In order to investigate the full potential of the Electrical Impedance Spectroscopy (EIS) when used to address various aspects of conductive and dielectric response within the field of Condensed Matter Physics and Electrochemistry, a new analysis of the electrical impedance experiments has been undertaken. Within the framework of quantum mechanical band structure and using the concept of electrochemical potential for each of the relevant energies, the problem of electrical response in condensed phase has been formulated, using augmented Maxwell equations of Classical Electrodynamics, as a boundary value problem of a set of coupled, non-linear parabolic equations in energy, space and time. The result of this numerical analysis is a principal possibility of a complete electrical characterisation of both monocrystals, glassy solids and liquids. The EIS has been put in this way on a new qualitative level and should be considered now as the most general electrical experimental characterisation tool available. In this article, a methodology of numerical simulations of electrical response in condensed matter systems at classical frequencies (from ~1 THz down to dc) is presented and the numerical simulation results are then discussed, using monocrystalline Silicon, chalcogenide glass ion conductor Agx(AsS2)1−x and simple aqueous chloride solution as experimental test cases. Some other unique results of the new EIS analysis will also be discussed. These include the possibility of a clear distinction between the contribution to the electrical response from bound and mobile electrical charges, the possibility of simultaneous and independent determination of the mobile electrical charges mobility and their density in one EIS experiment and incorporation of the interfacial regions of the system under test (SUT) as an essential part of the overall electrical response.

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A Stefan-type problem is considered. This is an initial-boundary value problem on a composite domain for a parabolic reaction-diffusion equation with a moving interface boundary. At the moving boundary between the two subdomains, an interface condition is prescribed for the solution of the problem and its derivatives. A finite difference scheme is constructed that approximates the initial-boundary value problem. An iterative Newton-type method for the solution of the difference scheme and a numerical method for the analysis of the errors of the computed discrete solutions are both developed. Keywords:Stefan-type problem, initial-boundary value problem, composite domain, parabolic reaction-diffusion equation, moving interface boundary, finite difference scheme, iterative Newton-type method.