反常扩散的广义积分方程构造理论及其数值应用

项目名称： 反常扩散的广义积分方程构造理论及其数值应用

项目编号： No.11301257

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 吴国成

作者单位： 内江师范学院

项目金额： 22万元

中文摘要： 多孔介质中的扩散常表现出历史依赖和全域相关性，分数阶导数能够准确描述这类扩散现象。有限差分法与有限单元法为常用的数值方法，随长时间计算，计算数据信息存储量急剧增加，这成为分数阶微分方程数值计算的难题之一。变分法是探索微分方程等价积分表示的有效工具。项目利用分数阶变分原理和变分迭代格式，将时间分数阶扩散方程化成等价的广义积分方程，消除由分数阶导数引起的计算复杂度，提高分数阶扩散方程的数值计算效率。基于积分方程理论和方法，如Taylor级数法、配置方法和预估校正等，进行分数阶扩散方程的解析和数值计算,给出收敛性证明和误差分析，力求在分数阶微分和广义积分方程等价构造理论和数值算法的精度、稳定性上取得一定的进展。

中文关键词： 分数阶导数；积分方程方法；时标；分数阶差分；快速分解方法

英文摘要： The fractional differnetial equation is an effficient tool to describe the diffusion problems in porous medium. The most often used methods are the finitie difference method and the finite volmue method. In the numerical simulation,the storage information becomes very large which greatly effects numerical methods' efficiencies. As is well known, variation approach plays a crucial role in the eqvailent integral representation of differential equations. Based on fractional calculus of variations and variational iteration formulae,this program suggests a strategy to construct the eqvailent integral equations of the nonlinear fractional diffusion equation which overcomes the drawbacks arising in the difference methods. Using theories and numerical methods of integral equations,i.e., Taylor series, spline collation method and predictor corrector approach,the numerically simulation of the fractional diffusion equation is then considered. The proofs of the convergence, the stability and the error analysis are also given.

英文关键词： fractional derivative；integral equation method；time scales；fractional differences；fast Adomian decomposition method