In this work, the Finite Element Method is used for finding the numerical solution of an elliptic problem with Henstock–Kurzweil integrable functions. In particular, Henstock–Kurzweil high oscillatory functions were considered. The weak formulation of the problem leads to integrals that are calculated using some special quadratures. Definitions and theorems were used to guarantee the existence of the integrals that appear in the weak formulation. This allowed us to apply the above formulation for the type of slope bounded variation functions. Numerical examples were developed to illustrate the ideas presented in this article.

We tested different optical elements placed in three different positions by applying the irradiance transport equation (ITE), obtaining the wavefront [ W ( x , y ) ] and aberration surface [ AS ( r , θ ) ]. The existing noise in the captures I as well as in the W and A S were analyzed applying several filters: first a filter based on Legendre polynomials (LP), generating the most probable points increasing the data resolution; second, a filter based on a 50 deg 2D-LP was used as a multilineal fit (multiple linear regression); and third, an ideal bandpass filter in the Fourier space after inducing a periodicity using Ronchi simulated masks with periods in x , y , x y was used to perform data scanning (similar to the four-step phase-shifting method). Signal-to-noise ratio values were obtained for each proposed filter along with the most probable image free from noise, determined from a linear combination of the original data and the applied filters.

In this paper, we give a solution to the irradiance transport equation (ITE) using the two-dimensional (2D) Legendre polynomials (LPs) and an interpolator (i-LP) based on the LP. In the first place, we analyze the experimental data; subsequently, we proceed to fit the most probable 2D LPs’ surface to the data in order to obtain the wavefront surface ($W(x,y)$ of the elements under test) as a solution of the ITE differential equation associated with a boundary problem; and finally, we interpolate the resulting fitting. The interpolation is built from LP to increase the resolution and sharpness of the data. We apply the ITE to these results in order to obtain the wavefront as a nondeterministic solution that increases the resolution of the ITE as an optical test, and we compare our results regarding the obtained aberration surfaces ($\mathrm{AS}(x,y)$).

In this work, we propose a solution to the problem of identification of sources in the brain from measurements of the electrical potential, recorded on the scalp EEG (electroencephalogram), where boundary problems are used to model the skull, brain region, and scalp, solving the inverse problem from the EEG measurements, the so-called Electroencephalographic Inverse Problem (EIP), which is ill-posed in the Hadamard sense since the problem has numerical instability. We focus on the identification of volumetric dipolar sources of the EEG by constructing and modeling a simplification to reduce the multilayer conductive medium (two layers or regions ${\Omega }_1$ and ${\Omega }_2$) to a problem of a single layer of a homogeneous medium with a null Neumann condition on the boundary. For this simplification purpose, we consider the Cauchy problem to be solved at each time. We compare the results we obtained solving the multiple layers problem with those obtained by our simplification proposal. In both cases, we solve the direct and inverse problems for two different sources, as synthetic results for dipolar sources resembling epileptic foci, and a similar case with an external stimulus (intense light, skin stimuli, sleep problems, etc). For the inverse problem, we use the Tikhonov regularization method to handle its numerical instability. Additionally, we build an algorithm to solve both models (multiple layers problem and our simplification) in time, showing optimization of the problem when considering 128 divisions in the time interval $[0,1]$ s, solving the inverse problem at each time (interval division) and comparing the recovered source with the initial one in the algorithm. We observed a significant decrease in the computation times when simplifying the numerical calculations, resulting in a decrease up to $50\%$ in the execution times, between the EIP multilayer model and our simplification proposal, to a single layer homogeneous problem of a homogeneous medium, which translates into a numerical efficiency in this type of problem.