分数阶微分方程的数值解法课件

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,Part1: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）,Stress:（应力）forceperunitareaofparticlesexertedStrain:（应变）measurementofdeformationSprings（弹簧）(HookesLaw)（胡克定律）=:Stress:StrainE:Elasticmodulus,Viscous（粘性）,Stress:（应力）forceperunitareaofparticlesexertedStrainrate:（应变率）therateatwhichstrainoccurs.ItisthetimerateofchangeofstrainDashpots:(NewtonianFluid)（牛顿流体）=():Stress,():Strainrate:Viscosity,Whatisstress（应力的定义）,Thestressacrossasurfaceelement(yellowdisk)istheforcethatthematerialononeside(topball)exertsonthematerialontheotherside(bottomball),dividedbytheareaofthesurface,Idealizedstressinastraightbarwithuniformcross-section.,Whatisthestrain?（应变的定义）,Astrain（应变）measureofdeformationrepresentingthedisplacementbetweenparticlesinthebodyrelativetoareferencelengthThedeformation（变形）ofathinstraightrodintoaclosedloop,Motionofacontinuumbody,Elasticity（弹簧）,Hookeslaw（胡克定律）,F=kXF:force（张力）X:distance（距离）k:springconstant（参数）,Springs（弹簧）,Viscosity（粘性）,NewtonianFluid（牛顿流体）=:sherestress（切向力）:shearviscosityderivativeofthevelocity（加速度）,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.Strainforalongrubber,Strain（形变）(deformation),=000:theoriginallength(原始长度）,L(t):itslengthateachtimet.（变形后的长度）,strainrate/（形变率）Rateofchangeinstrain,Springs（弹簧）anddashpot,Theelastic（弹性）componentscanbemodeledassprings（弹簧）ofelasticconstantE=,Theviscous（粘性）componentscanbemodeledasdashpotssuchthatthestressstrainrate（应力-应变率）=,Integer-orderModels,Maxwellmodel:（麦克斯韦尔模型）=1+(=,Notcorrespondtoexperimentalobservation)Voigtmodel:（Voigt模型）()=+(Notcorrespondtoexperimentalobservation),Fractional-ordermodels(分数阶模型）,Stress（应力）isproportionaltostrainderivative（形变导数）of“intermediate”(non-integer)order:(0,1)=Here:Caputofractionalderivative（分数阶导数）GeneralizedMaxwellmodel:+=0,Viscoelasticmaterials（粘弹性材料）,amorphouspolymerssemicrystallinepolymersBiopolymers（生物聚合物）Redbloodcells（红血细胞）,Fractionalderivatives（分数阶导数）,Let=,2=2/(2),Whatdoes12=?,Riemann-Liouvilleintegral(黎曼-刘维尔积分）,Riemann-LiouvilleIntegraloperator=0,With+()=1()011With=1()011,Fractionalderivative（分数阶导数）,=,0.integrationmeansantiderivative（积分=反导数）With0,1,wemaywrite=1Or=(1(),Fractionalderivatives,Riemann-Liouville=(1()Or=110Integrationfirst,derivativesecond（先积分，后导数）,Caputo=(1()Or=110Derivativefirstintegrationsecond（先导数，后积分）,=?=0,12,1,0=bluecurve12=212purplecurve1=1redcurve=+1+1,0,0,Fractionaldifferentialequations分数阶微分方程,Riemann-Liouville（黎曼刘维尔）,=,10=0Initialvaluehasnophysicalmeaning,buthasgoodmathematicsstructure,mathematicianusethis,Caputo,=,0=0Initialvaluehavephysicalmeaning,Engineersusethisformulation,Equivalentform（等价形式）,=,0=0Equivalentto0=,Equivalenttoy0=101,Mathematicalproblem,Existence（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),(Fractionalderivativeisapproximatedbyfinitedifferenceschemes)Fornumericalmethods,weneedtoconsiderStability(稳定性）2.Errorestimates（误差估计）3.Computationalcost（计算费用）,FinitedifferencemethodI(integraldiscretization),0=101,Let0=0121,Anomalousdiffusion,ALvyflightisarandomwalkinwhichthestep-lengthshaveaprobabilitydistributionthatisheavy-tailed.,Fractionalpartialdifferentialequations,Let(,)denotesthedensityofLevyflight,then(,)satisfies,=,+f(x,t)Here,denotestheCaputofractionalderivative,denotestheRieszfractionalderivative,Mathematicsproblem,Existence(Fractionalsobolevspace)uniquenessRegularity(Notavailableyet),Numericalmethods,Finitedifferencemethods,NevilleandYan,(2011),(2012)Finiteelementmethods,Nevill