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The membrane-fusion-based internalization without lysosomal entrapment is advantageous for intracellular delivery over endocytosis. However, protein corona formed on the membrane-fusogenic liposome surface converts its membrane-fusion performance to lysosome-dependent endocytosis, causing poorer delivery efficiency in biological conditions. Herein, we develop an antifouling membrane-fusogenic liposome for effective intracellular delivery in vivo. Leveraging specific lipid composition at an optimized ratio, such antifouling membrane-fusogenic liposome facilitates fusion capacity even in protein-rich conditions, attributed to the copious zwitterionic phosphorylcholine groups for protein-adsorption resistance. Consequently, the antifouling membrane-fusogenic liposome demonstrates robust membranefusion-mediated delivery in the medium with up to 38% fetal bovine serum, outclassing two traditional membrane-fusogenic liposomes effective at 4% and 6% concentrations. When injected into mice, antifouling membrane-fusogenic liposomes can keep their membrane-fusion-transportation behaviors, thereby achieving efficient luciferase transfection and enhancing gene-editingmediated viral inhibition. This study provides a promising tool for effective intracellular delivery under complex physiological environments, enlightening future nanomedicine design. [ABSTRACT FROM AUTHOR]

An artificial boundary method is developed for solving the one-dimensional advection diffusion equation in the real line. In order to construct a fully discrete fast numerical algorithm with rigorous error analysis, we start with the two-step backward difference formula for time discretization of the advection diffusion equation in the whole real line. Then, we use the discrete analogue of the Laplace transform to derive a second-order time-stepping scheme in a bounded domain equipped with a discrete artificial boundary condition (ABC). The Galerkin finite element method is used for spatial discretization. To expedite the evaluation of time convolution involved in the discrete ABC, we propose a fast algorithm based on the best rational approximation of square root function in subdomains of the complex plane. An estimate for this best rational approximation enables us to prove optimal-order convergence of the fully discrete numerical scheme (integrating the fast approximation algorithm). Several numerical examples are provided to illustrate the convergence of numerical solutions and the effectiveness of the proposed fast approximation algorithm. [ABSTRACT FROM AUTHOR]

The stability and error analysis of a second-order fast approximation are considered for the one-dimensional local and nonlocal diffusion equations in the unbounded spatial domain. We first use the conventional central difference scheme to discretize the local second-order spatial derivative operator and use an asymptotically compatible difference scheme to discretize the spatial nonlocal diffusion operator, and apply second-order backward differentiation formula (BDF2) to approximate the temporal derivative to achieve a fully discrete infinity system. To solve the resulting fully discrete systems, we develop a unified framework that is applicable to the discretization of both local and nonlocal problems. A key ingredient is to derive Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs). To do so, we apply the z-transform and solve an exterior problem using an iteration technique to derive a Dirichlet-to-Dirichlet (DtD)-type mapping as exact ABCs. After that, we use the Green formula to reformulate the DtD-type mapping equivalently as the DtN-type mapping. The resulting DtN-type mapping allows us to reduce the infinity discrete system into a finite discrete system in a truncated computational domain of interest, and also make it possible to present the stability and convergence analysis of the reduced problem under some open but reasonable assumptions. To efficiently implement the exact ABCs, we further develop a fast convolution algorithm based on approximation of the contour integral induced by the inverse z-transform. The stability and error analysis of the reduced finite discrete system based on the fast algorithm for exact ABCs are also established, and numerical examples are provided to demonstrate the effectiveness of our proposed approach. [ABSTRACT FROM AUTHOR]

A second-order Crank-Nicolson finite difference method, integrating a fast approximation of an exact discrete absorbing boundary condition, is proposed for solving the one-dimensional Schrodinger equation in the whole space. The fast approximation is based on Gaussian quadrature approximation of the convolution coefficients in the discrete absorbing boundary conditions. It approximates the time convolution in the discrete absorbing boundary conditions by a system of O(log² N) ordinary differential equations at each time step, where N denotes the total number of time steps. Stability and an error estimate are presented for the numerical solutions given by the proposed fast algorithm. Numerical experiments are provided, which agree with the theoretical results and show the performance of the proposed numerical method. [ABSTRACT FROM AUTHOR]

We are concerned with the numerical solution of a nonlocal wave equation in an infinite two-dimensional space. The contribution of this paper is the derivation of an absorbing boundary condition which allows the wave field defined on the finite computational domain to retain the same feature as that defined on the original infinite domain. We resort to the idea of a first-kind integral equation method and develop a solution formulation in terms of a potential summation on a surrounding ghost region. This new formulation can be taken as an absorbing boundary condition of generalized Dirichlet-to-Dirichlet type. The accuracy and effectiveness of our approach are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

Numerical computation of a nonlocal diffusion equation on the real axis is considered in this paper. We first apply an extensively studied quadrature scheme to obtain a discrete nonlocal diffusion system on an unbounded domain. Then we derive an alternative formulation of the discrete problem based on the spectral analysis of the z-transform. This new formulation can be seen as a system defined on a bounded domain with an artificial boundary condition, and it allows us to reformulate the original infinite domain problem into an equivalent bounded domain problem. To numerically implement the exact artificial boundary condition, we apply the trapezoidal quadrature rule to approximate the contour integral induced by the inverse z-transform. Numerical examples are presented to demonstrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

Gaussian beams are generally local asymptotic solutions to the linear wave equations in the high-frequency regime. Each Gaussian beam is concentrated around a specific ray path determined by the underlying Hamiltonian system. Expressed as some superposition of Gaussian beams, Gaussian beam approximation is expected to be a high-frequency asymptotic solution which remains globally valid even around caustics. We derive optimal first-order error estimates for firstorder Gaussian beam approximations, in both continuous and discrete levels, to the Schrodinger equation equipped with a WKB initial data. Our error estimates are valid for any spatial dimension and unaffected by the presence of caustics. Some numerical tests are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]