分数阶微分方程的数值解法

1、 分数阶微分方程的数值解 闫玉斌 数学系 切斯特大学, 英国 09/2014Outlines Part 1: 分数阶微分方程 1. Modelling (Viscoelasticity) 数学模型（粘弹性）2. Mathematics formulation （数学理论）3. Numerical method （数值方法） Part 2: 分数阶偏微分方程1.Modelling (anomalous diffusion)数学模型（不规则扩散）2.Mathematica formulation （数学理论） 3.Numerical methods （数值方法）Part 1: Viscoelast

2、icity （粘弹性）Elastic （弹性）（弹性） Viscous （粘性）（粘性）What is stress （应力的定义）The stress across a surface element (yellow disk) is the force that the material on one side (top ball) exerts on the material on the other side (bottom ball), divided by the area of the surfaceIdealized stress in a straight bar with

3、uniform cross-section. What is the strain? （应变的定义） A strain（应变）应变） measure of deformation representing the displacement between particles in the body relative to a reference lengthThe deformation （变形）（变形）of a thin straight rod into a closed loopMotion of a continuum bodyElasticity （弹簧） Hookes law （胡

4、克定律）（胡克定律） F= k X F: force （张力） X: distance （距离） k: spring constant （参数）Springs （弹簧）（弹簧）Viscosity （粘性） Viscous stress （粘性力）is proportional to the strain rate （形变率）Strain rate （形变率）= the time derivative of the strain（形变的时间导数）=gradient of the velocity of the material （速度的导数）e.g. Strain for a long rubb

5、erSprings （弹簧）and dashpot Integer-order ModelsFractional-order models (分数阶模型）Viscoelastic materials （粘弹性材料） amorphous polymers semicrystalline polymers Biopolymers （生物聚合物） Red blood cells （红血细胞）Fractional derivatives （分数阶导数）Riemann-Liouville integral (黎曼-刘维尔积分） Fractional derivative （分数阶导数）Fractiona

6、l derivativesFractional differential equations分数阶微分方程 Riemann- Liouville （黎曼刘维（黎曼刘维尔）尔） Caputo Equivalent form （等价形式）Mathematical problem Existence （存在性） (Fixed point theorem) Uniqueness （唯一性） Regularity （正则性） Numerical issues （数值问题） Numerical scheme （格式） Algorithm （算法） Programming （程序）Finite differ

7、ence method (有限差分法)( Fractional derivative is approximated by finite difference schemes) For numerical methods, we need to consider1. Stability (稳定性）2. Error estimates （误差估计）3. Computational cost （计算费用）Finite difference method I(integral discretization )Finite difference method-II (derivative discre

8、tization) Part 2: Fractional partial differential equations Normal Diffusion: Some particles are dissolved in a glass of water. Brownian motion: a big particle collides with a large set of smaller particles which move with different velocities in different random directionsNormal Diffusion equation

9、Anomalous diffusionAnomalous diffusion A Lvy flight is a random walk in which the step-lengths have a probability distribution that is heavy-tailed. Fractional partial differential equations Mathematics problem Existence ( Fractional sobolev space) uniqueness Regularity ( Not available yet) Numerical methods Finite difference methods, Neville and Yan, (2011), ( 2012) Finite element methods, Neville and Yan, (2012)Current works Numerical methods for time fractional PDE