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Finite difference approximate solutions for the Rosenau equation. (English) Zbl 0904.65093
Numerical solutions of a finite difference scheme are discussed for the KdV-like Rosenau equation \[ u_t+ u_{xxxxt} +uu_x +u_x=0, \quad (x,t)\in (0,1)\times (0,T] \] with an initial condition \[ u(x,0)= u_0(x), \quad x\in (0,1) \] and boundary conditions \[ u(0,t)= u(1,t)=0, \quad u_{xx} (0,t)= u_{xx} (1,t) =0. \] This equation was modelled by P. Rosenau [Dynamics of dense discrete systems, Prog. Theoretical Phys. 79, 1028-1042 (1988)] in order to describe the dynamics of dense discrete systems.
Existence and uniqueness of the solution for the scheme are shown by using the Brouwer fixed point theorem. An a priori bound and convergence of order \(O(h^2 +k^2)\) as well as conservation of energy of the finite difference approximate solutions are discussed with numerical examples.
Reviewer: S.K.Chung (Seoul)

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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References:
[1] Browder F. E., Esistence and uniqueness theorems for solutions of nonlinear boundary value problems, Applications of nonlinear partial differential equations (1965)
[2] DOI: 10.1080/00036819408840267 · Zbl 0830.65097
[3] DOI: 10.1016/0362-546X(93)90179-V · Zbl 0811.35142
[4] DOI: 10.1143/PTP.79.1028
[5] DOI: 10.1088/0031-8949/34/6B/020
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