Two fourth-order difference approximations for fractional derivatives based on Lubich-type second-order approximation with different shifts are derived. These approximations are applied to the space fractional diffusion equation with the Crank–Nicolson scheme. Here, we analyse the stability and convergence of these schemes and prove that they are unconditionally stable and convergent for a fractional order $\alpha $ ranging from $1$ to $2$. Numerical examples are presented to show that both schemes converge, and we obtain the correct convergence rates and unconditional stability.

This paper focuses on the Aw–Rascle model of traffic flow for the Born–Infeld equation of state with Coulomb-like friction, whose Riemann problem is solved with the variable substitution method. Four kinds of nonself-similar solutions are derived. The delta shock occurs in the solutions, although the system is strictly hyperbolic with a genuinely nonlinear characteristic field and a linearly degenerate characteristic field. The generalized Rankine–Hugoniot relation and entropy condition for the delta shock are clarified. The delta shock can be used to describe the serious traffic jam. Under the impact of the friction term, the rarefaction wave (R), shock wave (S), contact discontinuity (J) and delta shock ($\delta $) are bent into parabolic curves. Furthermore, it is proved that the $S+J$ solution and $\delta $ solution of the nonhomogeneous Aw–Rascle model tend to be the $\delta $ solution of the zero-pressure Euler system with friction; the $R+J$ solution and $R+\mbox {Vac}+J$ solution tend to be the vacuum solution of the zero-pressure Euler system with friction.

The modulational instability of weakly nonlinear capillary-gravity waves (CGWs) on the surface of infinitely deep water with uniform vorticity background shear is examined. Assuming a narrow band of waves, the fourth-order nonlinear Schrödinger equation (NSE) is derived from Zakharov’s integral equation (ZIE). The analysis is restricted to one horizontal dimension, parallel to the direction along the wave propagation to take advantage of a formulation using potential flow theory. It is to be noted that the dominant new effect introduced to the fourth order is the wave-induced mean flow response. The key point of this paper is that the present fourth-order analysis shows considerable deviation in the stability properties of CGWs from the third-order analysis and gives better results consistent with the exact results. It is found that the growth rate of instability increases for negative vorticity and decreases for positive vorticity, and the effect of capillarity is to reduce the growth rate of instability. Additionally, the effect of vorticity on the Peregrine breather, which can be considered as a prototype for freak waves, is investigated.

We study the long time dynamic properties of the nonlocal Kuramoto–Sivashinsky (KS) equation with multiplicative white noise. First, we consider the dynamic properties of the stochastic nonlocal KS equation via a transformation into the associated conjugated random differential equation. Next, we prove the existence and uniqueness of solution for the conjugated random differential equation in the theory of random dynamical systems. We also establish the existence and uniqueness of a random attractor for the stochastic nonlocal equation.

This research introduces an adapted multidimensional fractional optimal control problem, developed from a newly established framework that combines first-order partial differential equations (PDEs) with inequality constraints. We methodically establish and demonstrate the optimality conditions relevant to this framework. Moreover, we illustrate that, under certain generalized convexity assumptions, there exists a correspondence between the optimal solution of the multidimensional fractional optimal control problem and a saddle point related to the Lagrange functional of the revised formulation. To emphasize the significance and practical implications of our findings, we present several illustrative examples.