文档名：一类抛物型方程初值与源项同时反演问题的唯一性与数值计算
摘要：基于实际应用的驱动,利用局部观测数据研究了一类抛物型方程初值与源项同时反演问题.首先利用特征函数展开法得到问题的级数形式解,基于级数解构造性地证明基于2个观测时刻的含局部测量数据同时反演问题的非唯一性;其次利用抛物型方程解的解析延拓性,证明了基于3个观测时刻含局部测量数据的同时反演问题的唯一性;然后基于有限元插值技术和叠加原理,构造了同时反演问题的反演算法;最后通过构造具有解析解和不具有解析解的数值算例验证了反演算法的有效性.
Abstract：Theparabolicequation,aclassicdevelopingequation,iswidelyappliedinthescientificandengineeringfields,suchaspredictingthesolutetransportationingroundwater,simulatingtemperatureofthethermalconductivematerialsandanalyzingthepopulationmutualeffect.Generally,differentpracticalproblemsaresummarizedintodifferentmodels.However,therealwaysexistsomeunknownconditionsorparametersinthemodelswhentheparabolicmodelsareappliedtoaddresssomepracticalproblems.Theunknownconditionsusuallyneedreconstructionbysomeotheradditionaldatainanindirectway.Mathematically,theseidentificationproblemsarecalledasparabolicequationinverseproblems.Inapplication,theinitialstateortheinnersourceterminsomediffusionsystemalwaysneedsidentification.Theidentificationproblemsaregenerallymodeledasbackwardproblemandinversesourceproblemfortheparabolicequation,twoclassicinverseproblemsextensivelystudiedbyengineersandmathematicians.Intheoreticalaspect,theexistenceanduniquenessofthetwoinverseproblemsareprovenbyintegralequationtheories,Laplacetransformation,Kalermanestimateandfixedpointtheory.Inalgorithmaspect,quasi-boundaryregularizationmethod,quasi-reversibilitymethod,Tikhonovregularizationmethodandprojectionmethodareusuallyadoptedtosolvetheaboveinverseproblems.Accordingtopublishedworks,theadditionaldataforthebackwardproblemortheinversesourceproblemarerequiredinthewholespatialdomainonsometerminaltime.Studiesofinverseproblemsforparabolicequationarescarcewhentheobservationdatacomesfromlocalobservation,whilethesimultaneousidentificationproblemsfortheparabolicequationareevenscarcerwhentheadditionaldataaretakenfromthelocalmeasurement.Generally,theobservationinthewholespatialdomainisdifficult.Obtainingtheobservationdatainalocalspatiallydomainismorepractical.Drivenbytherealapplication,simultaneousidentificationofinitialvalueandsourcetermforakindofparabolicequationisstudiedinthispaperbasedonlocalmeasurements.First,theformalseriessolutiontothedirectproblemisobtainedbytheeigenfunctionexpansionmethod,andthenon-uniquenessofthesimultaneousidentificationisprovenwhentheadditionaldataaregiveninaspatialsub-domainattwoobservationtimes.Then,theuniquenessofthesimultaneousinversionproblemisprovenbasedonthelocalmeasurementsatthreeobservationtimesandtheresultofanalyticcontinuationfortheparabolicequation.Next,aneasilyparalleledinversionalgorithmisproposedbasedonthetechniqueoffiniteelementinterpolationandtheprincipleofsuperposition.Last,severalnumericalexamplesincludingthecasesofexistingandnon-existinganalyticalsolutionsaretestedtodemonstratetheefficiencyoftheinversionalgorithm.
作者：阮周生 万广红 陈振兴Author：RUANZhousheng WANGuanghong CHENZhenxing
作者单位：东华理工大学理学院,南昌330013
刊名：重庆理工大学学报
Journal：JournalofChongqingInstituteofTechnology
年，卷(期)：2024, 38(3)
分类号：O175.28
关键词：抛物型方程 同时反演问题 唯一性 解析延拓
Keywords：parabolicequation simultaneousinversion uniqueness analyticalcontinuation
机标分类号：O353TP391.9U447.1
在线出版日期：2024年3月25日
基金项目：国家自然科学基金，国家自然科学基金，江西省自然科学基金一类抛物型方程初值与源项同时反演问题的唯一性与数值计算[
期刊论文] 重庆理工大学学报--2024, 38(3)阮周生 万广红 陈振兴基于实际应用的驱动,利用局部观测数据研究了一类抛物型方程初值与源项同时反演问题.首先利用特征函数展开法得到问题的级数形式解,基于级数解构造性地证明基于2个观测时刻的含局部测量数据同时反演问题的非唯一性;其次...参考文献和引证文献
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