分数阶微分方程的数值解法ppt课件.ppt

《分数阶微分方程的数值解法ppt课件.ppt》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

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09/2014OutlinesPart1:分数阶微分方程1.Modelling(Viscoelasticity)数学模型（粘弹性）2.Mathematicsformulation（数学理论）3.Numericalmethod（数值方法）Part2:分数阶偏微分方程Modelling(anomalousdiffusion)数学模型（不规则扩散）Mathematicaformulation（数学理论）Numericalmethods（数值方法）Part1:Viscoelasti

2、city（粘弹性）Elastic（弹性）Viscous（粘性）Whatisstress（应力的定义）Thestressacrossasurfaceelement(yellowdisk)istheforcethatthematerialononeside(topball)exertsonthematerialontheotherside(bottomball),dividedbytheareaofthesurfaceIdealizedstressinastraightbarwithuniformcross-section.W

3、hatisthestrain?（应变的定义）Astrain（应变）measureofdeformationrepresentingthedisplacementbetweenparticlesinthebodyrelativetoareferencelengthThedeformation（变形）ofathinstraightrodintoaclosedloopMotionofacontinuumbodyElasticity（弹簧）Hooke’slaw（胡克定律）F=kXF:force（张力）X:distance（距离）k

4、:springconstant（参数）Springs（弹簧）Viscosity（粘性）Viscousstress（粘性力）isproportionaltothestrainrate（形变率）Strainrate（形变率）=thetimederivativeofthestrain（形变的时间导数）=gradientofthevelocityofthematerial（速度的导数）e.g.StrainforalongrubberSprings（弹簧）anddashpotInteger-orderModelsFractional

5、-ordermodels(分数阶模型）Viscoelasticmaterials（粘弹性材料）amorphouspolymerssemicrystallinepolymersBiopolymers（生物聚合物）Redbloodcells（红血细胞）Fractionalderivatives（分数阶导数）Riemann-Liouvilleintegral(黎曼-刘维尔积分）Fractionalderivative（分数阶导数）FractionalderivativesFractionaldifferentialequatio

6、ns分数阶微分方程Riemann-Liouville（黎曼刘维尔）CaputoEquivalentform（等价形式）MathematicalproblemExistence（存在性）(Fixedpointtheorem)Uniqueness（唯一性）Regularity（正则性）Numericalissues（数值问题）Numericalscheme（格式）Algorithm（算法）Programming（程序）Finitedifferencemethod(有限差分法)(Fractionalderivativeisapp

7、roximatedbyfinitedifferenceschemes)Fornumericalmethods,weneedtoconsiderStability(稳定性）2.Errorestimates（误差估计）3.Computationalcost（计算费用）FinitedifferencemethodI(integraldiscretization)Finitedifferencemethod-II(derivativediscretization)Part2:Fractionalpartialdifferent

8、ialequationsNormalDiffusion:Someparticlesaredissolvedinaglassofwater.Brownianmotion:abigparticlecollideswithalargesetofsmallerparticleswhichmovewithdiff