In this paper, we study a quasi-static frictional contact problem for a viscoelastic body with damage effect inside the body as well as normal compliance condition and multi-valued friction law on the contact boundary. The considered friction law generalizes Coulomb friction condition into multi-valued setting. The variational–hemi-variational formulation of the problem is derived and arguments of fixed point theory and surjectivity results for pseudo-monotone operators are applied, in order to prove the existence and uniqueness of solution.

Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short-time.

In this work, we analyse a broad class of generalized Poisson–Boltzmann equations and reveal a common mathematical structure. In the limit of a wide electrode, we show that a broad class of generalized Poisson–Boltzmann equations admits a reduction that affords an explicit connection between the functional form of the corresponding free energy and the associated differential capacitance data. We exploit the relation to we show that differential capacitance curves generically undergo an inflection transition with increasing salt concentration, shifting from a local minimum near the point of zero charge for dilute solutions to a local maximum point near the point of zero charge for concentrated solutions. In addition, we develop a robust numerical method for solving generalized Poisson–Boltzmann equations which is easily applicable to the broad class of generalized Poisson–Boltzmann equations with very few code adjustments required for each model

Outflow from a young star might be regarded as approximately equivalent to flow from a point source. If the fluid consists of charged particles, then the magnetic fields produced are governed by Faraday's law. This simple first approximation yields a linear partial differential equation in spherical polar coordinates, and its solution may be represented as the product of a Legendre polynomial with some function of the radial coordinate. This radial function is shown to involve orthogonal polynomials. Their properties are investigated and recurrence formulae for them are derived. Some of the magnetic fields generated by this simple model are illustrated.