Résumé:
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.
