Online parameter identification in time-dependent differential equations from time course observations related to the physical state can be understood as a non-linear inverse and ill-posed problem and appears in a variety of applications in science and engineering. The feature as well as the challenge of online identification is that sensor data have to be continuously processed during the operation of the real dynamic process in order to support simulation-based decision making. In this paper we present an online parameter identification method that is based on a non-linear parameter-to-output operator and, as opposed to methods available so far, works both for finite- and infinite-dimensional dynamical systems, e.g., both for ordinary differential equations and time-dependent partial differential equations. A further advantage of the method suggested is that it renders typical restrictive assumptions such as full state observability, linear parametrisation of the underlying model and data differentiation or filtering unnecessary. Assuming existence of a solution for exact data, a convergence analysis based on Lyapunov theory is presented. Numerical illustrations given are by means of online identification both of aerodynamic coefficients in a 3DoF-longitudinal aircraft model and of a (distributed) conductivity coefficient in a heat equation.

We study a scale of two-component composite structures of equal proportions with infinitely many microlevels. The structures are obtained recursively and we find that their effective conductivities are power means of the local conductivities.

We study the geodesic interpolating spline with a biharmonic regulariser for solving the landmark image registration problem. We show existence of solutions, discuss uniqueness and show how the problem can be efficiently solved numerically. The main advantage of the geodesic interpolating spline is that it provides a diffeomorphism and we show this is preserved under our numerical approximation.

Ion channels are proteins with a narrow hole down their middle that control a wide range of biological function by controlling the flow of spherical ions from one macroscopic region to another. Ion channels do not change their conformation on the biological time scale once they are open, so they can be described by a combination of Poisson and drift-diffusion (Nernst–Planck) equations called PNP in biophysics. We use singular perturbation techniques to analyse the steady-state PNP system for a channel with a general geometry and a piecewise constant permanent charge profile. We construct an outer solution for the case of a constant permanent charge density in three dimensions that is also a valid solution of the one-dimensional system. The asymptotical current–voltage (I–V) characteristic curve of the device (obtained by the singular perturbation analysis) is shown to be a very good approximation of the numerical I–V curve (obtained by solving the system numerically). The physical constraint of non-negative concentrations implies a unique solution, i.e., for each given applied potential there corresponds a unique electric current (relaxing this constraint yields non-physical multiple solutions for sufficiently large voltages).

A steady-state bubble solution to the constrained mass conserving Allen–Cahn equation in a two-dimensional domain is constructed in the limit of small diffusivity. The solution is asymptotically constant inside a circle of radius rb centred at some unknown location x0 and has a sharp interface at the bubble radius that allows for a transition to a different asymptotically constant state outside the bubble. In a study by M. J. Ward (Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56, 1996, 247–1279), the bubble centre was determined by a limiting solvability condition. The solution found by Ward suggests the existence of a corner type boundary layer where the normal derivative of the bubble solution readjusts to satisfy the no-flux condition at the boundary of the domain. This work is concerned with the details of the readjustment. A variational approach similar to the one of W. L. Kath, C. Knessl and B. J. Matkowsky (A variational approach to nonlinear singularly perturbed boundary-value problems. Stud. Appl. Math. 77, 1987, 61–88) shows the formation of a corner layer (for the derivative of the solution) which influences as a high-order correction the available determination of the bubble centre. This corner layer describes to leading order the readjustment of the level lines of the bubble to lines parallel to the boundary of the container; moreover, it provides to leading order a smooth solution across the corner layer.

In this paper, we study basic properties of the diffusive wave approximation of the shallow water equations (DSW). This equation is a doubly non-linear diffusion equation arising in shallow water flow models. It has been used as a model to simulate water flow driven mainly by gravitational forces and dominated by shear stress, that is, under uniform and fully developed turbulent flow conditions. The aim of this work is to present a survey of relevant results coming from the studies of doubly non-linear diffusion equations that can be applied to the DSW equation when topographic effects are ignored. In fact, we present proofs of the most relevant results existing in the literature using constructive techniques that directly lead to the implementation of numerical algorithms to obtain approximate solutions.