Généralisation des techniques de linéarisation pour approcher les systèmes d'équations intégrales non linéaires

Abstract

Theobjectiveofthisthesisistoimprovenumericallinearizationtechniquesforsolvingsystemsofnonlinearintegralequations,whichplayacrucialroleinmodelingvariousproblemsacrossdomainssuchasphysics,biology,andmachinelearning.Thestudyfocuses on Banach spaces.Traditionally, most methods involve discretizing the problemfirstandthenproceedingwiththelinearizationprocesstosolvenonlinearfunctionalproblems.However,weproposeanovelapproachthatreversesthisorder,startingwiththelinearizationprocessusingNewton’siterativemethod,andthendiscretizingtheiterativelinearsystemobtainedfromthefirstphase.Weapplythisapproachtosolveavarietyofproblems,startingwithasystemofnonlinearFredholmintegralequationsofthesecondkindwithregularkernels,usingtheNyströmmethodfordiscretization. Additionally,weintroduceanewschemewithadoublediscretizationprocessthatutilizestheKantorovichprojectionmethodtoapproximatesolutionsfornonlinearfunctionalequations,specificallynonlinearintegro-differentialequationsofthesecondkind.Finally,weapplyourapproachtosolveasystemofnonlinearintegro-differentialequationswithaweaklysingularkernel,estimatingallintegralswithweaklysingularkernelsusingproductintegrationrules.Wehaveprovidednecessaryconditionsforeachapplicationtoensuretheconvergenceofourmethods.