The discrete continuation in the boundary value problem for systems of nonlinear differential equations with deviation argument

The solution of the boundary value problems for system of nonlinear differential equations with argument delay is considered in the article. The solution is based on the shooting method. Within its framework the method of continuation with respect to parameter in the Lahaye form, method of the best parametrization and the Newton method are implemented that allow to find possible solutions. To solve the Cauchy problem at each step of the shooting method the discrete continuation method with respect to the best parameter combined with the Newton method is applied. This approach allows to build the solution in the case when singular limit points exist. That provides continuation of Newton iteration process. The algorithm is completed by calculating the Lagrange polynomial to obtain the values of function in the delay points. The example given in the article represents the advantages of the proposed method.

Imaging the Thoracic Distribution of Normal Breath Sounds

Summary Objective: To assess the feasibility to generate a confident image of normal breath sounds (BS) based on the quantitative analysis of multichannel sensors and imaging them in three known clinical classes, i.e., tracheal, bronchial and vesicular, identifying their spatial distribution with high resolution on the posterior thoracic surface. Methods: Three parametrization techniques, the percentile frequencies, the univariate AR modeling, and the eigenvalues of the covariance matrix were evaluated when applied to BS. These sounds were acquired in twelve healthy subjects by a 5 × 5 sensor array on the posterior thoracic surface plus the sound at the tracheal position, to obtain feature vectors that fed a supervised multilayer neural network. Based on BS classification rate, the spatial distribution of each BS class was obtained by constructing an image using deterministic interpolation. Results: The univariate AR modeling was the best parametrization technique producing a classification performance of 96% during the validation phase and just 4% of not classified feature vectors. Corresponding values for the percentile frequencies were 92% and 7.7%, whereas for the eigenvalues were 91% and 9.0%. Conclusion: This work shows that it is possible to generate confident images associated with the distribution of normal BS classes. Therefore, a detailed image about the spatial distribution of BS in humans might be helpful for detecting lung diseases.