林俊吉Lin, Chun-ChiTran The DungTran The Dung2023-12-082024-08-012023-12-082022https://etds.lib.ntnu.edu.tw/thesis/detail/a2d6c7611508b98bb265fcf864b9a20e/http://rportal.lib.ntnu.edu.tw/handle/20.500.12235/121120NoneIn this thesis, the method of geometric flow is applied to prove the existence of global solutions to the problem of nonlinear spline interpolations for closed/non-closed curves and the problem of area-constrained planar elasticae with free boundaries on a straight line. Among them, this method applies the theory of either fourth-order parabolic PDEs/PDE or second-order parabolic PDEs/PDE with certain imposed boundary conditions. The results of this study demonstrate the existence of global solutions and sub-convergence of the elastic flow. Furthermore, the geometric flow method provides a new approach to the problem of nonlinear spline interpolations.NoneGeometric flowelastic flowfourth-order parabolic equationsecond-order parabolic equationelastic splinespline interpolationcurve fittingpath planningfree boundary problemcontact anglesHolder spacesetd