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:Viscoelasticity（粘弹性）Elastic（弹性）Viscous（粘性）Whatisstress（应力的定义）Thestressacrossasurfaceelement(yellowdisk)istheforcethatthematerialononeside(topball)exertsonthematerialontheotherside(bottomball),dividedbytheareaofthesurfaceIdealizedstressinastraightbarwithuniformcross-section.Whatisthestrain?（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: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-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）FractionalderivativesFractionaldifferentialequations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
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快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题
）Numericalscheme（格式）Algorithm（算法）Programming（程序）Finitedifferencemethod(有限差分法)(Fractionalderivativeisapproximatedbyfinitedifferenceschemes)Fornumericalmethods,weneedtoconsiderStability(稳定性）2.Errorestimates（误差估计）3.Computationalcost（计算费用）FinitedifferencemethodI(integraldiscretization)Finitedifferencemethod-II(derivativediscretization)Part2:FractionalpartialdifferentialequationsNormalDiffusion:Someparticlesaredissolvedinaglassofwater.Brownianmotion:abigparticlecollideswithalargesetofsmallerparticleswhichmovewithdifferentvelocitiesindifferentrandomdirectionsNormalDiffusionequationAnomalousdiffusionAnomalousdiffusionALévyflightisarandomwalkinwhichthestep-lengthshaveaprobabilitydistributionthatisheavy-tailed.FractionalpartialdifferentialequationsMathematicsproblemExistence(Fractionalsobolevspace)uniquenessRegularity(Notavailableyet)NumericalmethodsFinitedifferencemethods,NevilleandYan,(2011),(2012)Finiteelementmethods,NevilleandYan,(2012)CurrentworksNumericalmethodsfortimefractionalPDEsNumericalmethodsforspace-fractionalPDEsNumericalmethodsforstochasticfractionalPDEsMATLABprogramsThanksforyourattention!Anyquestions?
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