In this study, we describe the fractional convection operator for first time and present its discrete form with second-order convergence. A numerical scheme fractional-convection–diffusion equation is also constructed in order to get insight into behavior visually. Then, study fractional-convection-dominated diffusion which has never been considered, where normal characterized by Laplacian. The interesting phenomena are observed through simulation. Moreover, investigate...

Compared to the classical first-order Gr\"unwald-Letnikov formula at time $t_{k+1} (\textmd{or}\, t_{k})$, we firstly propose a second-order numerical approximate scheme for discretizing Riemann-Liouvile derivative $t_{k+\frac{1}{2}}$, which is very suitable constructing Crank-Niclson technique applied time-fractional differential equations. The established has following form $$ \begin{array}{lll} \displaystyle...

The time-fractional Cattaneo equation is an where the fractional order α∈(1,2) has capacity to model anomalous dynamics of physical diffusion processes. In this paper, we consider efficient scheme for solving such in two space dimensions. First, obtain space’s semi-discrete numerical by using compact difference operator spatial direction. Then, converted a low-order system means reduction, and fully discrete presented applying L2-1σ formula. To improve computational efficiency, adopt fast...

In this paper, we are interested in the effective numerical schemes of time-fractional Black–Scholes equation. We convert original equation into an equivalent integral-differential and then discretize time-integral term form using piecewise linear interpolation, while compact difference formula is applied spatial direction. Thus, derive a fully discrete scheme with second-order accuracy time fourth-order space. Rigorous proofs corresponding stability convergence given. Furthermore, order to...

<abstract><p>In this paper, two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes equation. We first eliminate convection term in equivalent form of considered equation by using exponential transformation, then combine sixth-order/eighth-order method a temporal meshes-based trapezoidal formulation integral to obtain fully discrete schemes. The stability and convergence analysis studied applying Fourier analysis....

<abstract> In this paper, we consider the efficient numerical scheme for solving time-fractional mobile/immobile transport equation. By utilizing compact difference operator to approximate Laplacian, develop an Crank-Nicolson based on modified L1 method. It is proved that proposed stable with accuracy of $ O(\tau^{2-\alpha}+h^4) $, where \tau and h are respectively temporal spatial stepsizes, fractional order \alpha\in(0, 1) $. addition, improve computational performance non-smooth issue by...

Fine-tuning large language models (LLMs) with classic first-order optimizers entails prohibitive GPU memory due to the backpropagation process. Recent works have turned zeroth-order for fine-tuning, which save substantial by using two forward passes. However, these are plagued heterogeneity of parameter curvatures across different dimensions. In this work, we propose HiZOO, a diagonal Hessian informed optimizer is first work leverage enhance fine-tuning LLMs. What's more, HiZOO avoids...

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that original first transformed into a lower system, and then graded mesh-based fast L2-1σ formula second-order operator Caputo derivative differential are applied, respectively, to derive fully discrete scheme. By adding suitable perturbation terms, we construct ADI scheme, which significantly improves...

The objective of this paper is to discuss the action derivative spectra corn in determining spectral bands that are best suited for characterizing agronomic parameters. data study comes from filed and indoor reflectance measurements by a ASD FieldSpec Pro FR. Observed characteristics included leaf area index, aboveground fresh biomass chlorophyll concentration carotenoid concentration. shows can determine some characteristic wavelength eliminate influence soil background degree. Leaf...