Solution of nonlinearly curvature driven evolution of plane curves

Solution of nonlinear curvature driven evolution of plane convex curves

The numerical approximation scheme for solving the nonlinear initial value problem ∂tb(v) = ∂xxβ(x, v) + β(x, v) with periodic boundary conditions is presented. Local existence and uniqueness of a solution and convergence of approximations is a consequence of the results of Mikula and Kačur (1996), where the anisotropic curvature driven evolution of plane convex curves is studied. The considere...

Computational and qualitative aspects of evolution of curves driven by curvature and external force

We propose a direct method for solving the evolution of plane curves satisfying the geometric equation v= β(x, k, ν) where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ ⊂ R2 at a point x ∈ Γ . We derive and analyze the governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolv...

Solution and Application of Anisotropic Curvature Driven Evolution of Curves

Similarity solutions in the theory of curvature driven diffusion along planar curves: I. Symmetric curves expanding in time

A numerical method is developed for obtaining similarity solutions of the differential equations governing the evolution of a planar curve in the theory of diffusion along curves. The method is applied to cases in which the solution can be interpreted as describing the subsequent evolution (by curvature driven surface diffusion) of the boundary of a three-dimensional body that in the limit t--~...

A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

A new method for solution of the evolution of plane curves satisfying the geometric equation v= (x; k; ), where v is the normal velocity, k and are the curvature and tangential angle of a plane curve ⊂R2 at the point x∈ , is proposed. We derive a governing system of partial di erential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane...