A Nonlinear Volterra Integrodifferential Equation Describing the Stretching of Polymeric Liquids

摘要

We study a model equation for the elongation of filaments or sheets of Polymeric liquids under the influence of a force applied to the ends. Mathematically this equation has the form of a nonlinear Volterra integrodifferential equation with the kernel given by a finite sum of exponentials. The unknown function denotes the length of the filament or, respectively, the thickness of the sheet. We study the equation both analytically and numerically. The force is assumed to converge to zero exponentially as t approaches minus infinity and to vanish identically after a finite time t sub 0. It is shown that under this condition there is a unique solution which approaches a given limit as t approaches minus infinity; moreover, the solution also has a limit as t approaches plus infinity. A numerical scheme is analyzed and convergence uniformly in t is established. Particular attention is paid to the dependence of solutions on a parameter mu, which corresponds to a Newtonian contribution to the viscosity. It is proved that solutions converge uniformly in t as mu approaches, and that the convergence of the numerical scheme is also uniform in mu. (Author)