- Email: [email protected]
On symmetry-preserving difference scheme to a generalized Benjamin equation and third-order Burgers equation
AlgebraLetters and its50Applications 466 (2015) 102–116 Applied Linear Mathematics (2015) 146–152
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
Inverse eigenvalue problem matrix On symmetry-preserving difference schemeoftoJacobi a generalized with mixed data Benjamin equation and third-order Burgers equation✩ ∗ 1 Ying Wei Pan-Li Ma, Shou-Fu Tian , Tian-Tian Zhang ∗ Department of Mathematics and Center of Nonlinear Equations, China of University of Mining and Department of Mathematics, Nanjing University Aeronautics and Astronautics, Technology, Xuzhou 221116, People’s Republic of China Nanjing 210016, PR China
article
i n faor t i c l e
ianbfs ot r a c t
a b s t r a c t
Article history: Article history: In this of paper, the inverse eigenvalue of reconstructing In this paper, an exposition a method is presented for problem discretizing a generalized 2014 Received 21 May 2015 Received 16 January Benjamin Jacobi matrix from its eigenvalues, its leading principal equation anda third-order Burgers equation while preserving their Lie point Accepted 20 September 2014 Received in revised form 23 June and partlaws, of the of its of submatrix symmetries. By usingsubmatrix local conservation the eigenvalues potential systems original Available online 22 October 2014 2015 is which considered. The necessary and sufficient conditions for equation are obtained, can be used to construct the invariant difference Accepted 23 June 2015 Submitted by Y. Wei the existence and uniqueness of the equation, solution respectively. are derived. models and symmetry-preserving difference models of original Available online 9 July 2015 Furthermore, a numerical anddiscrete some high-order numerical Furthermore, this method is very effective and can algorithm be applied to MSC: examples are given. nonlinear evolution equations. 15A18 Keywords: 2014 Published Elsevier Inc. Ltd. All by rights reserved. © 2015©Elsevier 15A57 Generalized Benjamin equation Third-order Burgers equation Lie point symmetries Keywords: Symmetry-preserving Jacobi matrix Eigenvalue Invariant difference models Inverse problem Submatrix
1. Introduction It is well known that Lie symmetries play an important role in the analysis of nonlinear models. One of the main applications of the Lie group theory is to provide many powerful tools for solving ordinary and partial differential equations, specially nonlinear ones [1–3]. Based on Lie group theory, the standard way of solving ordinary and partial differential equations is to find the Lie point symmetry group G of the equation and then get invariant solutions. In addition, there are some effective methods to get group invariant solutions of differential equations, such as the classical Lie group approach and the non-classical Lie group approach [4–8]. Applications of the Lie group theory to difference equations are much more recent [9–25]. Lie group theory was originally invented as a systematic tool for obtaining exact analytical solutions of ordinary and partial E-mail address: [email protected]. differential equations1 (ODEs and PDEs). For differential equations the existence of a nontrivial symmetry Tel.: +86 13914485239. group makes it possible to reduce the order of the equation. All numerical methods for solving ODEs replace http://dx.doi.org/10.1016/j.laa.2014.09.031 the differential equation by a difference one, usually on a priori chosen lattice, either a regular one, or 0024-3795/© 2014 Published by Elsevier Inc.
Project supported by the Fundamental Research Funds for the Central Universities of China under the Grant No. 2015QNA53. ∗ Corresponding authors. E-mail addresses: [email protected], [email protected] (S.F. Tian), [email protected] (T.T. Zhang).
✩
http://dx.doi.org/10.1016/j.aml.2015.06.017 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
147
P.-L. Ma et al. / Applied Mathematics Letters 50 (2015) 146–152
one adapted to some known or expected behavior of solutions. In this paper, we give a method to obtain the difference models completely preserving the symmetries of the original partial differential equation. The difference of this way to others is that we construct the symmetry-preserving difference models of the potential system instead of the original equation. The paper is organized as follows: In Section 2, in order to make our presentation closed and self-contained, we briefly recall the overview of the discretization Procedure. In Section 3, the difference model of the generalized Benjamin equation is constructed which completely preserving the symmetries of the original partial differential equation. In Section 4, the same way is applied to the third-order Burgers equation and the difference model is constructed that the whole symmetries of original equation are preserved. 2. Preliminaries on discretization procedure In order to make our presentation closed and self-contained, in this section, we briefly recall the required elementary notations [20–25] about the finite difference operators and Lie transformation groups. of formal series, consider the formal transformation group whose infinitesimal operator is In the space Z the total derivative operator as follows ∂ ∂ ∂ ∂ + u1 + u2 + · · · + us+1 + ··· . (2.1) ∂x ∂u ∂u1 ∂us For simplicity, in the following, we only consider the case of one independent variable x and one dependent variable u. Let us fix an arbitrary parameter h > 0 and use the tangent field (2.1) of the Taylor group to form a pair of operators, D=
hD S =e ≡
+h
∞ hs s=0
Ds ,
s!
−hD ≡ S =e
−h
∞ (−h)s s=0
s!
Ds ,
(2.2)
the above operators be called the right and left discrete shift operators respectively, in which D is a derivation By means of S and S , we can form a pair of right and left discrete (finite-difference) differentiation in Z. +h
−h
operators as follows ∞
s−1
h 1 Ds , D = ( S −1) ≡ h +h s! +h s=1
∞
(−h) 1 D = (1 − S ) ≡ h s! −h −h s=1
s−1
Ds .
(2.3)
Suppose ∂ ∂ ∂ + ξx +η + ··· , ∂t ∂x ∂u is the generator of a one-parameter transformation group. For an evolution differential equation, X = ξt
F (x, t, u, ut , ux , uxx , uxxx ) = 0,
(2.4)
(2.5)
the group generated by (2.4) transformations a point (x, t, u, ut , ux , uxx , uxxx ) to a new one (x∗ , t∗ , u∗ , u∗t , u∗x , u∗xx , u∗xxx ) together with Eq. (2.5). When applying Lie point transformations to the difference equations, this situation be changed. Proposition 1.1. For an mesh ω to preserve uniform (h+ = h− ) under the action of the transformation group h
: G1 , the following condition should be satisfied at each point z ∈ Z h
D D (ξ(z)) = 0.
+h −h
The meshes satisfying criterion (2.6) are said to be invariantly uniform.
(2.6)
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P.-L. Ma et al. / Applied Mathematics Letters 50 (2015) 146–152
Proposition 1.2. For an orthogonal mesh ω to converse its orthogonality in the plane (t, x) under any transh
formation, it is necessary and sufficient that the following condition should be satisfied at each point: t x D (ξ ) = − D (ξ ).
(2.7)
F (z) = 0,
(2.8)
+τ
+h
Assume difference equation
is written on finitely many point of the difference mesh ω , which may be uniform or nonuniform. h
Let Ω (z) = 0,
(2.9)
be an equation determines the mesh. Proposition 1.3. Suppose G1 is a one-parameter group in Z with operator h
X=ξ
∂ ∂ ∂ +η + · · · + h+ D (ξ) + . ∂x ∂u ∂h +h
(2.10)
For the difference equation (2.8) to admit the group G1 with operator (2.10) on the mesh (2.9), it is necessary and sufficient that the following condition be satisfied: XF (z)|(2.8), (2.9) = 0,
XΩ (z)|(2.8), (2.9) = 0.
(2.11)
Theorem 1. If the difference models of potential system (2.8), i.e., Fi (z) = 0, i = 1, 2 have the symmetries (2.4), then the difference model D1 F1 (z) + D2 F2 (z) = 0 preserves the whole symmetries of original +h
+h
equation. 3. Invariant model for the generalized Benjamin equation We consider nonlinear generalized Benjamin equation which is given by utt + α(un ux )x + βuxxxx = 0,
(3.1)
where α and β are constants. This kind of equation is one of the most important nonlinear differential equations, used in the analysis of long waves in shallow water [26]. This yields n=2
(3.2)
so that the generalized Benjamin equation given by Eq. (3.1) modifies to utt + α(u2 ux )x + βuxxxx = 0.
(3.3)
We can obtain the following potential system by means of the local conservation law of the generalized Benjamin equation (3.3), α (3.4) wx = ut , vx = w, vt = − u3 − βuxx , 3 in which w(x, t), v(x, t) are potential variables. One can obtain the five-parameter translation group of Eq. (3.4) by using the Lie group method. The group is defined by the following infinitesimal operators
P.-L. Ma et al. / Applied Mathematics Letters 50 (2015) 146–152
149
Fig. 1. The difference stencil for Eq. (3.10).
∂ ∂ ∂ ∂ ∂ , X2 = , X3 = , X4 = x + , ∂t ∂x ∂v ∂v ∂w x ∂ ∂ u ∂ v ∂ ∂ X5 = +t − − −w . (3.5) 2 ∂x ∂t 2 ∂u 2 ∂v ∂w It is obviously that the system (3.5) satisfies conditions (2.6), (2.7), it implies that the grid orthogonality and in both x and t directions is uniform. So one can take an orthogonal mesh which is uniform in space. The stencil of this mesh is shown in Fig. 1. The corresponding subspace of difference variables is thirteen-dimensional, i.e. M ∼ (t, x, τ, h, u, v, w, u+ , u− , u ˆ, v+ , vˆ, w+ ), in which τ = tˆ−t. The symmetry operator (3.5) prolonged to the difference stencil variables has the following form X1 =
∂ ∂ ∂ ∂ ∂ x = ξ t ∂ + ξ x ∂ + (ξˆt − ξ t ) ∂ + (ξ+ X − ξx) +η + η+ + η− +ϕ ∂t ∂x ∂τ ∂h ∂u ∂u+ ∂u− ∂w + ϕ+
∂ ∂ ∂ ∂ ∂ + φ+ + ϕ− +φ + φˆ , ∂w+ ∂w− ∂v ∂v+ ∂ˆ v
(3.6)
where we have used the notations fˆ = f (t + τ, x, u), f± = f (t, x ± h, u), and fˇ = f (t − τ, x, u) represent the time and space shifts. By solving the following linear equations i I(t, x, τ, h, u, w, u+ , u− , w+ , w− , w, X ˆ w ˆ− ) = 0,
i = 1, 2, . . . , n,
(3.7)
one obtains eight difference invariants as follows, I1 = uh,
I2 = wh2 ,
I3 = w+ h2 ,
I5 = u+ h + u− h − 2uh,
h2 , τ I7 = v+ h − vh,
I4 =
I6 = u ˆh − uh,
I8 = vˆh − vh.
By utilizing the invariants (3.8), the discrete counterpart of Eq. (3.4) can be rewritten as follows, α I3 − I2 = I4 I6 , I7 = I2 , I8 I4 = − I13 − βI5 , 3 i.e. α w x = ut , vx = w, vt = −β u2 − u3 . 3 τ τ h h h
(3.8)
(3.9)
(3.10)
Utilizing the compatibility as follows vxt = vtx ,
wxt = wtx ,
uxt = utx ,
h
h
h
h
h
(3.11)
h
one obtains the invariant difference model of Eq. (3.3) α u2 + τ u3 = − [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] − β(u4 + h u5 ). h
h
h
h
(3.12)
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P.-L. Ma et al. / Applied Mathematics Letters 50 (2015) 146–152
We can obtain the following operators in Z by extending X5 to h and τ , h
∂ ∂ X¯1 = , X¯2 = , ∂t ∂x ∂ ∂ ∂ X¯3 = , X¯4 = x + , ∂v ∂v ∂w ∂ u ∂ 3 x ∂ ∂ ∂ +t − − ux − uxx X¯5 = 2 ∂x ∂t 2 ∂u h ∂ ux 2 h ∂ uxx h
− 2 uxxx h
h
5 ∂ h ∂ v ∂ ∂ ∂ ∂ − uxxxx +τ + − −w . ∂ uxxx 2 h ∂ uxxxx ∂τ 2 ∂h 2 ∂v ∂w h
(3.13)
h
We check the conditions (2.11) for Eq. (3.12) and the operators (3.13) α X¯1 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0, h
h
h
h
α X¯2 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0, h
h
h
h
α X¯3 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0, h
h
h
h
α X¯4 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0, h
h
h
h
α X¯5 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0. h
h
h
(3.14)
h
From above equations, we know that Eq. (3.12) is the discrete difference model preserving the whole symmetry of given equation. 4. Invariant model for the third-order Burgers equation It is well known that the Burgers hierarchy is of the form: ut = Km (u) ≡ Φ m−1 ux ,
m = 1, 2, . . . ,
(4.1)
where Φ = D + u + ux D−1 is the recursive operator. In particular, if m = 3, then we have the third-order Burgers’ equation ut = K3 (u) ≡ Φ 2 ux , that is ut = uxxx + 3u2x + 3uuxx + 3u2 ux ,
(4.2)
where u = u(x, t) denotes the unknown function of the space variable x and time t. Burgers’ equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow. It is named for Johannes Martinus Burgers (1895–1981). It relates to the Navier–Stokes equation for incompressible flow with the pressure term removed [27].
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P.-L. Ma et al. / Applied Mathematics Letters 50 (2015) 146–152
Fig. 2. The difference stencil for Eq. (4.7).
We can obtain the following potential system by means of the local conservation law of the third-order Burgers equation wx = u,
wt = uxx + u3 + 3uux ,
(4.3)
in which w(x, t) is a potential variable. In the following, we will construct the difference models of the potential system (4.3). By using the Lie group method, we can obtain the four-parameter point transformation group of system (4.3) ∂ ∂ ∂ x ∂ ∂ u ∂ , X2 = , X3 = , X4 = +t − . (4.4) ∂t ∂x ∂w 3 ∂x ∂t 3 ∂u It is easy to check that the system (4.4) satisfies conditions (2.6), (2.7), it implies that the grid orthogonality and in both x and t directions is uniform. So one can take an orthogonal mesh which is uniform in space. The stencil of this mesh is shown in Fig. 2. The corresponding subspace of difference variables is eleven-dimensional, i.e. M ∼ (t, x, τ, h, u, w, u+ , u− , w+ , w− , w). ˆ One obtains the seven difference invariants by solving the linear equations (3.7) X1 =
I1 = uh,
I2 = u+ h,
I4 = w+ − w,
I3 = u− h,
I5 = w ˆ − w,
I6 =
h3 , τ
I7 = u3 τ. (4.5)
By utilizing the invariants (4.5), the discrete counterpart of Eq. (4.3) can be rewritten as follows I4 = I1 ,
I5 =
I2 + I3 − 2I1 I2 − I1 + I7 + 3I1 , I6 I6
(4.6)
i.e. wx = u, h
wt = u2 +u3 + 3u u1 .
τ
h
(4.7)
h
Utilizing the difference operators D , D onto Eq. (4.7), respectively, one can obtain the invariant difference +τ +h
equation of Eq. (4.2) ut = u3 +3u2 u1 +3huu1 2 + h2 u1 3 + 3u1 2 + 3u u2 , τ
h
h
h
h
h
(4.8)
h
in which u ˆ−u u+ − u , ut = . (4.9) h τ τ h It is easy to check that Eq. (4.8) satisfies the conditions (2.11), it means that the scheme (4.8) is the discrete difference model preserving the whole symmetry of Eq. (4.2). u1 =
5. Conclusions and discussions The generalized Benjamin equation and third-order Burgers equation are of considerable physical interests. For example, this kind of nonlinear Benjamin equation is one of the most important nonlinear
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P.-L. Ma et al. / Applied Mathematics Letters 50 (2015) 146–152
PDEs in the analysis of long waves in shallow water. In this paper, our attention is focused on the problem of obtaining the difference equations completely preserving the symmetries of the original partial differential equation. By taking the generalized Benjamin equation and third-order Burgers equation for examples, one constructs the difference models of potential system instead of the original equation. Then, the difference models of original differential equation are obtained by means of a theorem and it is easy to check the symmetries are also completely preserved. Acknowledgments We would like to thank the Editor and Referee for their timely and valuable comments and suggestions. This work is supported by the Natural Sciences Foundation of China under the grant 11301527. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
P.J. Olver, Applications of Lie Groups to Differential Equations, second ed., Springer-Verlag, New York, 1993. G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1989. N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985. S.Y. Lou, H.Y. Ruan, D.F. Chen, W.Z. Chen, J. Phys. A: Math. Gen. 24 (1991) 1455–1467. S.Y. Lou, X.R. Hu, Y. Chen, J. Phys. A: Math. Theor. 45 (2012) 155209–155212. D. Levi, P. Winternitz, J. Phys. A: Math. Gen. 22 (1989) 2915–2924. Y. Chen, X.R. Hu, Z. Naturforsch A. 62 (2009) 8–14. C.Z. Qu, C.R. Zhang, J. Phys. A: Math. Theor. 42 (2009) 475201–27. S.F. Shen, Commun. Theor. Phys. 44 (2005) 964–966. R.O. Popovych, C. Sophocleous, O.O. Vaneeva, Appl. Math. Lett. 21 (2008) 209–214. Y.F. Liu, D.S. Wang, Appl. Math. Lett. 24 (2011) 481–486. S.F. Tian, T.T. Zhang, P.L. Ma, X.Y. Zhang, J. Nonlinear Math. Phys. 22 (2) (2015) 180–193. A.G. Johnpillai, A.H. Kara, Anjan Biswas, Appl. Math. Lett. 26 (2013) 376–381. A.M. Wazwaz, Appl. Math. Lett. 25 (10) (2012) 1495–1499. D. Levi, P. Winternitz, J. Math. Phys. 34 (1993) 3713–3730. D. Levi, L. Vinet, P. Winternitz, J. Phys. A 30 (1997) 633–649. D. Levi, R. Hernandez Heredero, P. Winternitz, J. Phys. A 32 (1999) 2685–2695. G.R.W. Quispel, H.W. Capel, R. Sahadevan, Phys. Lett. A 170 (1992) 379–383. R. Floreanini, L. Vinet, J. Math. Phys. 36 (1995) 7024–7042. V. Dorodnitsyn, J. Sovi. Math. 55 (1991) 1490–1517. M.I. Bakirova, V.A. Dorodnitsyn, R.V. Kozlov, J. Phys. A 30 (1997) 8139–8155. V. Dorodnitsyn, Appl. Numer. Math. 39 (2001) 307–321. V. Dorodnitsyn, R. Kozlov, P. Winternitz, J. Nonlinear Math. Phys. 2 (2003) 41–56. V. Dorodnitsyn, P. Winternitz, Nonlinear Dyn. 22 (2000) 49–59. X.P. Xin, Y. Chen, Commun. Theor. Phys. 59 (2013) 573–578. W. Hereman, P.P. Banerjee, A. Korpel, G. Assanto, A. VanImmerzeele, A. Meerpole, J. Phys. A: Math. Gen. 19 (5) (1986) 607–628. [27] E. Hopf, Comm. Pure Appl. Math. 3 (3) (1950) 201–230.
Contents lists at ScienceDirect Contents lists available at available ScienceDirect
LinearMathematics Algebra andLetters its Applications Applied www.elsevier.com/locate/laa www.elsevier.com/locate/aml
Inverse eigenvalue problem matrix On symmetry-preserving difference schemeoftoJacobi a generalized with mixed data Benjamin equation and third-order Burgers equation✩ ∗ 1 Ying Wei Pan-Li Ma, Shou-Fu Tian , Tian-Tian Zhang ∗ Department of Mathematics and Center of Nonlinear Equations, China of University of Mining and Department of Mathematics, Nanjing University Aeronautics and Astronautics, Technology, Xuzhou 221116, People’s Republic of China Nanjing 210016, PR China
article
i n faor t i c l e
ianbfs ot r a c t
a b s t r a c t
Article history: Article history: In this of paper, the inverse eigenvalue of reconstructing In this paper, an exposition a method is presented for problem discretizing a generalized 2014 Received 21 May 2015 Received 16 January Benjamin Jacobi matrix from its eigenvalues, its leading principal equation anda third-order Burgers equation while preserving their Lie point Accepted 20 September 2014 Received in revised form 23 June and partlaws, of the of its of submatrix symmetries. By usingsubmatrix local conservation the eigenvalues potential systems original Available online 22 October 2014 2015 is which considered. The necessary and sufficient conditions for equation are obtained, can be used to construct the invariant difference Accepted 23 June 2015 Submitted by Y. Wei the existence and uniqueness of the equation, solution respectively. are derived. models and symmetry-preserving difference models of original Available online 9 July 2015 Furthermore, a numerical anddiscrete some high-order numerical Furthermore, this method is very effective and can algorithm be applied to MSC: examples are given. nonlinear evolution equations. 15A18 Keywords: 2014 Published Elsevier Inc. Ltd. All by rights reserved. © 2015©Elsevier 15A57 Generalized Benjamin equation Third-order Burgers equation Lie point symmetries Keywords: Symmetry-preserving Jacobi matrix Eigenvalue Invariant difference models Inverse problem Submatrix
1. Introduction It is well known that Lie symmetries play an important role in the analysis of nonlinear models. One of the main applications of the Lie group theory is to provide many powerful tools for solving ordinary and partial differential equations, specially nonlinear ones [1–3]. Based on Lie group theory, the standard way of solving ordinary and partial differential equations is to find the Lie point symmetry group G of the equation and then get invariant solutions. In addition, there are some effective methods to get group invariant solutions of differential equations, such as the classical Lie group approach and the non-classical Lie group approach [4–8]. Applications of the Lie group theory to difference equations are much more recent [9–25]. Lie group theory was originally invented as a systematic tool for obtaining exact analytical solutions of ordinary and partial E-mail address: [email protected]. differential equations1 (ODEs and PDEs). For differential equations the existence of a nontrivial symmetry Tel.: +86 13914485239. group makes it possible to reduce the order of the equation. All numerical methods for solving ODEs replace http://dx.doi.org/10.1016/j.laa.2014.09.031 the differential equation by a difference one, usually on a priori chosen lattice, either a regular one, or 0024-3795/© 2014 Published by Elsevier Inc.
Project supported by the Fundamental Research Funds for the Central Universities of China under the Grant No. 2015QNA53. ∗ Corresponding authors. E-mail addresses: [email protected], [email protected] (S.F. Tian), [email protected] (T.T. Zhang).
✩
http://dx.doi.org/10.1016/j.aml.2015.06.017 0893-9659/© 2015 Elsevier Ltd. All rights reserved.
147
P.-L. Ma et al. / Applied Mathematics Letters 50 (2015) 146–152
one adapted to some known or expected behavior of solutions. In this paper, we give a method to obtain the difference models completely preserving the symmetries of the original partial differential equation. The difference of this way to others is that we construct the symmetry-preserving difference models of the potential system instead of the original equation. The paper is organized as follows: In Section 2, in order to make our presentation closed and self-contained, we briefly recall the overview of the discretization Procedure. In Section 3, the difference model of the generalized Benjamin equation is constructed which completely preserving the symmetries of the original partial differential equation. In Section 4, the same way is applied to the third-order Burgers equation and the difference model is constructed that the whole symmetries of original equation are preserved. 2. Preliminaries on discretization procedure In order to make our presentation closed and self-contained, in this section, we briefly recall the required elementary notations [20–25] about the finite difference operators and Lie transformation groups. of formal series, consider the formal transformation group whose infinitesimal operator is In the space Z the total derivative operator as follows ∂ ∂ ∂ ∂ + u1 + u2 + · · · + us+1 + ··· . (2.1) ∂x ∂u ∂u1 ∂us For simplicity, in the following, we only consider the case of one independent variable x and one dependent variable u. Let us fix an arbitrary parameter h > 0 and use the tangent field (2.1) of the Taylor group to form a pair of operators, D=
hD S =e ≡
+h
∞ hs s=0
Ds ,
s!
−hD ≡ S =e
−h
∞ (−h)s s=0
s!
Ds ,
(2.2)
the above operators be called the right and left discrete shift operators respectively, in which D is a derivation By means of S and S , we can form a pair of right and left discrete (finite-difference) differentiation in Z. +h
−h
operators as follows ∞
s−1
h 1 Ds , D = ( S −1) ≡ h +h s! +h s=1
∞
(−h) 1 D = (1 − S ) ≡ h s! −h −h s=1
s−1
Ds .
(2.3)
Suppose ∂ ∂ ∂ + ξx +η + ··· , ∂t ∂x ∂u is the generator of a one-parameter transformation group. For an evolution differential equation, X = ξt
F (x, t, u, ut , ux , uxx , uxxx ) = 0,
(2.4)
(2.5)
the group generated by (2.4) transformations a point (x, t, u, ut , ux , uxx , uxxx ) to a new one (x∗ , t∗ , u∗ , u∗t , u∗x , u∗xx , u∗xxx ) together with Eq. (2.5). When applying Lie point transformations to the difference equations, this situation be changed. Proposition 1.1. For an mesh ω to preserve uniform (h+ = h− ) under the action of the transformation group h
: G1 , the following condition should be satisfied at each point z ∈ Z h
D D (ξ(z)) = 0.
+h −h
The meshes satisfying criterion (2.6) are said to be invariantly uniform.
(2.6)
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P.-L. Ma et al. / Applied Mathematics Letters 50 (2015) 146–152
Proposition 1.2. For an orthogonal mesh ω to converse its orthogonality in the plane (t, x) under any transh
formation, it is necessary and sufficient that the following condition should be satisfied at each point: t x D (ξ ) = − D (ξ ).
(2.7)
F (z) = 0,
(2.8)
+τ
+h
Assume difference equation
is written on finitely many point of the difference mesh ω , which may be uniform or nonuniform. h
Let Ω (z) = 0,
(2.9)
be an equation determines the mesh. Proposition 1.3. Suppose G1 is a one-parameter group in Z with operator h
X=ξ
∂ ∂ ∂ +η + · · · + h+ D (ξ) + . ∂x ∂u ∂h +h
(2.10)
For the difference equation (2.8) to admit the group G1 with operator (2.10) on the mesh (2.9), it is necessary and sufficient that the following condition be satisfied: XF (z)|(2.8), (2.9) = 0,
XΩ (z)|(2.8), (2.9) = 0.
(2.11)
Theorem 1. If the difference models of potential system (2.8), i.e., Fi (z) = 0, i = 1, 2 have the symmetries (2.4), then the difference model D1 F1 (z) + D2 F2 (z) = 0 preserves the whole symmetries of original +h
+h
equation. 3. Invariant model for the generalized Benjamin equation We consider nonlinear generalized Benjamin equation which is given by utt + α(un ux )x + βuxxxx = 0,
(3.1)
where α and β are constants. This kind of equation is one of the most important nonlinear differential equations, used in the analysis of long waves in shallow water [26]. This yields n=2
(3.2)
so that the generalized Benjamin equation given by Eq. (3.1) modifies to utt + α(u2 ux )x + βuxxxx = 0.
(3.3)
We can obtain the following potential system by means of the local conservation law of the generalized Benjamin equation (3.3), α (3.4) wx = ut , vx = w, vt = − u3 − βuxx , 3 in which w(x, t), v(x, t) are potential variables. One can obtain the five-parameter translation group of Eq. (3.4) by using the Lie group method. The group is defined by the following infinitesimal operators
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149
Fig. 1. The difference stencil for Eq. (3.10).
∂ ∂ ∂ ∂ ∂ , X2 = , X3 = , X4 = x + , ∂t ∂x ∂v ∂v ∂w x ∂ ∂ u ∂ v ∂ ∂ X5 = +t − − −w . (3.5) 2 ∂x ∂t 2 ∂u 2 ∂v ∂w It is obviously that the system (3.5) satisfies conditions (2.6), (2.7), it implies that the grid orthogonality and in both x and t directions is uniform. So one can take an orthogonal mesh which is uniform in space. The stencil of this mesh is shown in Fig. 1. The corresponding subspace of difference variables is thirteen-dimensional, i.e. M ∼ (t, x, τ, h, u, v, w, u+ , u− , u ˆ, v+ , vˆ, w+ ), in which τ = tˆ−t. The symmetry operator (3.5) prolonged to the difference stencil variables has the following form X1 =
∂ ∂ ∂ ∂ ∂ x = ξ t ∂ + ξ x ∂ + (ξˆt − ξ t ) ∂ + (ξ+ X − ξx) +η + η+ + η− +ϕ ∂t ∂x ∂τ ∂h ∂u ∂u+ ∂u− ∂w + ϕ+
∂ ∂ ∂ ∂ ∂ + φ+ + ϕ− +φ + φˆ , ∂w+ ∂w− ∂v ∂v+ ∂ˆ v
(3.6)
where we have used the notations fˆ = f (t + τ, x, u), f± = f (t, x ± h, u), and fˇ = f (t − τ, x, u) represent the time and space shifts. By solving the following linear equations i I(t, x, τ, h, u, w, u+ , u− , w+ , w− , w, X ˆ w ˆ− ) = 0,
i = 1, 2, . . . , n,
(3.7)
one obtains eight difference invariants as follows, I1 = uh,
I2 = wh2 ,
I3 = w+ h2 ,
I5 = u+ h + u− h − 2uh,
h2 , τ I7 = v+ h − vh,
I4 =
I6 = u ˆh − uh,
I8 = vˆh − vh.
By utilizing the invariants (3.8), the discrete counterpart of Eq. (3.4) can be rewritten as follows, α I3 − I2 = I4 I6 , I7 = I2 , I8 I4 = − I13 − βI5 , 3 i.e. α w x = ut , vx = w, vt = −β u2 − u3 . 3 τ τ h h h
(3.8)
(3.9)
(3.10)
Utilizing the compatibility as follows vxt = vtx ,
wxt = wtx ,
uxt = utx ,
h
h
h
h
h
(3.11)
h
one obtains the invariant difference model of Eq. (3.3) α u2 + τ u3 = − [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] − β(u4 + h u5 ). h
h
h
h
(3.12)
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We can obtain the following operators in Z by extending X5 to h and τ , h
∂ ∂ X¯1 = , X¯2 = , ∂t ∂x ∂ ∂ ∂ X¯3 = , X¯4 = x + , ∂v ∂v ∂w ∂ u ∂ 3 x ∂ ∂ ∂ +t − − ux − uxx X¯5 = 2 ∂x ∂t 2 ∂u h ∂ ux 2 h ∂ uxx h
− 2 uxxx h
h
5 ∂ h ∂ v ∂ ∂ ∂ ∂ − uxxxx +τ + − −w . ∂ uxxx 2 h ∂ uxxxx ∂τ 2 ∂h 2 ∂v ∂w h
(3.13)
h
We check the conditions (2.11) for Eq. (3.12) and the operators (3.13) α X¯1 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0, h
h
h
h
α X¯2 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0, h
h
h
h
α X¯3 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0, h
h
h
h
α X¯4 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0, h
h
h
h
α X¯5 {u2 + τ u3 + [6hu1 3 + 3u2 (u2 + h u3 ) + 6hu u1 (u2 + h u3 ) + 3h2 u1 2 (u2 + h u3 ) 3 τ τ h h h h h h h h h + h4 (u2 + h u3 )3 ] + β(u4 + h u5 )}|(3.12) = 0. h
h
h
(3.14)
h
From above equations, we know that Eq. (3.12) is the discrete difference model preserving the whole symmetry of given equation. 4. Invariant model for the third-order Burgers equation It is well known that the Burgers hierarchy is of the form: ut = Km (u) ≡ Φ m−1 ux ,
m = 1, 2, . . . ,
(4.1)
where Φ = D + u + ux D−1 is the recursive operator. In particular, if m = 3, then we have the third-order Burgers’ equation ut = K3 (u) ≡ Φ 2 ux , that is ut = uxxx + 3u2x + 3uuxx + 3u2 ux ,
(4.2)
where u = u(x, t) denotes the unknown function of the space variable x and time t. Burgers’ equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow. It is named for Johannes Martinus Burgers (1895–1981). It relates to the Navier–Stokes equation for incompressible flow with the pressure term removed [27].
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Fig. 2. The difference stencil for Eq. (4.7).
We can obtain the following potential system by means of the local conservation law of the third-order Burgers equation wx = u,
wt = uxx + u3 + 3uux ,
(4.3)
in which w(x, t) is a potential variable. In the following, we will construct the difference models of the potential system (4.3). By using the Lie group method, we can obtain the four-parameter point transformation group of system (4.3) ∂ ∂ ∂ x ∂ ∂ u ∂ , X2 = , X3 = , X4 = +t − . (4.4) ∂t ∂x ∂w 3 ∂x ∂t 3 ∂u It is easy to check that the system (4.4) satisfies conditions (2.6), (2.7), it implies that the grid orthogonality and in both x and t directions is uniform. So one can take an orthogonal mesh which is uniform in space. The stencil of this mesh is shown in Fig. 2. The corresponding subspace of difference variables is eleven-dimensional, i.e. M ∼ (t, x, τ, h, u, w, u+ , u− , w+ , w− , w). ˆ One obtains the seven difference invariants by solving the linear equations (3.7) X1 =
I1 = uh,
I2 = u+ h,
I4 = w+ − w,
I3 = u− h,
I5 = w ˆ − w,
I6 =
h3 , τ
I7 = u3 τ. (4.5)
By utilizing the invariants (4.5), the discrete counterpart of Eq. (4.3) can be rewritten as follows I4 = I1 ,
I5 =
I2 + I3 − 2I1 I2 − I1 + I7 + 3I1 , I6 I6
(4.6)
i.e. wx = u, h
wt = u2 +u3 + 3u u1 .
τ
h
(4.7)
h
Utilizing the difference operators D , D onto Eq. (4.7), respectively, one can obtain the invariant difference +τ +h
equation of Eq. (4.2) ut = u3 +3u2 u1 +3huu1 2 + h2 u1 3 + 3u1 2 + 3u u2 , τ
h
h
h
h
h
(4.8)
h
in which u ˆ−u u+ − u , ut = . (4.9) h τ τ h It is easy to check that Eq. (4.8) satisfies the conditions (2.11), it means that the scheme (4.8) is the discrete difference model preserving the whole symmetry of Eq. (4.2). u1 =
5. Conclusions and discussions The generalized Benjamin equation and third-order Burgers equation are of considerable physical interests. For example, this kind of nonlinear Benjamin equation is one of the most important nonlinear
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PDEs in the analysis of long waves in shallow water. In this paper, our attention is focused on the problem of obtaining the difference equations completely preserving the symmetries of the original partial differential equation. By taking the generalized Benjamin equation and third-order Burgers equation for examples, one constructs the difference models of potential system instead of the original equation. Then, the difference models of original differential equation are obtained by means of a theorem and it is easy to check the symmetries are also completely preserved. Acknowledgments We would like to thank the Editor and Referee for their timely and valuable comments and suggestions. This work is supported by the Natural Sciences Foundation of China under the grant 11301527. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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