Integral Equation Formulations and Related Numerical Solution Methods in Some Biomedical Applications of Electromagnetic Fields

Abstract

The paper reviews certain integral equation approaches and related numerical methods used in studies of biomedical applications of electromagnetic fields pertaining to transcranial magnetic stimulation (TMS) and nerve fiber stimulation. TMS is analyzed by solving the set of coupled surface integral equations (SIEs), while the numerical solution of governing equations is carried out via Method of Moments (MoM) scheme. A myelinated nerve fiber, stimulated by a current source, is represented by a straight thin wire antenna. The model is based on the corresponding homogeneous Pocklington integro-differential equation solved by means of the Galerkin Bubnov Indirect Boundary Element Method (GB-IBEM). Some illustrative numerical results for the TMS induced fields and intracellular current distribution along the myelinated nerve fiber (active and passive), respectively, are presented in the paper.

On the solvability of a general class of a coupled system of stochastic functional integral equations

“…In other words, integral equations can successfully describe the evolution of many real-world systems, such as systems in quantum mechanics, Newtonian mechanics, electrical engineering, electromagnetic theory, and many other fields, (see e.g. Poljak et al, 2018). Also, in ordinary and partial differential equations we can convert several initial and boundary value problems to an equivalent integral equations.…”