A posteriori error estimation for a singularly perturbed Volterra integro-differential equation

Numerical Algorithms - A singularly perturbed Volterra integro-differential equation with an integrable singularity in the integral term is considered. The upwind difference method is used to...     

Tumor Microenvironment Stimuli-Responsive Fluorescence Imaging and Synergistic Cancer Therapy by Carbon-Dot–Cu2+ Nanoassemblies

A method is developed to fabricate tumor microenvironment (TME) stimuli-responsive nanoplatform for fluorescence (FL) imaging and synergistic cancer therapy via assembling photosensitizer (chlorine e6, Ce6) modified carbon dots (CDs-Ce6) and Cu²⁺. The as-obtained nanoassemblies (named Cu/CC nanoparticles, NPs) exhibit quenched FL and photosensitization due to the aggregation of CDs-Ce6. Their FL imaging and photodynamic therapy (PDT) functions are recovered efficiently once they entering tumor sites by the stimulation of TME. Introducing of Cu²⁺ not only provides extra chemodynamic therapy (CDT) function through reaction with hydrogen peroxide (H₂O₂), but also depletes GSH in tumors by a redox reaction, thus amplifying the intracellular oxidative stress and enhancing the efficacy of reactive oxygen species (ROS) based therapy. Cu/CC NPs can act as a FL imaging guided trimodal synergistic cancer treatment agent by photothermal therapy (PTT), PDT, and thermally amplified CDT.     

A second-order difference scheme for a parameterized singular perturbation problem

In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results.     

An almost second order uniformly convergent scheme for a singularly perturbed initial value problem

In this paper a nonlinear singularly perturbed initial problem is considered. The behavior of the exact solution and its derivatives is analyzed, and this leads to the construction of a Shishkin-type mesh. On this mesh a hybrid difference scheme is proposed, which is a combination of the second order difference schemes on the fine mesh and the midpoint upwind scheme on the coarse mesh. It is proved that the scheme is almost second-order convergent, in the discrete maximum norm, independently of singular perturbation parameter. Numerical experiment supports these theoretical results.     

A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems

A system of coupled singularly perturbed initial value problems with two small parameters is considered. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solution of the system has boundary layers that overlap and interact. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh a hybrid finite difference scheme is proved to be almost second-order accurate, uniformly in both small parameters. Numerical results supporting the theory are presented.     

四阶奇异摄动边值问题在自适应网格上的一致收敛分析

we study a difference scheme for the fourth-order singular pertur-bation differential equation on the Bakhvalov-Shishkin grid by Green‘‘s function.The method is shown to be uniformly convergent with respect to the perturbation parameter,of order N^-2 in the maxmum norm on Bakhvalov-Shishkin meshes.Numerical results support our theoretical results.     

A robust adaptive grid method for a nonlinear singularly perturbed differential equation with integral boundary condition

Numerical Algorithms - In this paper, the numerical solution of a nonlinear first-order singularly perturbed differential equation with integral boundary condition is considered. The discrete...     

On the hybrid finite difference scheme for a singularly perturbed Riccati equation

In this paper a singularly perturbed Riccati equation is considered. The problem is solved by using a hybrid finite difference method on a Shishkin mesh. The method is composed of the midpoint upwind scheme and the backward Euler scheme based on the relation between the local mesh width and the perturbation parameter. It is proved that the scheme is almost second-order convergent, in the global maximum norm, independently of the singular perturbation parameter. Numerical experiments support the theoretical results.     

由电热水袋引发的学生课外探究活动

教师在引导学生探究电热水袋故障这个真实情境的科学问题时,发现温控开关是引起电热水袋故障的主要因素,并深入探究了电热水袋的工作原理。但更重要的是学生在一步步揭开疑团的过程中,不自觉地使用现有知识,超越现有知识,对科学探究有了更深刻的体验和理解。     

A robust and accurate finite difference method for a generalized Black-Scholes equation

In this paper we present a numerical method for a generalized Black-Scholes equation, which is used for option pricing. The method is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique. Our scheme is stable for arbitrary volatility and arbitrary interest rate, and is second-order convergent with respect to the spatial variable. Furthermore, the present paper efficiently treats the singularities of the non-smooth payoff function. Numerical results support the theoretical results.