分數(shù)階微分方程的數(shù)值解法

1、 分數(shù)階微分方程的數(shù)值解 閆玉斌 數(shù)學(xué)系 切斯特大學(xué), 英國 09/2014Outlines Part 1: 分數(shù)階微分方程 1. Modelling (Viscoelasticity) 數(shù)學(xué)模型（粘彈性）2. Mathematics formulation （數(shù)學(xué)理論）3. Numerical method （數(shù)值方法） Part 2: 分數(shù)階偏微分方程1.Modelling (anomalous diffusion)數(shù)學(xué)模型（不規(guī)則擴散）2.Mathematica formulation （數(shù)學(xué)理論） 3.Numerical methods （數(shù)值方法）Part 1: Viscoelast

2、icity （粘彈性）Elastic （彈性）（彈性） Viscous （粘性）（粘性）What is stress （應(yīng)力的定義）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? （應(yīng)變的定義） A strain（應(yīng)變）應(yīng)變） measure of deformation representing the displacement between particles in the body relative to a reference lengthThe deformation （變形）（變形）of a thin straight rod into a closed loopMotion of a continuum bodyElasticity （彈簧） Hookes law （胡

4、克定律）（胡克定律） F= k X F: force （張力） X: distance （距離） k: spring constant （參數(shù)）Springs （彈簧）（彈簧）Viscosity （粘性） Viscous stress （粘性力）is proportional to the strain rate （形變率）Strain rate （形變率）= the time derivative of the strain（形變的時間導(dǎo)數(shù)）=gradient of the velocity of the material （速度的導(dǎo)數(shù)）e.g. Strain for a long rubb

5、erSprings （彈簧）and dashpot Integer-order ModelsFractional-order models (分數(shù)階模型）Viscoelastic materials （粘彈性材料） amorphous polymers semicrystalline polymers Biopolymers （生物聚合物） Red blood cells （紅血細胞）Fractional derivatives （分數(shù)階導(dǎo)數(shù)）Riemann-Liouville integral (黎曼-劉維爾積分） Fractional derivative （分數(shù)階導(dǎo)數(shù)）Fractiona

6、l derivativesFractional differential equations分數(shù)階微分方程 Riemann- Liouville （黎曼劉維（黎曼劉維爾）爾） Caputo Equivalent form （等價形式）Mathematical problem Existence （存在性） (Fixed point theorem) Uniqueness （唯一性） Regularity （正則性） Numerical issues （數(shù)值問題） 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 (穩(wěn)定性）2. Error estimates （誤差估計）3. Computational cost （計算費用）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