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Adomian method for a class of hyperbolic equations with an integral condition
Applied Mathematics and Computation 177 (2006) 545–552 www.elsevier.com/locate/amc
Adomian method for a class of hyperbolic equations with an integral condition Lazhar Bougoffa Faculty of Computer Science and Information, Al-Imam Muhammad Ibn Saud Islamic University, P.O. Box 84880, Riyadh 11681, Saudi Arabia
Abstract In this paper Adomian Decomposition Method (ADM) is applied to the solvability of a boundary problem with a nonlocal condition for a class of hyperbolic equations and illustrated with a few simple examples. 2005 Elsevier Inc. All rights reserved. Keywords: Nonlocal condition; Adomian decomposition method
1. Introduction Various problems arising in heat conduction [6,7,9], chemical engineering [8], thermo-elasticity [12], and plasma physics [11] can be reduced to the nonlocal problems with integral boundary conditions. Boundary value problems with integral conditions constitute a very interesting and important class of problems. For comments on their importance, we refer the reader to the above papers. This type of boundary value problems has been investigated in [3,5–9,12,13] for parabolic equations and problems for hyperbolic equations were the subject of few papers [4,10,12]. Such problems with nonlocal boundary conditions become more difficult and are usually not easy to handle and one must frequently resort to Fourier series method, Riesz representation theorem and numerical treatment. The objective of this paper is to solve analytically a boundary value problem with a nonlocal integral condition for a class of hyperbolic equations by using the Adomian decomposition method. 2. The homogeneous equation Consider the homogeneous hyperbolic equation o2 u o2 u þ ku ¼ 0; ot2 ox2
k 2 R;
E-mail address: bougoff[email protected] 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.07.069
ð2:1Þ
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L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
in the rectangular domain X = (0, T) · (0, 1). To Eq. (2.1) we attach the initial conditions uð0; xÞ ¼ uðxÞ; ut ð0; xÞ ¼ wðxÞ;
ð2:2Þ ð2:3Þ
Dirichlet boundary condition uðt; 1Þ uðt; 0Þ ¼ 0;
ð2:4Þ
and the nonlocal boundary condition Z 1 uðt; xÞ dx ¼ 0.
ð2:5Þ
0
We assume that the following requirements: u(x), w(x) 2 L2(0, 1) are known functions and satisfy the compatibility conditions Z 1 Z 1 uð1Þ uð0Þ ¼ 0; wð1Þ wð0Þ ¼ 0 and uðxÞ dx ¼ wðxÞ dx ¼ 0. 0
0
Firstly, as in [4] we reduce (2.1)–(2.5) to an equivalent problem. Lemma 1. Problem (2.1)–(2.5) is equivalent to the following problem: 8 2 2 o u > ooxu2 þ ku ¼ 0; > ot2 > > > > > uð0; xÞ ¼ uðxÞ; > > < ðPrÞ1 ut ð0; xÞ ¼ wðxÞ; > > > > > uðt; 1Þ uðt; 0Þ ¼ 0; > > > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0. Proof. Let u(t, x) be a solution of (2.1)–(2.5). Integrating (2.1) with respect to x over (0, 1), and taking in account (2.5), we obtain ux ðt; 1Þ ux ðt; 0Þ ¼ 0.
ð2:6Þ
Let now u(t, x) be a solution of (Pr)1, we are required to show that Z 1 uðt; xÞ dx ¼ 0; 8t 2 ð0; T Þ. 0
For this end we integrate (2.1) with respect to x we obtain Z 1 Z 1 d2 uðt; xÞ dx þ k uðt; xÞ dx ¼ 0; 8t 2 ð0; T Þ; dt2 0 0 by virtue of the compatibility conditions Z 1 Z 1 uð0; xÞ dx ¼ 0 and ut ð0; xÞ dx ¼ 0; 0
we get Z
0
1
uðt; xÞ dx ¼ 0.
0
In summary, the presence of integral condition complicates the application of standard methods. For this, Lemma 1 eliminates the nonlocal condition (integral condition) and reduces it to the classical boundary condition. Therefore it is usually to have recourse to ordinary methods.
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
547
3. Adomian decomposition method Here we shall use the Adomian’s decomposition method [1,2] for solving (Pr)1. Consider first equation of (Pr)1 in an operator form as Ltt u Lxx u þ ku ¼ 0; 2
ð3:1Þ 2
where Ltt u ¼ oot2u and Lxx u ¼ ooxu2 . It is clear that Ltt and Lxx are invertible, and they are defined as Rt Rt Rx Rx 1 L1 tt ð.Þ ¼ 0 0 ð.Þ dt dt and Lxx ð.Þ ¼ 0 0 ð.Þ dx dx. Following the Adomian decomposition method [1,2] the unknown solution u is assumed to be given by a series of the form 1 X un ðt; xÞ; ð3:2Þ uðt; xÞ ¼ n¼0
where the components un(t, x) are going to be determined recurrently. Therefore, we can write the solutions in t and x directions as u0 ðt; xÞ ¼ uðxÞ þ twðxÞ;
ð3:3Þ
2
unþ1 ðt; xÞ ¼ L1 tt
ou L1 tt kun ; ox2
n P 0;
ð3:4Þ
and u0 ðt; xÞ ¼ uðt; 0Þ þ xux ðt; 0Þ;
ð3:5Þ
2
unþ1 ðt; xÞ ¼ L1 xx
ou þ L1 xx kun ; ot2
n P 0;
ð3:6Þ
respectively. Using ADM described above, we obtain series solutions for (Pr)1. In order to demonstrate the feasibility and efficiency of the ADM, two examples with closed form solutions are studied carefully. Example 1. Consider problem (2.1)–(2.5) with the initial conditions uð0; xÞ ¼ uðxÞ ¼ cosð2pxÞ; ut ð0; xÞ ¼ wðxÞ ¼ cosð2pxÞ; and boundary conditions uðt; 1Þ uðt; 0Þ ¼ 0; Z 1 uðt; xÞ dx ¼ 0; 0
where uðt; 1Þ ¼ uðt; 0Þ ¼ expðtÞ
and
k ¼ ð1 þ 4p2 Þ.
By Lemma 1 this nonlocal boundary problem can be transformed into the form of (Pr)1
ðPrÞ1
8 2 2 o u ooxu2 þ ku ¼ 0; > > ot2 > > > > > > uð0; xÞ ¼ cosð2pxÞ; <
ut ð0; xÞ ¼ cosð2pxÞ; > > > > > uðt; 1Þ uðt; 0Þ ¼ 0; > > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0;
where ux ðt; 1Þ ¼ ux ðt; 0Þ ¼ 0.
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L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
The exact solution of this problem is u ðt; xÞ ¼ expðtÞ cosð2pxÞ. To obtain the solution in t direction by ADM. Writing uðt; xÞ ¼ of ADM as
P1
n¼0 un ðt; xÞ,
we express the recurrent scheme
u0 ¼ ð1 tÞ cosð2pxÞ; 2 t t3 u1 ¼ cosð2pxÞ; 2! 3! 4 t t5 u2 ¼ cosð2pxÞ; 4! 5! .. . Then the terms of u(t, x) can be written as n t2 n t þ . uðt; xÞ ¼ cosð2pxÞ 1 t þ þ ð1Þ 2! n!
ð3:7Þ
It can be easily observed that (3.7) is equivalent to the exact solution. Similarly, to obtain the solution in x direction, we express the recurrent scheme of ADM as u0 ¼ expðtÞ; 2
ð2pÞ x2 expðtÞ; 2! 4 4 ð2pÞ x expðtÞ; u2 ¼ 4! .. . u1 ¼
Then the terms of u(t, x) can be written as ! 2n 2n ð2pÞ2 x2 ð2pÞ4 x4 n ð2pÞ x þ þ ð1Þ þ ; uðt; xÞ ¼ expðtÞ 1 2! 4! ð2nÞ! we again obtain the above exact solution. Example 2. Consider problem (2.1)–(2.5) with the initial conditions uð0; xÞ ¼ uðxÞ ¼ 0; ut ð0; xÞ ¼ wðxÞ ¼ sinð2pxÞ; and boundary conditions uðt; 1Þ uðt; 0Þ ¼ 0; Z 1 uðt; xÞ dx ¼ 0; 0
where uðt; 1Þ ¼ uðt; 0Þ ¼ 0 and
k ¼ 1 4p2 .
ð3:8Þ
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
549
It can be easily observed that this problem is equivalent to the following problem: 8 2 2 o u > ooxu2 þ ku ¼ 0; > ot2 > > > > < uð0; xÞ ¼ 0; ðPrÞ1 ut ð0; xÞ ¼ sinð2pxÞ; > > > > uðt; 1Þ uðt; 0Þ ¼ 0; > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0; where ux ðt; 1Þ ¼ ux ðt; 0Þ ¼ 2p sin t. The exact solution of this problem is u ðt; xÞ ¼ sin t sinð2pxÞ. To obtain the solution in t direction by ADM. Writing uðt; xÞ ¼ of ADM as
P1
n¼0 un ðt; xÞ,
we express the recurrent scheme
u0 ¼ t sinð2pxÞ; u1 ¼ u2 ¼
t3 sinð2pxÞ; 3!
t5 sinð2pxÞ; 5!
.. . un ¼ ð1Þ
n
t2nþ1 sinð2pxÞ. ð2n þ 1Þ!
Then the terms of u(t, x) can be written as 1 X un ðt; xÞ ¼ sin t sinð2pxÞ. uðt; xÞ ¼
ð3:9Þ
n¼0
For the solution of this problem in the x direction, we express the recurrent scheme of ADM as u0 ¼ 2px sin t; 3
ð2pxÞ sin t; 3! ð2pxÞ5 sin t; u2 ¼ 5! .. . u1 ¼
2nþ1
un ¼ ð1Þn
ð2pxÞ sin t. ð2n þ 1Þ!
Then the terms of u(t, x) can be written as 1 X uðt; xÞ ¼ un ðt; xÞ ¼ sin t sinð2pxÞ.
ð3:10Þ
n¼0
4. The inhomogeneous hyperbolic equation In this section, we consider the inhomogeneous hyperbolic equation o2 U o2 U 2 þ kU ¼ F ðt; xÞ; ot2 ox
ð4:1Þ
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L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
in the rectangular domain X = (0,T) · (0, 1). To Eq. (4.1) we attach the initial conditions U ð0; xÞ ¼ UðxÞ; U t ð0; xÞ ¼ WðxÞ.
ð4:2Þ ð4:3Þ
Dirichlet boundary condition U ðt; 0Þ ¼ 0;
ð4:4Þ
and the nonlocal boundary condition Z 1 U ðt; xÞ dx ¼ 0;
ð4:5Þ
0
where U(x), W(x) 2 L2(0, 1) are known functions and satisfy the compatibility conditions Uð0Þ ¼ 0;
Wð0Þ ¼ 0 and
Z
Z
1
UðxÞ dx ¼
0
1
WðxÞ dx ¼ 0.
0
In a similar way, we can reduce (4.1)–(4.5) to an equivalent problem. Lemma 2. Problem (4.1)–(4.5) is equivalent to the following problem:
ðPRÞ
8 2 2 o U ooxU2 þ kU ¼ F ðt; xÞ; > > ot2 > > > > > < U ð0; xÞ ¼ UðxÞ; U t ð0; xÞ ¼ WðxÞ; > > > > U ðt; 0Þ ¼ 0; > > > R1 : U x ðt; 1Þ U x ðt; 0Þ ¼ 0 F ðt; xÞ dx.
Introduce now the new unknown function u(t, x) = U(t, x) + w(t, x), where wðt; xÞ ¼ xðx1Þ 2 (PR) is transformed into
ðPrÞ2
R1 0
F ðt; xÞ dx. Then
8 2 2 o u > ooxu2 þ ku ¼ f ðt; xÞ; > ot2 > > > > < uð0; xÞ ¼ uðxÞ; ut ð0; xÞ ¼ wðxÞ; > > > > uðt; 0Þ ¼ 0; > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0;
where Z
1
xðx 1Þ f ðt; xÞ ¼ F ðt; xÞ F ðt; xÞ dx þ 2 0 Z xðx 1Þ 1 uðxÞ ¼ UðxÞ þ F ð0; xÞ dx; 2 0
Z 0
1
xðx 1Þ F tt ðt; xÞ dx þ k 2
Z
1
F ðt; xÞ dx;
0
and wðxÞ ¼ WðxÞ þ
xðx 1Þ 2
Z
1
F t ð0; xÞ dx.
0
In order to illustrate the technique discussed above, we shall give in the following an example in which we can apply our technique and ADM.
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
551
Example 3. Consider the following problem:
ðPRÞ
8 o2 U o2 U ox2 U ¼ expðtÞ; > ot2 > > > > xðx23Þ > > > < U ð0; xÞ ¼ 2 ; xðx2Þ U t ð0; xÞ ¼ 2 3 ; > > > > > U ðt; 0Þ ¼ 0; > > > :R1 U ðt; xÞ dx ¼ 0. 0
The analytic solution of this problem is x x 23 U ðt; xÞ ¼ expðtÞ. 2 Hence using Lemma 2 and this transformation u(t, x) = U(t, x) + w(t, x), where wðt; xÞ ¼ xðx1Þ expðtÞ, our 2 problem will be transformed to the following standard problem: 8 2 2 o u > ooxu2 u ¼ 0; > ot2 > > x > > < uð0; xÞ ¼ 6 ; ðPrÞ2 ut ð0; xÞ ¼ 6x ; > > > > uðt; 0Þ ¼ 0; > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0. Simple calculation leads to x uðt; xÞ ¼ expðtÞ; 6 which is P the analytic solution of (Pr)2. To obtain the solution of (Pr)2 in t direction by ADM. Writing 1 uðt; xÞ ¼ n¼0 un ðt; xÞ, we express the recurrent scheme of ADM as x u0 ¼ ð1 þ tÞ; 6 x t2 t3 u1 ¼ þ ; 6 2! 3! x t4 t5 þ u2 ¼ ; 6 4! 5! .. . Then the terms of u(t, x) can be written as 1 X x uðt; xÞ ¼ un ðt; xÞ ¼ expðtÞ. 6 n¼0
ð4:6Þ
Similarly, the solution of this problem in the x direction, can be determined as x u0 ¼ expðtÞ; 6 unþ1 ¼ 0; n P 0. Then the terms of u(t, x) can be written as 1 X x uðt; xÞ ¼ un ðt; xÞ ¼ expðtÞ. 6 n¼0
ð4:7Þ
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L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
5. Conclusion The Adomian decomposition method was used successfully to solve a boundary value problem for a class of hyperbolic equations with a nonlocal boundary condition. Some examples with closed form solutions are studied carefully in order to illustrate the possible practical use of this method, and the results obtained are just the same as those given from applying the ADM. References [1] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, Orlando, FL, 1986. [2] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994. [3] G.W. Batten Jr., Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Math. Comput. 17 (1963) 405–413. [4] S.A. Belin, Existence of solutions for one-dimensional wave equations with nonlocal conditions, Electron. J. Differ. Equ. 2001 (76) (2001) 1–8. [5] L. Bougoffa, Weak solution for hyperbolic equations with a nonlocal condition, Aust. J. Math. Anal. Appl. (AJMAA) 2 (1) (2005) 1–7. [6] B. Cahlon, D.M. Kulkarni, P. Shi, Stepwise stability for the heat equation with a nonlocal constraint, SIAM J. Numer. Anal. 32 (1995) 571–593. [7] J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963) 155–160. [8] Y.S. Choi, K.Y. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal. 18 (1992) 317–331. [9] N.I. Kamynin, A boundary value problem in the theory of the heat conduction with non-classical boundary condition, USSR Comput. Math. Math. Phys. 4 (1964) 33–59. [10] L.S. Pulkina, A nonlocal problem with integral conditions for hyperbolic equations, Electron. J. Differ. Equ. 1999 (45) (1999) 1–6. [11] A.A. Samarski, Some problems in the modern theory of differential equations, Differ. Uraven. 16 (1980) 1221–1228. [12] P. Shi, Weak solution to evolution problem with a nonlocal constraint, SIAM J. Anal. 24 (1993) 46–58. [13] N.I. Yurchuk, Mixed problem with an integral condition for certain parabolic equations, Differ. Equ. 22 (1986) 1457–1463.
Adomian method for a class of hyperbolic equations with an integral condition Lazhar Bougoffa Faculty of Computer Science and Information, Al-Imam Muhammad Ibn Saud Islamic University, P.O. Box 84880, Riyadh 11681, Saudi Arabia
Abstract In this paper Adomian Decomposition Method (ADM) is applied to the solvability of a boundary problem with a nonlocal condition for a class of hyperbolic equations and illustrated with a few simple examples. 2005 Elsevier Inc. All rights reserved. Keywords: Nonlocal condition; Adomian decomposition method
1. Introduction Various problems arising in heat conduction [6,7,9], chemical engineering [8], thermo-elasticity [12], and plasma physics [11] can be reduced to the nonlocal problems with integral boundary conditions. Boundary value problems with integral conditions constitute a very interesting and important class of problems. For comments on their importance, we refer the reader to the above papers. This type of boundary value problems has been investigated in [3,5–9,12,13] for parabolic equations and problems for hyperbolic equations were the subject of few papers [4,10,12]. Such problems with nonlocal boundary conditions become more difficult and are usually not easy to handle and one must frequently resort to Fourier series method, Riesz representation theorem and numerical treatment. The objective of this paper is to solve analytically a boundary value problem with a nonlocal integral condition for a class of hyperbolic equations by using the Adomian decomposition method. 2. The homogeneous equation Consider the homogeneous hyperbolic equation o2 u o2 u þ ku ¼ 0; ot2 ox2
k 2 R;
E-mail address: bougoff[email protected] 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.07.069
ð2:1Þ
546
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
in the rectangular domain X = (0, T) · (0, 1). To Eq. (2.1) we attach the initial conditions uð0; xÞ ¼ uðxÞ; ut ð0; xÞ ¼ wðxÞ;
ð2:2Þ ð2:3Þ
Dirichlet boundary condition uðt; 1Þ uðt; 0Þ ¼ 0;
ð2:4Þ
and the nonlocal boundary condition Z 1 uðt; xÞ dx ¼ 0.
ð2:5Þ
0
We assume that the following requirements: u(x), w(x) 2 L2(0, 1) are known functions and satisfy the compatibility conditions Z 1 Z 1 uð1Þ uð0Þ ¼ 0; wð1Þ wð0Þ ¼ 0 and uðxÞ dx ¼ wðxÞ dx ¼ 0. 0
0
Firstly, as in [4] we reduce (2.1)–(2.5) to an equivalent problem. Lemma 1. Problem (2.1)–(2.5) is equivalent to the following problem: 8 2 2 o u > ooxu2 þ ku ¼ 0; > ot2 > > > > > uð0; xÞ ¼ uðxÞ; > > < ðPrÞ1 ut ð0; xÞ ¼ wðxÞ; > > > > > uðt; 1Þ uðt; 0Þ ¼ 0; > > > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0. Proof. Let u(t, x) be a solution of (2.1)–(2.5). Integrating (2.1) with respect to x over (0, 1), and taking in account (2.5), we obtain ux ðt; 1Þ ux ðt; 0Þ ¼ 0.
ð2:6Þ
Let now u(t, x) be a solution of (Pr)1, we are required to show that Z 1 uðt; xÞ dx ¼ 0; 8t 2 ð0; T Þ. 0
For this end we integrate (2.1) with respect to x we obtain Z 1 Z 1 d2 uðt; xÞ dx þ k uðt; xÞ dx ¼ 0; 8t 2 ð0; T Þ; dt2 0 0 by virtue of the compatibility conditions Z 1 Z 1 uð0; xÞ dx ¼ 0 and ut ð0; xÞ dx ¼ 0; 0
we get Z
0
1
uðt; xÞ dx ¼ 0.
0
In summary, the presence of integral condition complicates the application of standard methods. For this, Lemma 1 eliminates the nonlocal condition (integral condition) and reduces it to the classical boundary condition. Therefore it is usually to have recourse to ordinary methods.
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
547
3. Adomian decomposition method Here we shall use the Adomian’s decomposition method [1,2] for solving (Pr)1. Consider first equation of (Pr)1 in an operator form as Ltt u Lxx u þ ku ¼ 0; 2
ð3:1Þ 2
where Ltt u ¼ oot2u and Lxx u ¼ ooxu2 . It is clear that Ltt and Lxx are invertible, and they are defined as Rt Rt Rx Rx 1 L1 tt ð.Þ ¼ 0 0 ð.Þ dt dt and Lxx ð.Þ ¼ 0 0 ð.Þ dx dx. Following the Adomian decomposition method [1,2] the unknown solution u is assumed to be given by a series of the form 1 X un ðt; xÞ; ð3:2Þ uðt; xÞ ¼ n¼0
where the components un(t, x) are going to be determined recurrently. Therefore, we can write the solutions in t and x directions as u0 ðt; xÞ ¼ uðxÞ þ twðxÞ;
ð3:3Þ
2
unþ1 ðt; xÞ ¼ L1 tt
ou L1 tt kun ; ox2
n P 0;
ð3:4Þ
and u0 ðt; xÞ ¼ uðt; 0Þ þ xux ðt; 0Þ;
ð3:5Þ
2
unþ1 ðt; xÞ ¼ L1 xx
ou þ L1 xx kun ; ot2
n P 0;
ð3:6Þ
respectively. Using ADM described above, we obtain series solutions for (Pr)1. In order to demonstrate the feasibility and efficiency of the ADM, two examples with closed form solutions are studied carefully. Example 1. Consider problem (2.1)–(2.5) with the initial conditions uð0; xÞ ¼ uðxÞ ¼ cosð2pxÞ; ut ð0; xÞ ¼ wðxÞ ¼ cosð2pxÞ; and boundary conditions uðt; 1Þ uðt; 0Þ ¼ 0; Z 1 uðt; xÞ dx ¼ 0; 0
where uðt; 1Þ ¼ uðt; 0Þ ¼ expðtÞ
and
k ¼ ð1 þ 4p2 Þ.
By Lemma 1 this nonlocal boundary problem can be transformed into the form of (Pr)1
ðPrÞ1
8 2 2 o u ooxu2 þ ku ¼ 0; > > ot2 > > > > > > uð0; xÞ ¼ cosð2pxÞ; <
ut ð0; xÞ ¼ cosð2pxÞ; > > > > > uðt; 1Þ uðt; 0Þ ¼ 0; > > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0;
where ux ðt; 1Þ ¼ ux ðt; 0Þ ¼ 0.
548
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
The exact solution of this problem is u ðt; xÞ ¼ expðtÞ cosð2pxÞ. To obtain the solution in t direction by ADM. Writing uðt; xÞ ¼ of ADM as
P1
n¼0 un ðt; xÞ,
we express the recurrent scheme
u0 ¼ ð1 tÞ cosð2pxÞ; 2 t t3 u1 ¼ cosð2pxÞ; 2! 3! 4 t t5 u2 ¼ cosð2pxÞ; 4! 5! .. . Then the terms of u(t, x) can be written as n t2 n t þ . uðt; xÞ ¼ cosð2pxÞ 1 t þ þ ð1Þ 2! n!
ð3:7Þ
It can be easily observed that (3.7) is equivalent to the exact solution. Similarly, to obtain the solution in x direction, we express the recurrent scheme of ADM as u0 ¼ expðtÞ; 2
ð2pÞ x2 expðtÞ; 2! 4 4 ð2pÞ x expðtÞ; u2 ¼ 4! .. . u1 ¼
Then the terms of u(t, x) can be written as ! 2n 2n ð2pÞ2 x2 ð2pÞ4 x4 n ð2pÞ x þ þ ð1Þ þ ; uðt; xÞ ¼ expðtÞ 1 2! 4! ð2nÞ! we again obtain the above exact solution. Example 2. Consider problem (2.1)–(2.5) with the initial conditions uð0; xÞ ¼ uðxÞ ¼ 0; ut ð0; xÞ ¼ wðxÞ ¼ sinð2pxÞ; and boundary conditions uðt; 1Þ uðt; 0Þ ¼ 0; Z 1 uðt; xÞ dx ¼ 0; 0
where uðt; 1Þ ¼ uðt; 0Þ ¼ 0 and
k ¼ 1 4p2 .
ð3:8Þ
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
549
It can be easily observed that this problem is equivalent to the following problem: 8 2 2 o u > ooxu2 þ ku ¼ 0; > ot2 > > > > < uð0; xÞ ¼ 0; ðPrÞ1 ut ð0; xÞ ¼ sinð2pxÞ; > > > > uðt; 1Þ uðt; 0Þ ¼ 0; > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0; where ux ðt; 1Þ ¼ ux ðt; 0Þ ¼ 2p sin t. The exact solution of this problem is u ðt; xÞ ¼ sin t sinð2pxÞ. To obtain the solution in t direction by ADM. Writing uðt; xÞ ¼ of ADM as
P1
n¼0 un ðt; xÞ,
we express the recurrent scheme
u0 ¼ t sinð2pxÞ; u1 ¼ u2 ¼
t3 sinð2pxÞ; 3!
t5 sinð2pxÞ; 5!
.. . un ¼ ð1Þ
n
t2nþ1 sinð2pxÞ. ð2n þ 1Þ!
Then the terms of u(t, x) can be written as 1 X un ðt; xÞ ¼ sin t sinð2pxÞ. uðt; xÞ ¼
ð3:9Þ
n¼0
For the solution of this problem in the x direction, we express the recurrent scheme of ADM as u0 ¼ 2px sin t; 3
ð2pxÞ sin t; 3! ð2pxÞ5 sin t; u2 ¼ 5! .. . u1 ¼
2nþ1
un ¼ ð1Þn
ð2pxÞ sin t. ð2n þ 1Þ!
Then the terms of u(t, x) can be written as 1 X uðt; xÞ ¼ un ðt; xÞ ¼ sin t sinð2pxÞ.
ð3:10Þ
n¼0
4. The inhomogeneous hyperbolic equation In this section, we consider the inhomogeneous hyperbolic equation o2 U o2 U 2 þ kU ¼ F ðt; xÞ; ot2 ox
ð4:1Þ
550
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
in the rectangular domain X = (0,T) · (0, 1). To Eq. (4.1) we attach the initial conditions U ð0; xÞ ¼ UðxÞ; U t ð0; xÞ ¼ WðxÞ.
ð4:2Þ ð4:3Þ
Dirichlet boundary condition U ðt; 0Þ ¼ 0;
ð4:4Þ
and the nonlocal boundary condition Z 1 U ðt; xÞ dx ¼ 0;
ð4:5Þ
0
where U(x), W(x) 2 L2(0, 1) are known functions and satisfy the compatibility conditions Uð0Þ ¼ 0;
Wð0Þ ¼ 0 and
Z
Z
1
UðxÞ dx ¼
0
1
WðxÞ dx ¼ 0.
0
In a similar way, we can reduce (4.1)–(4.5) to an equivalent problem. Lemma 2. Problem (4.1)–(4.5) is equivalent to the following problem:
ðPRÞ
8 2 2 o U ooxU2 þ kU ¼ F ðt; xÞ; > > ot2 > > > > > < U ð0; xÞ ¼ UðxÞ; U t ð0; xÞ ¼ WðxÞ; > > > > U ðt; 0Þ ¼ 0; > > > R1 : U x ðt; 1Þ U x ðt; 0Þ ¼ 0 F ðt; xÞ dx.
Introduce now the new unknown function u(t, x) = U(t, x) + w(t, x), where wðt; xÞ ¼ xðx1Þ 2 (PR) is transformed into
ðPrÞ2
R1 0
F ðt; xÞ dx. Then
8 2 2 o u > ooxu2 þ ku ¼ f ðt; xÞ; > ot2 > > > > < uð0; xÞ ¼ uðxÞ; ut ð0; xÞ ¼ wðxÞ; > > > > uðt; 0Þ ¼ 0; > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0;
where Z
1
xðx 1Þ f ðt; xÞ ¼ F ðt; xÞ F ðt; xÞ dx þ 2 0 Z xðx 1Þ 1 uðxÞ ¼ UðxÞ þ F ð0; xÞ dx; 2 0
Z 0
1
xðx 1Þ F tt ðt; xÞ dx þ k 2
Z
1
F ðt; xÞ dx;
0
and wðxÞ ¼ WðxÞ þ
xðx 1Þ 2
Z
1
F t ð0; xÞ dx.
0
In order to illustrate the technique discussed above, we shall give in the following an example in which we can apply our technique and ADM.
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
551
Example 3. Consider the following problem:
ðPRÞ
8 o2 U o2 U ox2 U ¼ expðtÞ; > ot2 > > > > xðx23Þ > > > < U ð0; xÞ ¼ 2 ; xðx2Þ U t ð0; xÞ ¼ 2 3 ; > > > > > U ðt; 0Þ ¼ 0; > > > :R1 U ðt; xÞ dx ¼ 0. 0
The analytic solution of this problem is x x 23 U ðt; xÞ ¼ expðtÞ. 2 Hence using Lemma 2 and this transformation u(t, x) = U(t, x) + w(t, x), where wðt; xÞ ¼ xðx1Þ expðtÞ, our 2 problem will be transformed to the following standard problem: 8 2 2 o u > ooxu2 u ¼ 0; > ot2 > > x > > < uð0; xÞ ¼ 6 ; ðPrÞ2 ut ð0; xÞ ¼ 6x ; > > > > uðt; 0Þ ¼ 0; > > : ux ðt; 1Þ ux ðt; 0Þ ¼ 0. Simple calculation leads to x uðt; xÞ ¼ expðtÞ; 6 which is P the analytic solution of (Pr)2. To obtain the solution of (Pr)2 in t direction by ADM. Writing 1 uðt; xÞ ¼ n¼0 un ðt; xÞ, we express the recurrent scheme of ADM as x u0 ¼ ð1 þ tÞ; 6 x t2 t3 u1 ¼ þ ; 6 2! 3! x t4 t5 þ u2 ¼ ; 6 4! 5! .. . Then the terms of u(t, x) can be written as 1 X x uðt; xÞ ¼ un ðt; xÞ ¼ expðtÞ. 6 n¼0
ð4:6Þ
Similarly, the solution of this problem in the x direction, can be determined as x u0 ¼ expðtÞ; 6 unþ1 ¼ 0; n P 0. Then the terms of u(t, x) can be written as 1 X x uðt; xÞ ¼ un ðt; xÞ ¼ expðtÞ. 6 n¼0
ð4:7Þ
552
L. Bougoffa / Applied Mathematics and Computation 177 (2006) 545–552
5. Conclusion The Adomian decomposition method was used successfully to solve a boundary value problem for a class of hyperbolic equations with a nonlocal boundary condition. Some examples with closed form solutions are studied carefully in order to illustrate the possible practical use of this method, and the results obtained are just the same as those given from applying the ADM. References [1] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, Orlando, FL, 1986. [2] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994. [3] G.W. Batten Jr., Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Math. Comput. 17 (1963) 405–413. [4] S.A. Belin, Existence of solutions for one-dimensional wave equations with nonlocal conditions, Electron. J. Differ. Equ. 2001 (76) (2001) 1–8. [5] L. Bougoffa, Weak solution for hyperbolic equations with a nonlocal condition, Aust. J. Math. Anal. Appl. (AJMAA) 2 (1) (2005) 1–7. [6] B. Cahlon, D.M. Kulkarni, P. Shi, Stepwise stability for the heat equation with a nonlocal constraint, SIAM J. Numer. Anal. 32 (1995) 571–593. [7] J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963) 155–160. [8] Y.S. Choi, K.Y. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal. 18 (1992) 317–331. [9] N.I. Kamynin, A boundary value problem in the theory of the heat conduction with non-classical boundary condition, USSR Comput. Math. Math. Phys. 4 (1964) 33–59. [10] L.S. Pulkina, A nonlocal problem with integral conditions for hyperbolic equations, Electron. J. Differ. Equ. 1999 (45) (1999) 1–6. [11] A.A. Samarski, Some problems in the modern theory of differential equations, Differ. Uraven. 16 (1980) 1221–1228. [12] P. Shi, Weak solution to evolution problem with a nonlocal constraint, SIAM J. Anal. 24 (1993) 46–58. [13] N.I. Yurchuk, Mixed problem with an integral condition for certain parabolic equations, Differ. Equ. 22 (1986) 1457–1463.