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Abstract This study is devoted to the numerical simulation of vector solitons described by the time-fractional coupled sine–Gordon equations in the sense of Caputo fractional derivative, where the problem is defined on the multidimensional unbounded domains R d (d = 2 , 3). For this purpose, we employ the Hermite–Galerkin spectral method with scaling factor for the spatial approximation to avoid the errors introduced by the domain truncation, and we apply the finite difference method based on the Crank–Nicolson method for the temporal discretization. Comprehensive numerical studies are carried out to verify the accuracy and the stability of our method, which shows that the method is convergent with (3 − max { α 1 , α 2 }) -order accuracy in time and spectral accuracy in space. Here, α i (1 < α i < 2 , i = 1 , 2) are the orders of the Caputo fractional derivative. In addition, the effect of the Caputo fractional derivative on the evolutions of the vector solitons is numerically studied. Finally, several numerical simulations for both two- and three-dimensional cases of the problem are performed to illustrate the robustness of the method as well as to investigate the collisions of circular and elliptical ring vector solitons. [ABSTRACT FROM AUTHOR]

In this paper, we construct a novel linearized finite difference/spectral-Galerkin scheme for three-dimensional distributed-order time–space fractional nonlinear reaction–diffusion-wave equation. By using Gauss–Legendre quadrature rule to discretize the distributed integral terms in both the spatial and temporal directions, we first approximate the original distributed-order fractional problem by the multi-term time–space fractional differential equation. Then, we employ the finite difference method for the discretization of the multi-term Caputo fractional derivatives and apply the Legendre–Galerkin spectral method for the spatial approximation. The main advantage of the proposed scheme is that the implementation of the iterative method is avoided for the nonlinear term in the fractional problem. Additionally, numerical experiments are conducted to validate the accuracy and stability of the scheme. Our approach is show-cased by solving several three-dimensional Gordon-type models of practical interest, including the fractional versions of sine-, sinh-, and Klein–Gordon equations, together with the numerical simulations of the collisions of the Gordon-type solitons. The simulation results can provide a deeper understanding of the complicated nonlinear behaviors of the 3D Gordon-type solitons. [ABSTRACT FROM AUTHOR]

To assess whether the model misspecification matters for hedging accuracy, we carefully select six increasingly complicated asset models, i.e., the Black–Scholes (BS) model, the Merton (M) model, the Heston (H) model, the Heston jump-diffusion (HJ) model, the double Heston (dbH) model and the double Heston jump-diffusion (dbHJ) model, and then impartially evaluate their performances in mitigating the risk of an option, under a controllable experimental market. In experiments, the ℙ measure asset paths are piecewisely simulated by a hybrid-model (including the Black–Scholes-type and the (double) Heston-type, with or without jump-diffusion term) with randomly given properly defined parameters. We access the hedging accuracy of six models within the operational dynamic hedging framework proposed by sun (2015), and apply the Fourier-COS-expansion method (i.e., the COS formula, Fang and Oosterlee (2008) to price options and to calculate the Greeks). Extensive numerical results indicate that the model misspecification shows no significant impact on hedging accuracy, but the market fit does matter critically for hedging.