High accuracy two-level implicit compact difference scheme for 1D unsteady biharmonic problem of first kind: application to the generalized Kuramoto–Sivashinsky equation

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

“…Later, various types of B-spline methods like quintic B-spline collocation scheme proposed by Mittal and Arora (2010), quintic B-spline with differential quadrature scheme implemented by Mittal and Dahiya (2017), and septic B-spline collocation approach adopted by Zarebnia and Parvaz (2013) are successfully used to solve the approximate solution of KSE. Furthermore, the numerical approach based on Bspline basis functions is introduced by Lakestani and Dehghan (2012), two-level implicit compact difference scheme applied by Mohanty and Kaur (2019), modified Kudryashov scheme proposed by Kilicman and Silambarasan (2018) are used to solve the generalized Kuramoto-Sivashinsky equation.…”

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

“…Nonlinear wave phenomena play an important role in engineering and sciences. In the past, many scientists have studied about different mathematical models to explain the wave behavior, such as the KdV equation (Korteweg and Vries 1895;Ozer and Kutluay 2005;Skogestad and Kalisch 2009;Kim et al 2012;Yan et al 2016), the Rosenau equation (Rosenau 1986(Rosenau , 1988Park 1992), the Rosenau-KdV equation (Zuo 2009;Esfahani 2011;Triki and Biswas 2013;Zheng and Zhou 2014), the Rosenau-RLW equation (Pan and Zhang 2012;Wongsaijai et al , 2019, and many others (Lu and Chen 2015;Coclite and Ruvob 2017;Mohanty and Kaur 2019;Kaur and Mohanty 2019).…”

Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation

“…Introduction. The deterministic Kuramoto-Sivashinsky (KS) equation just describes pattern formation phenomena with the phase turbulence, which was first introduced by Kuramoto[16], with developments even in the recent papers [15,20] (deterministic) or [7,8,10,27] (stochastic).…”

“…where ρ and R f are defined by (19) and (20) respectively. In fact, by (16), K is backward absorbing in the following sense…”

“…D = {D(τ, ω)} ∈ D if and only if lim t→+∞ e −εt D(τ − t, θ −t ω) 2 = 0, ∀ε > 0, τ ∈ R, ω ∈ Ω,(39)where we have omitted the supremum in the definition (11) of B. Then, by the same method as in Lemma 3.1, one can prove that Φ has a D-pullback absorbing set given byK D (τ, ω) = {w ∈ H o : w 2 ≤ c(1 + ρ(ω) + R D (τ, ω))} βr+ Dg ∞ 0 r |z(θrω)|dr f (r + τ ) 2 dr such that sup s≤τ R D (s, ω) = R f (τ, ω) in view of the definition of R f in(20). As an integral of random variables, R D (τ, •) is measurable (although we do not know the measurability of R f ).…”