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This paper discusses an electrostatic, homogeneous field in a uniform two‐dimensional domain with Neumann's boundary conditions. The boundary conditions are known only at some segments of the boundary. The synthesis is understood as the computation of the remaining boundary conditions which would ensure the required potential distribution in some subdomains within the boundary. The introduction of a single‐layer potential leads to Fredholm's equation of the second order. Stepwise approximation of the source distribution along the boundary rearranges Fredholm's equation and the requirements concerning the single layer potential distribution. It leads to a matrix equation with a rectangular coefficient matrix. In order to solve approximately this equation, in the sense of the least squares minimization, the singular value decomposition (SVD) method is used. The choice of subdomains with determined potential distribution influences significantly the conditioning of the equation. Easy selection of an acceptable solution among all possible solutions proves the suitability of the SVD method in the above problem. The numerical experiments reported in the paper are a good illustration of this.

The electric stimulation of the vagus nerve is used to obtain therapeutic results in epilepsy, depression and Alzheimer diseases. The purpose of this paper is to show numerical model of stimulation, focusing on the mathematical approach to modeling a phenomenon of neural cells activation and its propagation in the nerve.

The paper presents a model based on the bidomain theory. It uses two continuous, averaged domains which depict the intra‐ and extra‐cellular domains and are connected with the membrane ionic currents. The numerical model uses 3D cylindrical model approximating the anatomical shape of the neck. The simulator is based on a time domain finite element method.

The presented approach allows to model the discrete behaviour of the membrane potential in the macroscopic, realistic model of the nerve. The validation of the parameters with the velocity of activation propagations suggests the strong disscussion on physical interpretation to the bidomain theory parameters. To obtain realistic results the parameters needed to be unrealistic.

The paper presents the combination of bidomain model of neural tissue with the time domain finite element method along with the atributes of bidomain model for realistic modeling of the process of propagtion of activation.

Two new benchmark inverse problems for eddy currents are proposed. The first originates in the optimal design of the tubular inductive heater. The authors’ goal is to offer a simple problem which will check whether new software is able to minimize a multimodal objective function. The second benchmark is a simplified model of a hardening device. The purpose of this problem is to test the ability of the software to deal with different design parameters. This benchmark can be easily extended for more complicated, coupled fields problems. For both benchmarks reference standard solutions are presented. They were obtained using a finite element package for electromagnetics, developed by the authors.