分数阶微分方程的数值解法

Fractional partial differential equations
Mathematics problem
• Existence ( Fractional sobolev space)
• uniqueness
• Regularity ( Not available yet)
Numerical methods
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 directions
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）
Elastic （弹性） Viscous （粘性）
பைடு நூலகம்
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 surface
Fractional derivatives
Fractional differential equations 分数阶微分方程
Riemann- Liouville （黎曼刘维 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 methods for space-fractional PDEs • Numerical methods for stochastic fractional PDEs • MATLAB programs
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Numerical issues （数值问题）
• Numerical scheme （格式）
• Algorithm （算法） • Programming （程序）
Finite difference method (有限差分法)
( Fractional derivative is approximated by finite difference schemes) For numerical methods, we need to consider 1. Stability (稳定性）
The deformation （变形）of a thin straight rod into a closed loop
Motion of a continuum body
Elasticity （弹簧）
Hooke’s law （胡克定律） • F= k X • F: force （张力） • X: distance （距离） • k: spring constant （参数） Springs （弹簧）
• 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）
分数阶微分方程的数值解
闫玉斌
数学系 切斯特大学, 英国
09/2014
Outlines
• 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）
e.g. Strain for a long rubber
Springs （弹簧）and dashpot
Integer-order Models
Fractional-order models (分数阶模型）
Viscoelastic materials （粘弹性材料）
• amorphous polymers
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 discretization)
Idealized stress in a straight bar with uniform crosssection.
What is the strain? （应变的定义）
A strain（应变） measure of deformation representing the displacement between particles in the body relative to a reference length
• 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: Viscoelasticity （粘弹性）
• Finite difference methods, Neville and Yan, (2011), ( 2012) • Finite element methods, Neville and Yan, (2012)
Current works
• Numerical methods for time fractional PDEs
Normal Diffusion equation
Anomalous diffusion
Anomalous diffusion
• A Lévy flight is a random walk in which the step-lengths have a probability distribution that is heavy-tailed.