In this first part of our two-part article, we present some theoretical background along with descriptions of some numerical techniques for solving a particular semilinear elliptic eigenproblem of Lane-Emden type on a triangular domain without any lines of symmetry. For solving the principal (1st) eigenproblem, we describe an operator splitting method applied to the corresponding time-dependent...

Abstract. A thin, flexible, one side-built-in rod under a concentrated terminal force is studied in its elastic equilibrium configuration. In order to make the problem more tractable, a proper set of state variables is chosen, facing with a second order, nonlinear, but autonomous boundary value problem, in the rotation φ(s) pertaining to each s-section. The search of the free end rotation φ0, f...

A modified parabolic equation for adaptive monotone difference schemes based on equal-arclength mesh, applied to the linear convection equation, is derived and its convergence analysis shows that solutions of the modified equation approach a discontinuous (piecewise smooth) solution of the linear convection equation at order one rate in the L1-norm. It is well known that solutions of the monoto...

An algorithm is presented to numerically treat a free boundary problem arising in the theory of phase transition. The problem is one in which a collection of simple closed curves (particles) evolves in such a way that the enclosed area remains constant while the total arclength decreases. Material is transported between particles and within particles by diiusion, driven by curvature which expre...

In this work, we consider an optimization problem described on a surface. The approach is illustrated on the problem of finding a closed curve whose arclength is as small as possible while the area enclosed by the curve is fixed. This problem exemplifies a class of optimization and inverse problems that arise in diverse applications. In our approach, we assume that the surface is given parametr...