The use of interpolating element free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate

چکیده مقاله

InthispaperanumericaltechniqueisproposedforsolvingthenonlineargeneralizedBen-jamin–Bona–Mahony–Burgers and regularized long-wave equations. Firstly, we obtain a timediscreteschemebyapproximatingtimederivativeviaafinitedifferenceformula,then weusetheinterpolatingelement-freeGalerkinapproachtoapproximatethespatialderiva-tives.Theelement-freeGalerkinmethodusesaweakformoftheconsideredequationthat issimilartothefiniteelementmethodwiththedifferencethatintheelement-freeGalerkin methodtestandtrialfunctionsaremovingleastsquaresapproximationshapefunctions. Also,intheelement-freeGalerkinmethod,wedonotuseanytriangular,quadrangularor othertypeofmeshes.Itisaglobalmethodwhilefiniteelementmethodisalocalone.The elementfreeGalerkinmethodisnotatrulymeshlessmethodandforintegrationemploys abackgroundmesh.Weprovethatthetimediscreteschemeisunconditionallystableand convergentintimevariableusingtheenergymethod.Weshowthatconvergenceorderof thetimediscreteschemeisO (τ).Sincetheshapefunctionsofmovingleastsquaresapprox-imationdonothaveKroneckerdeltaproperty,wecannotimplementtheessentialbound-arycondition,directly.Thus,theimprovedmovingleastsquaresshapefunctionsthathave thementionedpropertyareemployed.Anerrorestimateforthemethodproposedinthe currentpaperisobtained.Also,thetwo-dimensionalversionofbothequationsondifferent complexgeometriesissolved.Theaimofthispaperistoshowthatthemeshlessmethod basedontheweakformisalsosuitableforthetreatmentofthenonlinearpartialdiffer-entialequationsandtoobtainanerrorboundforthenewmethod.Numericalexamples confirmtheefficiencyoftheproposedscheme.