A new conservative finite difference scheme for the generalized Rosenau–KdV–RLW equation

Dai, Weizhong

doi:10.1007/s40314-020-01280-x

Cited by 3 publications

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“…Two typical equations are

u_{t} + u_{x x x x t} + u_{x} + u u_{x} = f {false(u false)}_{x},

and

u_{t t} - c u_{x x} + h u_{x x x x} + ϱ u_{x x x x t t} = f {false(u false)}_{x x},

where c , h , ϱ are positive constants. So far, much work has been done on the existence and uniqueness of the solutions to the Rosenau equation (), as well as numerical schemes by Galerkin method; see Atouani, Barreto et al, Park, and Wang and Dai 16–19 for examples. As for (), there are few results concerning the existence and nonexistence of solutions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global existence, asymptotic behavior, and regularity of solutions for time–space fractional Rosenau equations

Zhang

2021

Math Methods in App Sciences

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In this paper, we investigate initial boundary value problems for the following time-space fractional Rosenau equationwith the Caputo time fractional derivatives and the spectral fractional Laplacian operators. To the best of our knowledge, there are few results concerning the time-space fractional Rosenau equations. The main difficulties are the nonlocal effects generated by the operators 𝜕 𝛽 t and (− Δ) 𝛾 . First, we establish some rough and rigorous decay estimates of weak solutions to the corresponding linear equations, respectively. Based on the decay estimates, under small initial value condition, we prove the global existence and asymptotic behavior of weak solutions in the time-weighted Sobolev spaces by the contraction mapping principle. Furthermore, we discuss the regularity of weak solutions when initial value data are strengthened from H s (Ω) to H s+1 (Ω).

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“…Two typical equations are