arXiv subjects

Leilei Wei

Publications and source records attributed to Leilei Wei.

5 recordsLinked to original sources

arXiv2025

$C$-existence families, $C$-semigroups and their associated abstract Cauchy problems in complete random normed modules

In this paper, we first introduce the notion of a (mild) $C$-existence family in complete random normed modules, then we prove that a (mild) $C$-existence family can guarantee the existence of the (mild) solutions of the associated abstract Cauchy problem in the random setting. Second, we investigate several important properties peculiar to locally almost surely bounded $C$-semigroups in complete random normed modules, which are not involved in the classical theory of $C$-semigroups. Finally, based on the above work, some relations among $C$-existence families, $C$-semigroups and their associated abstract Cauchy problems in complete random normed modules are established, which extend and improve some known results. Besides, an application to a type of stochastic differential equations is also given.

math.FA↗

arXiv2025

A Laplace transform approach to $C$-semigroups on a $\mathcal{T}_{\varepsilon, \lambda}$-complete random normed module

In this paper, we first introduce the notion of the Laplace transform for an abstract-valued function from $[0, \infty)$ to a $\mathcal{T}_{\varepsilon, \lambda}$-complete random normed module $S$. Then, combining respective advantages of the $(\varepsilon, \lambda)$-topology and the locally $L^0$-convex topology on $S$, we prove the differentiability, Post-Widder inversion formula and uniqueness of such a Laplace transform. Second, based on the above work, we establish the Hille-Yosida theorem for an exponentially bounded $C$-semigroup on $S$, considering both the dense and nondense cases of the range of $C$, respectively, which extends and improves several important results. Finally, we also apply such a Laplace transform to abstract Cauchy problems in the random setting.

math.FA↗

arXiv2020

A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation

The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an efficient finite difference scheme to approximate the tempered fractional derivatives and then a fully discrete LDG method for the tempered fractional diffusion equation. We prove that the scheme is unconditionally stable and convergent with the order $O(h^{k+1}+τ^{2-α})$, where $h, τ$ and $k$ are the step size in space, time and the degree of piecewise polynomials, respectively. Finally numerical experimets are performed to show the effectiveness and testify the accuracy of the method.

math.NA↗

arXiv2015

Finite difference/local discontinuous Galerkin method for solving the fractional diffusion-wave equation

In this paper a finite difference/local discontinuous Galerkin method for the fractional diffusion-wave equation is presented and analyzed. We first propose a new finite difference method to approximate the time fractional derivatives, and give a semidiscrete scheme in time with the truncation error $O((\Delta t)^2)$, where $\Delta t$ is the time step size. Further we develop a fully discrete scheme for the fractional diffusion-wave equation, and prove that the method is unconditionally stable and convergent with order $O(h^{k+1}+(\Delta t)^{2})$, where $k$ is the degree of piecewise polynomial. Extensive numerical examples are carried out to confirm the theoretical convergence rates.

math.NA↗

arXiv2012

Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation

In this paper, a fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(\Delta t)^2+(\Delta t)^\frac{\alpha}{2}h^{k+1/2})$. Numerical examples are presented to show the efficiency and accuracy of our scheme.

math.NA↗