iteration method; Lagrange multipliers; Nonlinear boundary value problems; Troesch’s problem
1. Introduction In this paper, we extend the applications of the variational iteration method [1–6] to find approximate solutions for nonlinear boundary value problems. The variational iteration method, which proposed by Ji-Huan He [1–6], is effectively and easily used to solve some classes of nonlinear problems. For linear problems, its exact solution can be obtained by only one iteration step due to the fact that the Lagrange multiplier can be exactly identified. The variational iteration method is proposed to solve the generalized normalized diode equation, by suitable choice of the initial trial-function, one-step iteration leads to an high accurate solution, which is valid for the whole solution domain [6]. Draˇgaˇnescu and Caˇpaˇlnaˇsan [7] applied the variational iteration method to nonlinear anelastic model describing the acceleration of the relaxation process in the presence of the vibrations. The combination of a perturbation method, variational iteration method, method of variation of constants and averaging method to establish an approximate solution of one degree of freedom weakly nonlinear systems was proposed in [8]. Moreover, the method was successfully applied to delay differential equations in [1], to autonomous ordinary differential systems [5], to Helmholtz equation [9], and other fields [3,4,10]. *
Corresponding author. E-mail addresses: [email protected] (S. Momani), [email protected] (Z. Odibat).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.138
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Recently, the application of He’s variational iteration method is extended to fractional differential equations [2,11–13]. The objective of the present paper is to extend the application of the variational iteration method to provides an approximate solutions for a class of boundary value problems and to make comparison with that obtained by Adomian decomposition method. 2. Variational iteration method To illustrate the basic concepts of the variational iteration method, we consider the following general nonlinear differential equation: LuðtÞ þ NuðtÞ ¼ gðtÞ;
ð2:1Þ
where L is a linear operator, N is a nonlinear operator and g(t) is a known analytical function. According to the variational iteration method, we can construct a correction functional as follows: Z t unþ1 ðtÞ ¼ un ðtÞ þ kðLun ðnÞ þ N u~n ðnÞ gðnÞÞ dn; ð2:2Þ 0
where k is a general Lagrange multiplier [14], which can be identified optimally via the variational theory [14– 22], the subscript n denotes the nth approximation, and ~un is considered as a restricted variation [1–6,17], i.e. d~ un ¼ 0. For convenience of the reader we will present the following example. Example 2.1. Consider the second order linear boundary value problem [5] d2 T þ T þ x ¼ 0; dx2
ð2:3Þ
0 < x < 1;
with boundary conditions T ð0Þ ¼ T ð1Þ ¼ 0;
ð2:4Þ
where T is temperature. Its correction functional can be written down as follows: Z x kðT 00n ðsÞ þ T n ðsÞ þ sÞ ds: T nþ1 ðxÞ ¼ T n ðxÞ þ
ð2:5Þ
0
Taking variation with respect to the independent variable Tn, noticing that dTn(0) = 0, Z x dT nþ1 ðxÞ ¼ dT n ðxÞ þ d kðT 00n ðsÞ þ T n ðsÞ þ sÞ ds; 0 Z x 0 0 ðk00 þ kÞdT n ds ¼ 0; ¼ dT n ðxÞ þ kdT n ðsÞjs¼x k dT n ðsÞjs¼x þ 0
for all variations dTn and dT 0n , implying the following stationary conditions: dT n : k00 ðsÞjs¼x þ kðsÞjs¼x ¼ 0; dT 0n : kðsÞjs¼x ¼ 0; dT n : 1 k0 ðsÞjs¼x ¼ 0: The general Lagrange multiplier, therefore, can be readily identified k ¼ sinðs xÞ; as a result, we obtain the following iteration formula: Z x sinðs xÞðT 00n ðsÞ þ T n ðsÞ þ sÞ ds: T nþ1 ðxÞ ¼ T n ðxÞ þ 0
ð2:6Þ
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If we use its complementary solution T0 = A cos x + B sin x as an initial approximation, using the iteration formula, we get Z x T 1 ¼ A cos x þ B sin x þ s sinðs xÞ ds ¼ A cos x þ B sin x x sin x: 0
By imposing the boundary conditions (2.4) yields A = 1 and B ¼ sin1 1 1, as a result, we have T 1 ðxÞ ¼ sin x x which is the exact solution. sin 1 3. Applying the variational iteration method to boundary value problems In this section we consider boundary value problems of the form u00 ¼ bF ðuÞ;
uð0Þ ¼ a;
uð1Þ ¼ c;
ð3:1Þ
where b is a real number and the nonlinear function F(u) is assumed to have a power series representation. To solve (3.1) by means of the variational iteration method, we construct a correction functional as follows: Z x unþ1 ðxÞ ¼ un ðxÞ þ kðu00n ðnÞ þ bF ð~ un ðnÞÞÞ dn; ð3:2Þ 0
where k is a general Lagrange multiplier, u0 is an initial approximation or trial-function with possible unknowns, ~ un is considered as a restricted variation, i.e. d~un ¼ 0. Making the above correction functional (3.2) stationary, noticing that d~ un ¼ 0, dunþ1 ðxÞ ¼ dun ðxÞ þ d dunþ1 ðxÞ ¼ dun ðxÞ þ d
Z
x
Z0 x
kðu00n ðnÞ þ bF ð~ un ðnÞÞÞ dn;
ð3:3Þ
kðu00n ðnÞÞ dn;
ð3:4Þ
0
yields the following stationary conditions: dun ðnÞ : 1 k0 ðnÞ ¼ 0; du0n ðnÞ : kðnÞ ¼ 0; dun ðnÞ : k00 ðnÞ ¼ 0: The Lagrange multiplier, therefore, can be readily identified as kðnÞ ¼ n x; as a result, we obtain the following iteration formula: Z x unþ1 ðxÞ ¼ un ðxÞ þ ðn xÞðu00n ðnÞ þ bF ðun ðnÞÞÞ dn:
ð3:5Þ
0
To illustrate the scheme, let the nonlinear operator F(un) be a nonlinear function of un, say g(un). Assume that the Taylor expansion of g(un) around u0 is 1 ð2Þ 2 g ðu0 Þðu u0 Þ þ 2! We begin with the initial approximation u0(x) = (c a)x + a and we approximate the solution u(x), using the iteration formula (3.5), by the nth term un(x). gðuÞ ¼ gðu0 Þ þ gð1Þ ðu0 Þðu u0 Þ þ
4. Numerical applications Example 4.1 (Troesch’s problem). In this example we apply the algorithm described in the previous section to approximate the nonlinear boundary value problem, Troesch’s problem. u00 ðxÞ ¼ b sinhðb uðxÞÞ;
0 6 x 6 1;
ð4:1Þ
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with the boundary conditions uð0Þ ¼ 0;
uð1Þ ¼ 1:
Troesch’s problem was described and solved by Weibel [23]. Also, solved in [24] using the Adomian decomposition method. It arises from a system of nonlinear ordinary differential equations which occur in an investigation of the confinement of a plasma column by radiation pressure. The problem has been studied extensively. Troesch found its numerical solution by the shooting method (see [25]). The closed form solution to this problem in terms of the Jacobian elliptic function has been given in [26] as _ 2 1 uð0Þ 2 ðkxj1 1=4u_ ð0ÞÞ ; uðxÞ ¼ sinh ð4:2Þ k 2 _ _ where uð0Þ, the derivative of u at 0, is given by the expression uð0Þ ¼ 2ð1 mÞ the transcendental equation sinhðk=2Þ 1=2
ð1 mÞ
1=2
, with m being the solution of
¼ scðkjmÞ;
ð4:3Þ
where sc(kjm) is the Jacobi elliptic function (see, for example, [27,28]). From (4.2), it was noted in [26] that the pole occurs at 1 16 ln x : ð4:4Þ 2k 1m In view of (3.2) the correction functional for (4.1) can be expressed as follows: Z x unþ1 ðxÞ ¼ un ðxÞ þ kðu00n ðnÞ b sinhðb~ un ðnÞÞÞ dn;
ð4:5Þ
0
where k are general Lagrange multipliers, and can be identified optimally by the variational theory, u0 is an initial approximation or trial-function with possible unknowns, ~un are considered as restricted variations, i.e. d~ un = 0. Making the correction functional (4.5) stationary, noticing that d~un = 0, Z x dunþ1 ðxÞ ¼ dun ðxÞ þ d kðu00n ðnÞ b sinhðb~ un ðnÞÞÞ dn; ð4:6Þ 0 Z x dunþ1 ðxÞ ¼ dun ðxÞ þ d kðu00n ðnÞÞ dn; ð4:7Þ 0 Z x dunþ1 ðxÞ ¼ dun ðxÞ þ kdu0n ðxÞ k0 dun ðxÞ dun ðnÞk00 dn; ð4:8Þ 0
yields the following stationary conditions: dun ðnÞ : 1 k0 ðnÞ ¼ 0; du0n ðnÞ : kðnÞ ¼ 0; dun ðnÞ : k00 ðnÞ ¼ 0: The Lagrange multiplier, therefore, can be readily identified, kðnÞ ¼ n x: As a result, we obtain the following iteration formula: Z x ðn xÞðu00n ðnÞ b sinhðbun ðnÞÞÞ dn: unþ1 ðxÞ ¼ un ðxÞ þ 0
ð4:9Þ
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Now we begin with the initial approximation u0 = x. Using Taylor series expansion of sinh(bun(n)) around u0(n) = n, we have sinhðbun ðnÞÞ sinhðbnÞ þ b coshðbnÞðun nÞ þ
b2 sinhðbnÞ 2 ðun nÞ þ 2!
ð4:10Þ
Substitute Eq. (4.10) into Eq. (4.9), we get Z x b2 sinhðbnÞ 2 00 ðun ðnÞ nÞ unþ1 ðxÞ ¼ un ðxÞ þ ðn xÞ un ðnÞ b sinhðbnÞ þ b coshðbnÞðun ðnÞ nÞ þ dn: 2! 0 ð4:11Þ By the above variational iteration formula (4.11), we can obtain following result: Z
x
u1 ðxÞ ¼ u0 ðxÞ ðn xÞðb sinhðbnÞÞ dn; 0 x sinhðxbÞ u1 ðxÞ ¼ x b : b b2
ð4:12Þ
Continuing in this manner, we obtain the first few components of un(x) u0 ðxÞ ¼ x;
x sinhðxbÞ u1 ðxÞ ¼ x b ; b b2 1 ð147xb þ 6x3 b3 216xb coshðxbÞ 9xb coshð2xbÞ þ 405 sinhðxbÞ u2 ðxÞ ¼ 72b þ 36x2 b2 sinhðxbÞ þ 18 sinhð2xbÞ þ sinhð3xbÞÞ .. . and so on, in the same manner the rest of components of the iteration formula (4.11) were obtained using the Mathematica Package. Example 4.2. In this example we consider the nonlinear boundary value problem [29] 2
u00 ðxÞ þ 2ðu0 ðxÞÞ þ 8uðxÞ ¼ 0;
0 < x < 1;
uð0Þ ¼ uð1Þ ¼ 0:
ð4:13Þ
Its correction variational functionals can be expressed as follows: unþ1 ðxÞ ¼ un ðxÞ þ
Z
x
kðu00n ðnÞ þ 2ð~ u0n ðnÞÞ2 þ 8un ðnÞÞ dn;
ð4:14Þ
0
where k are general Lagrange multiplier, and can be identified optimally by the variational theory, u0 is an initial approximation or trial-function with possible unknowns, ~un are considered as restricted variations, i.e. d~ un = 0. Making the correction functional (4.14) stationary, noticing that d~un = 0, dunþ1 ðxÞ ¼ dun ðxÞ þ d
Z
x
kðu00n ðnÞ þ 2ð~ u0n ðnÞÞ2 þ 8un ðnÞÞ dn;
ð4:15Þ
kðu00n ðnÞ þ 8un ðnÞÞ dn;
ð4:16Þ
0
dunþ1 ðxÞ ¼ dun ðxÞ þ d
Z
x
0
dunþ1 ðxÞ ¼ dun ðxÞ þ kdu0n ðnÞjn¼x k0 dun ðnÞjn¼x þ
Z
x
dun ðnÞð8 k00 Þ dn; 0
ð4:17Þ
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yields the following stationary conditions: dun : 1 k0 ðnÞjn¼x ¼ 0; du0n : kðnÞjn¼x ¼ 0; dun : 8 k00 ðnÞjn¼x ¼ 0: To find the Lagrange multiplier we solve the second order differential equation 8 k00 (n)jn=x = 0 subject to the given stationary conditions, we get pffiffiffi pffiffiffi 1 1 kðnÞ ¼ pffiffiffi exp½2 2ðn xÞ pffiffiffi exp½2 2ðn xÞ: ð4:18Þ 4 2 4 2 As a result, we obtain the following iteration formula: Z x pffiffiffi pffiffiffi 1 1 pffiffiffi exp½2 2ðn xÞ pffiffiffi exp½2 2ðn xÞ ðu00n ðnÞ þ 2ðu0n ðnÞÞ2 þ 8un ðnÞÞ dn: unþ1 ðxÞ ¼ un ðxÞ þ 4 2 4 2 0 ð4:19Þ Now, we begin with initial approximation u0(x) = x. By the above variational iteration formula (4.19), we can obtain following result: Z x pffiffiffi pffiffiffi 1 1 pffiffiffi exp½2 2ðn xÞ pffiffiffi exp½2 2ðn xÞ ðu000 ðnÞ þ 2ðu00 ðnÞÞ2 þ 8u0 ðnÞÞ dn; u1 ðxÞ ¼ u0 ðxÞ þ 4 2 4 2 0 ð4:20Þ Z x p ffiffi ffi p ffiffi ffi 1 u1 ðxÞ ¼ x þ pffiffiffi ðexpð2 2ðn xÞÞ exp½2 2ðn xÞÞ dnð0 þ 2 þ 8nÞ dn; ð4:21Þ 4 2 0 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 u1 ðxÞ ¼ x þ pffiffiffi exp½2 2xð2 þ 2 þ 2 exp½4 2x þ 2 exp½4 2xÞ 2 4 2 p ffiffi ffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 þ exp½4 2xð2 2 exp½4 2x þ 8 2 exp½4 2xÞ : ð4:22Þ 2 Continuing in this manner, we obtain the first few components of un(x) u0 ðxÞ ¼ x;
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 u1 ðxÞ ¼ x þ pffiffiffi exp½2 2xð2 þ 2 þ 2 exp½4 2x þ 2 exp½4 2xÞ 2 4 2 p ffiffi ffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 þ exp½4 2xð2 2 exp½4 2x þ 8 2 exp½4 2xÞ ; 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 exp½4 2x 3 þ 2 2 ð3 þ 2 2Þ exp½8 2x u2 ðxÞ ¼ 96 pffiffiffi pffiffiffi pffiffiffi pffiffiffi þ 6 exp½4 2xð25 þ 64xÞ 8 exp½2 2xð9 7 2 þ 3ð4 þ 3 2ÞxÞ pffiffiffi pffiffiffi pffiffiffi þ 8 exp½6 2xð9 7 2 þ 3ð4 þ 3 2ÞxÞ ; and so on, in the same manner the rest of components of the iteration formula (4.19) were obtained using the Mathematica Package. 5. Numerical evaluations In order to verify numerically whether the proposed method leads to higher accuracy, we can evaluate the approximate solution using the nth-order approximation un(x). Tables 1–3 show the exact solution, the approximate solution obtained by the variational iteration method and the Adomian decomposition method. It is to be noted that only the second-order approximate were used
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Table 1 Numerical results for Troesch’s problem (b = 0.5) x
uexact
uvar.
udecom.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.095176902 0.1906338691 0.286653403 0.3835229288 0.4815373854 0.5810019749 0.6822351326 0.7855717867 0.8913669875 0.9999999999
0.100042 0.200334 0.301128 0.402677 0.505241 0.609082 0.71447 0.821682 0.931008 1.04274
0.0959478 0.192135 0.288803 0.386196 0.484559 0.584144 0.685211 0.788023 0.892858 1
Table 2 Numerical results for Troesch’s problem (b = 1) x
uexact
uvar.
udecom.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0817969966 0.1645308709 0.2491673608 0.3367322092 0.428347161 0.5252740296 0.6289711434 0.7411683782 0.8639700206 1.00000000020
0.100167 0.201339 0.304541 0.410841 0.521373 0.637362 0.760162 0.891287 1.03246 1.18565
0.0849253 0.170679 0.258105 0.348078 0.441523 0.539438 0.642918 0.753195 0.871676 1
Table 3 Numerical results for BVP (4.13) x
uexact
uvar.
udecom.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.09 0.16 0.21 0.24 0.25 0.24 0.21 0.16 0.09
0.0899901 0.159705 0.207928 0.231962 0.227756 0.191372 0.115792 0.0902297 1.23397
0.089966 0.15885 0.200776 0.199056 0.118651 0.102914 0.566259 1.42252 2.88742
in evaluating the approximate solution using the variational iteration method for Tables 1 and 2 and thirdorder approximate for Table 3, two terms of the decomposition series [24] for Tables 1 and 2, three terms of the decomposition series [29] for Table 3. It is evident that the overall error can be made smaller by computing more terms using the variational iteration method. Tables 1–3 show that the decomposition method is more effective in solving Troesch’s problem (Example 4.1), rather than the variational iteration method, while in Example 4.2 the variational iteration method is more effective than the other method. We can conclude that the variational iteration method is powerful and efficient technique in finding exact and approximate solutions for linear and nonlinear problems. This works shows that one of the most advantages of variational iteration method over the Adomian’s decomposition method that the first overcomes the difficulty arising in calculating Adomain polynomials. Many of the
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results obtained in this work can be extended to significantly more general classes of linear and nonlinear differential equations. References [1] J.H. He, Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simulat. 2 (4) (1997) 235–236. [2] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Meth. Appl. Mech. Eng. 167 (1998) 57–68. [3] J.H. He, Approximate solution of non linear differential equations with convolution product nonlinearities, Comput. Meth. Appl. Mech. Eng. 167 (1998) 69–73. [4] J.H. He, Variational iteration method- a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech. 34 (1999) 699–708. [5] J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput. 114 (2000) 115–123. [6] J.H. He, Y.Q. Wan, Q. Guo, An iteration formulation for normalized diode characteristics, Int. J. Circuit Theory Appl. 32 (6) (2004) 629–632. [7] G.E. Draˇgaˇnescu, V. Caˇpaˇlnaˇsan, Nonlinear relaxation phenomena in polycrystalline solids, Int. J. Nonlinear Sci. Numer. Simulat. 4 (3) (2003) 219–226. [8] V. Marinca, An approximate solution for one-dimensional weakly nonlinear oscillations, Int. J. Nonlinear Sci. Numer. Simulat. 3 (2) (2002) 107–110. [9] S. Momani, S. Abuasad, Application of He’s variational iteration method to Helmholtz equation, Choas, Solitons & Fractals 27 (5) (2006) 1119–1123. [10] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burgers and coupled Burger’s equations, J. Comp. Appl. Math. 181 (2) (2005) 245–251. [11] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simulat. 7 (1) (2006) 27–34. [12] S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals, in press. [13] S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355 (2006) 271–279. [14] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nasser (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, Oxford, 1978, pp. 156–162. [15] J.H. He, Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int. J. Turbo. Jet-Engines 14 (1) (1997) 23–28. [16] J.H. He, Variational theory for linear magneto-electro-elasticity, Int. J. Nonlinear Sci. Numer. Simulat. 2 (4) (2001) 309–316. [17] J.H. He, Generalized Variational Principles in Fluids (in Chinese), Science and Culture publishing house of China, Hong Kong, 2003, pp. 222–230. [18] J.H. He, Variational principle for Nano thin film lubrication, Int. J. Nonlinear Sci. Numer. Simulat. 4 (3) (2003) 313–314. [19] J.H. He, Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos Solitons & Fractals 19 (4) (2004) 847–851. [20] H.M. Liu, Variational approach to nonlinear electrochemical system, Int. J. Nonlinear Sci. Numer. Simulat. 5 (1) (2004) 95–96. [21] H.M. Liu, Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method, Chaos Solitons & Fractals 23 (2) (2005) 573–576. [22] T.H. Hao, Search for variational principles in electrodynamics by Lagrange method, Int. J. Nonlinear Sci. Numer. Simulat. 6 (2) (2005) 209–210. [23] E.S. Weibel, in: R.K.M. Landshoff (Ed.), The Plasma in Magnetic Field, Stanford University Press, Stanford, 1958. [24] E. Deeba, S.A. Khuri, S. Xie, An algorithm for solving boundary value problems, J. Comput. Phys. 159 (2000) 125–138. [25] B.A. Troesch, A simple approach to a sensitive two-point boundary value problem, J. Comput. Phys. 21 (1976) 279. [26] S.M. Robets, J.S. Shipman, On the closed form solution of Troesch’s problem, J. Comput. Phys. 21 (1976) 291. [27] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New york, 1972. [28] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendetal Functions, McGraw-Hill, New York, 1953. [29] A.M. Wazwaz, A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Comput. Math. Appl. 41 (2001) 1237–1244.
Variational iteration method for solving nonlinear boundary value problems Shaher Momani
a,*
, Salah Abuasad a, Zaid Odibat
b
a
b
Department of Mathematics, Mutah University, P.O. Box 7, Mutah, Jordan Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balq’a Applied University, Salt, Jordan
Abstract In this paper, He’s variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems. The variational iteration method, which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. Numerical examples are given and comparisons are made with the Adomian decomposition method. The fact that this method solves nonlinear equations without using Adomian polynomials can be considered as an advantage of this method over Adomian decomposition method. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Variational iteration method; Lagrange multipliers; Nonlinear boundary value problems; Troesch’s problem
1. Introduction In this paper, we extend the applications of the variational iteration method [1–6] to find approximate solutions for nonlinear boundary value problems. The variational iteration method, which proposed by Ji-Huan He [1–6], is effectively and easily used to solve some classes of nonlinear problems. For linear problems, its exact solution can be obtained by only one iteration step due to the fact that the Lagrange multiplier can be exactly identified. The variational iteration method is proposed to solve the generalized normalized diode equation, by suitable choice of the initial trial-function, one-step iteration leads to an high accurate solution, which is valid for the whole solution domain [6]. Draˇgaˇnescu and Caˇpaˇlnaˇsan [7] applied the variational iteration method to nonlinear anelastic model describing the acceleration of the relaxation process in the presence of the vibrations. The combination of a perturbation method, variational iteration method, method of variation of constants and averaging method to establish an approximate solution of one degree of freedom weakly nonlinear systems was proposed in [8]. Moreover, the method was successfully applied to delay differential equations in [1], to autonomous ordinary differential systems [5], to Helmholtz equation [9], and other fields [3,4,10]. *
Corresponding author. E-mail addresses: [email protected] (S. Momani), [email protected] (Z. Odibat).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.05.138
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S. Momani et al. / Applied Mathematics and Computation 183 (2006) 1351–1358
Recently, the application of He’s variational iteration method is extended to fractional differential equations [2,11–13]. The objective of the present paper is to extend the application of the variational iteration method to provides an approximate solutions for a class of boundary value problems and to make comparison with that obtained by Adomian decomposition method. 2. Variational iteration method To illustrate the basic concepts of the variational iteration method, we consider the following general nonlinear differential equation: LuðtÞ þ NuðtÞ ¼ gðtÞ;
ð2:1Þ
where L is a linear operator, N is a nonlinear operator and g(t) is a known analytical function. According to the variational iteration method, we can construct a correction functional as follows: Z t unþ1 ðtÞ ¼ un ðtÞ þ kðLun ðnÞ þ N u~n ðnÞ gðnÞÞ dn; ð2:2Þ 0
where k is a general Lagrange multiplier [14], which can be identified optimally via the variational theory [14– 22], the subscript n denotes the nth approximation, and ~un is considered as a restricted variation [1–6,17], i.e. d~ un ¼ 0. For convenience of the reader we will present the following example. Example 2.1. Consider the second order linear boundary value problem [5] d2 T þ T þ x ¼ 0; dx2
ð2:3Þ
0 < x < 1;
with boundary conditions T ð0Þ ¼ T ð1Þ ¼ 0;
ð2:4Þ
where T is temperature. Its correction functional can be written down as follows: Z x kðT 00n ðsÞ þ T n ðsÞ þ sÞ ds: T nþ1 ðxÞ ¼ T n ðxÞ þ
ð2:5Þ
0
Taking variation with respect to the independent variable Tn, noticing that dTn(0) = 0, Z x dT nþ1 ðxÞ ¼ dT n ðxÞ þ d kðT 00n ðsÞ þ T n ðsÞ þ sÞ ds; 0 Z x 0 0 ðk00 þ kÞdT n ds ¼ 0; ¼ dT n ðxÞ þ kdT n ðsÞjs¼x k dT n ðsÞjs¼x þ 0
for all variations dTn and dT 0n , implying the following stationary conditions: dT n : k00 ðsÞjs¼x þ kðsÞjs¼x ¼ 0; dT 0n : kðsÞjs¼x ¼ 0; dT n : 1 k0 ðsÞjs¼x ¼ 0: The general Lagrange multiplier, therefore, can be readily identified k ¼ sinðs xÞ; as a result, we obtain the following iteration formula: Z x sinðs xÞðT 00n ðsÞ þ T n ðsÞ þ sÞ ds: T nþ1 ðxÞ ¼ T n ðxÞ þ 0
ð2:6Þ
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If we use its complementary solution T0 = A cos x + B sin x as an initial approximation, using the iteration formula, we get Z x T 1 ¼ A cos x þ B sin x þ s sinðs xÞ ds ¼ A cos x þ B sin x x sin x: 0
By imposing the boundary conditions (2.4) yields A = 1 and B ¼ sin1 1 1, as a result, we have T 1 ðxÞ ¼ sin x x which is the exact solution. sin 1 3. Applying the variational iteration method to boundary value problems In this section we consider boundary value problems of the form u00 ¼ bF ðuÞ;
uð0Þ ¼ a;
uð1Þ ¼ c;
ð3:1Þ
where b is a real number and the nonlinear function F(u) is assumed to have a power series representation. To solve (3.1) by means of the variational iteration method, we construct a correction functional as follows: Z x unþ1 ðxÞ ¼ un ðxÞ þ kðu00n ðnÞ þ bF ð~ un ðnÞÞÞ dn; ð3:2Þ 0
where k is a general Lagrange multiplier, u0 is an initial approximation or trial-function with possible unknowns, ~ un is considered as a restricted variation, i.e. d~un ¼ 0. Making the above correction functional (3.2) stationary, noticing that d~ un ¼ 0, dunþ1 ðxÞ ¼ dun ðxÞ þ d dunþ1 ðxÞ ¼ dun ðxÞ þ d
Z
x
Z0 x
kðu00n ðnÞ þ bF ð~ un ðnÞÞÞ dn;
ð3:3Þ
kðu00n ðnÞÞ dn;
ð3:4Þ
0
yields the following stationary conditions: dun ðnÞ : 1 k0 ðnÞ ¼ 0; du0n ðnÞ : kðnÞ ¼ 0; dun ðnÞ : k00 ðnÞ ¼ 0: The Lagrange multiplier, therefore, can be readily identified as kðnÞ ¼ n x; as a result, we obtain the following iteration formula: Z x unþ1 ðxÞ ¼ un ðxÞ þ ðn xÞðu00n ðnÞ þ bF ðun ðnÞÞÞ dn:
ð3:5Þ
0
To illustrate the scheme, let the nonlinear operator F(un) be a nonlinear function of un, say g(un). Assume that the Taylor expansion of g(un) around u0 is 1 ð2Þ 2 g ðu0 Þðu u0 Þ þ 2! We begin with the initial approximation u0(x) = (c a)x + a and we approximate the solution u(x), using the iteration formula (3.5), by the nth term un(x). gðuÞ ¼ gðu0 Þ þ gð1Þ ðu0 Þðu u0 Þ þ
4. Numerical applications Example 4.1 (Troesch’s problem). In this example we apply the algorithm described in the previous section to approximate the nonlinear boundary value problem, Troesch’s problem. u00 ðxÞ ¼ b sinhðb uðxÞÞ;
0 6 x 6 1;
ð4:1Þ
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with the boundary conditions uð0Þ ¼ 0;
uð1Þ ¼ 1:
Troesch’s problem was described and solved by Weibel [23]. Also, solved in [24] using the Adomian decomposition method. It arises from a system of nonlinear ordinary differential equations which occur in an investigation of the confinement of a plasma column by radiation pressure. The problem has been studied extensively. Troesch found its numerical solution by the shooting method (see [25]). The closed form solution to this problem in terms of the Jacobian elliptic function has been given in [26] as _ 2 1 uð0Þ 2 ðkxj1 1=4u_ ð0ÞÞ ; uðxÞ ¼ sinh ð4:2Þ k 2 _ _ where uð0Þ, the derivative of u at 0, is given by the expression uð0Þ ¼ 2ð1 mÞ the transcendental equation sinhðk=2Þ 1=2
ð1 mÞ
1=2
, with m being the solution of
¼ scðkjmÞ;
ð4:3Þ
where sc(kjm) is the Jacobi elliptic function (see, for example, [27,28]). From (4.2), it was noted in [26] that the pole occurs at 1 16 ln x : ð4:4Þ 2k 1m In view of (3.2) the correction functional for (4.1) can be expressed as follows: Z x unþ1 ðxÞ ¼ un ðxÞ þ kðu00n ðnÞ b sinhðb~ un ðnÞÞÞ dn;
ð4:5Þ
0
where k are general Lagrange multipliers, and can be identified optimally by the variational theory, u0 is an initial approximation or trial-function with possible unknowns, ~un are considered as restricted variations, i.e. d~ un = 0. Making the correction functional (4.5) stationary, noticing that d~un = 0, Z x dunþ1 ðxÞ ¼ dun ðxÞ þ d kðu00n ðnÞ b sinhðb~ un ðnÞÞÞ dn; ð4:6Þ 0 Z x dunþ1 ðxÞ ¼ dun ðxÞ þ d kðu00n ðnÞÞ dn; ð4:7Þ 0 Z x dunþ1 ðxÞ ¼ dun ðxÞ þ kdu0n ðxÞ k0 dun ðxÞ dun ðnÞk00 dn; ð4:8Þ 0
yields the following stationary conditions: dun ðnÞ : 1 k0 ðnÞ ¼ 0; du0n ðnÞ : kðnÞ ¼ 0; dun ðnÞ : k00 ðnÞ ¼ 0: The Lagrange multiplier, therefore, can be readily identified, kðnÞ ¼ n x: As a result, we obtain the following iteration formula: Z x ðn xÞðu00n ðnÞ b sinhðbun ðnÞÞÞ dn: unþ1 ðxÞ ¼ un ðxÞ þ 0
ð4:9Þ
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Now we begin with the initial approximation u0 = x. Using Taylor series expansion of sinh(bun(n)) around u0(n) = n, we have sinhðbun ðnÞÞ sinhðbnÞ þ b coshðbnÞðun nÞ þ
b2 sinhðbnÞ 2 ðun nÞ þ 2!
ð4:10Þ
Substitute Eq. (4.10) into Eq. (4.9), we get Z x b2 sinhðbnÞ 2 00 ðun ðnÞ nÞ unþ1 ðxÞ ¼ un ðxÞ þ ðn xÞ un ðnÞ b sinhðbnÞ þ b coshðbnÞðun ðnÞ nÞ þ dn: 2! 0 ð4:11Þ By the above variational iteration formula (4.11), we can obtain following result: Z
x
u1 ðxÞ ¼ u0 ðxÞ ðn xÞðb sinhðbnÞÞ dn; 0 x sinhðxbÞ u1 ðxÞ ¼ x b : b b2
ð4:12Þ
Continuing in this manner, we obtain the first few components of un(x) u0 ðxÞ ¼ x;
x sinhðxbÞ u1 ðxÞ ¼ x b ; b b2 1 ð147xb þ 6x3 b3 216xb coshðxbÞ 9xb coshð2xbÞ þ 405 sinhðxbÞ u2 ðxÞ ¼ 72b þ 36x2 b2 sinhðxbÞ þ 18 sinhð2xbÞ þ sinhð3xbÞÞ .. . and so on, in the same manner the rest of components of the iteration formula (4.11) were obtained using the Mathematica Package. Example 4.2. In this example we consider the nonlinear boundary value problem [29] 2
u00 ðxÞ þ 2ðu0 ðxÞÞ þ 8uðxÞ ¼ 0;
0 < x < 1;
uð0Þ ¼ uð1Þ ¼ 0:
ð4:13Þ
Its correction variational functionals can be expressed as follows: unþ1 ðxÞ ¼ un ðxÞ þ
Z
x
kðu00n ðnÞ þ 2ð~ u0n ðnÞÞ2 þ 8un ðnÞÞ dn;
ð4:14Þ
0
where k are general Lagrange multiplier, and can be identified optimally by the variational theory, u0 is an initial approximation or trial-function with possible unknowns, ~un are considered as restricted variations, i.e. d~ un = 0. Making the correction functional (4.14) stationary, noticing that d~un = 0, dunþ1 ðxÞ ¼ dun ðxÞ þ d
Z
x
kðu00n ðnÞ þ 2ð~ u0n ðnÞÞ2 þ 8un ðnÞÞ dn;
ð4:15Þ
kðu00n ðnÞ þ 8un ðnÞÞ dn;
ð4:16Þ
0
dunþ1 ðxÞ ¼ dun ðxÞ þ d
Z
x
0
dunþ1 ðxÞ ¼ dun ðxÞ þ kdu0n ðnÞjn¼x k0 dun ðnÞjn¼x þ
Z
x
dun ðnÞð8 k00 Þ dn; 0
ð4:17Þ
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yields the following stationary conditions: dun : 1 k0 ðnÞjn¼x ¼ 0; du0n : kðnÞjn¼x ¼ 0; dun : 8 k00 ðnÞjn¼x ¼ 0: To find the Lagrange multiplier we solve the second order differential equation 8 k00 (n)jn=x = 0 subject to the given stationary conditions, we get pffiffiffi pffiffiffi 1 1 kðnÞ ¼ pffiffiffi exp½2 2ðn xÞ pffiffiffi exp½2 2ðn xÞ: ð4:18Þ 4 2 4 2 As a result, we obtain the following iteration formula: Z x pffiffiffi pffiffiffi 1 1 pffiffiffi exp½2 2ðn xÞ pffiffiffi exp½2 2ðn xÞ ðu00n ðnÞ þ 2ðu0n ðnÞÞ2 þ 8un ðnÞÞ dn: unþ1 ðxÞ ¼ un ðxÞ þ 4 2 4 2 0 ð4:19Þ Now, we begin with initial approximation u0(x) = x. By the above variational iteration formula (4.19), we can obtain following result: Z x pffiffiffi pffiffiffi 1 1 pffiffiffi exp½2 2ðn xÞ pffiffiffi exp½2 2ðn xÞ ðu000 ðnÞ þ 2ðu00 ðnÞÞ2 þ 8u0 ðnÞÞ dn; u1 ðxÞ ¼ u0 ðxÞ þ 4 2 4 2 0 ð4:20Þ Z x p ffiffi ffi p ffiffi ffi 1 u1 ðxÞ ¼ x þ pffiffiffi ðexpð2 2ðn xÞÞ exp½2 2ðn xÞÞ dnð0 þ 2 þ 8nÞ dn; ð4:21Þ 4 2 0 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 u1 ðxÞ ¼ x þ pffiffiffi exp½2 2xð2 þ 2 þ 2 exp½4 2x þ 2 exp½4 2xÞ 2 4 2 p ffiffi ffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 þ exp½4 2xð2 2 exp½4 2x þ 8 2 exp½4 2xÞ : ð4:22Þ 2 Continuing in this manner, we obtain the first few components of un(x) u0 ðxÞ ¼ x;
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 u1 ðxÞ ¼ x þ pffiffiffi exp½2 2xð2 þ 2 þ 2 exp½4 2x þ 2 exp½4 2xÞ 2 4 2 p ffiffi ffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 þ exp½4 2xð2 2 exp½4 2x þ 8 2 exp½4 2xÞ ; 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 exp½4 2x 3 þ 2 2 ð3 þ 2 2Þ exp½8 2x u2 ðxÞ ¼ 96 pffiffiffi pffiffiffi pffiffiffi pffiffiffi þ 6 exp½4 2xð25 þ 64xÞ 8 exp½2 2xð9 7 2 þ 3ð4 þ 3 2ÞxÞ pffiffiffi pffiffiffi pffiffiffi þ 8 exp½6 2xð9 7 2 þ 3ð4 þ 3 2ÞxÞ ; and so on, in the same manner the rest of components of the iteration formula (4.19) were obtained using the Mathematica Package. 5. Numerical evaluations In order to verify numerically whether the proposed method leads to higher accuracy, we can evaluate the approximate solution using the nth-order approximation un(x). Tables 1–3 show the exact solution, the approximate solution obtained by the variational iteration method and the Adomian decomposition method. It is to be noted that only the second-order approximate were used
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Table 1 Numerical results for Troesch’s problem (b = 0.5) x
uexact
uvar.
udecom.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.095176902 0.1906338691 0.286653403 0.3835229288 0.4815373854 0.5810019749 0.6822351326 0.7855717867 0.8913669875 0.9999999999
0.100042 0.200334 0.301128 0.402677 0.505241 0.609082 0.71447 0.821682 0.931008 1.04274
0.0959478 0.192135 0.288803 0.386196 0.484559 0.584144 0.685211 0.788023 0.892858 1
Table 2 Numerical results for Troesch’s problem (b = 1) x
uexact
uvar.
udecom.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0817969966 0.1645308709 0.2491673608 0.3367322092 0.428347161 0.5252740296 0.6289711434 0.7411683782 0.8639700206 1.00000000020
0.100167 0.201339 0.304541 0.410841 0.521373 0.637362 0.760162 0.891287 1.03246 1.18565
0.0849253 0.170679 0.258105 0.348078 0.441523 0.539438 0.642918 0.753195 0.871676 1
Table 3 Numerical results for BVP (4.13) x
uexact
uvar.
udecom.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.09 0.16 0.21 0.24 0.25 0.24 0.21 0.16 0.09
0.0899901 0.159705 0.207928 0.231962 0.227756 0.191372 0.115792 0.0902297 1.23397
0.089966 0.15885 0.200776 0.199056 0.118651 0.102914 0.566259 1.42252 2.88742
in evaluating the approximate solution using the variational iteration method for Tables 1 and 2 and thirdorder approximate for Table 3, two terms of the decomposition series [24] for Tables 1 and 2, three terms of the decomposition series [29] for Table 3. It is evident that the overall error can be made smaller by computing more terms using the variational iteration method. Tables 1–3 show that the decomposition method is more effective in solving Troesch’s problem (Example 4.1), rather than the variational iteration method, while in Example 4.2 the variational iteration method is more effective than the other method. We can conclude that the variational iteration method is powerful and efficient technique in finding exact and approximate solutions for linear and nonlinear problems. This works shows that one of the most advantages of variational iteration method over the Adomian’s decomposition method that the first overcomes the difficulty arising in calculating Adomain polynomials. Many of the
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