Abstract.We study a nonlinear boundary value problem arising from electrochemistry. The essential difficulties are due to the strong nonlinear nature of part of the boundary condition. This part of the boundary condition is of an exponential type and is normally in the corrosion literature associated with the names of Butler and Volmer. We examine the questions of existence and uniqueness of solutions to this boundary value problem. In a numerical example we compare the behaviour of the solutions to the nonlinear problem with the behaviour of the solutions to a corresponding linearized problem. In contrast to earlier studies we put a major emphasis on studying parameter values that may be relevant for the case in which part of the boundary is in a transition to passivity-in practice most likely because it is nearly covered by an oxide layer.

Oscillatory and periodic solutions of retarded functional differential equations are investigated.The study concerns equations with piecewise constant arguments which found applications in certain biomédical problems.1. The study of oscillatory solutions of differential equations with deviating arguments has been the subject of many recent investigations.Of particular importance, however, has been the study of oscillations which are caused by the deviating arguments and which do not appear in the corresponding ordinary differential equation, see [1,[4][5][6][7][8][9][10][11]13].In this paper we study oscillatory properties of solutions of the linear delay differential equations with piecewise constant deviating argument of the typewhere a(t) and b(t) are continuous functions on [0, oo), and [•] designates the greatest integer function. Such equations are similar in structure to those found in certain "sequential-continuous" models of disease dynamics as treated by Busenberg and Cooke [2]. We give sufficient condition under which equation (1) has oscillatory solutions. We emphasize the fact that our condition is the "best possible" in the sense that when a and b are constants the condition reduces to b > ae~a/4(ea -1) which is a necessary and sufficient condition. In case of constant coefficients we find conditions under which oscillatory solutions are periodic. As it is customary, a solution is said to be oscillatory if it has arbitrarily large zeros.2. In this section we give a sufficient condition under which equation (1) has oscillatory solutions.DEFINITION. A solution of equation (1) on [0, oo) is a function y(t) that satisfies the initial data y(0) = Co, y(-l) = C-x, and the conditions: (i) y(t) is continuous on [0, oo); (ii) the derivative y'(t) exists at each point t G [0, oo), with the possible exception of the points [zj] G [0, oo) where one-sided derivatives exist;(iii) equation (1) is satisfied on each interval [n,n+ 1) C [0, oo) with integral endpoints.

We give a formal derivation of an effective nonlinear boundary condition associated with a steady state voltage potential on a corroding metal surface. Subsequently we use this boundary condition to analyze the determination of corrosion damage based on measurements of the boundary voltage potential and the associated current flux.Mathematics Subject Classification (1991). 35J65, 35R30.

No abstract