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Finding the optimal parameters and functions of iterative methods is among the main problems of the Numerical Analysis. For this aim, a technique of the stochastic arithmetic (SA) is used to control of accuracy on Taylor-collocation method for solving first kind weakly regular integral equations (IEs). Thus, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method is applied and instead of usual mathematical softwares the CADNA (Control of Accuracy and Debugging for Numerical Applications) library is used. Also, the convergence theorem of presented method is illustrated. In order to apply the CESTAC method we will prove a theorem that it will be our licence to use the new termination criterion instead of traditional absolute error. By using this theorem we can show that number of common significant digits (NCSDs) between two successive approximations are almost equal to NCSDs between exact and numerical solution. Finally, some examples are solved by using the Taylor-collocation method based on the CESTAC method. Several tables of numerical solutions based on the both arithmetics are presented. Comparison between number of iterations are demonstrated by using the floating point arithmetic (FPA) for different values of $\varepsilon$., Comment: 24 pages

An inverse problem in spectroscopy is considered. The objective is to restore the discrete spectrum from observed spectrum data, taking into account the spectrometer's line spread function. The problem is reduced to solution of a system of linear-nonlinear equations (SLNE) with respect to intensities and frequencies of the discrete spectral lines. The SLNE is linear with respect to lines' intensities and nonlinear with respect to the lines' frequencies. The integral approximation algorithm is proposed for the solution of this SLNE. The algorithm combines solution of linear integral equations with solution of a system of linear algebraic equations and avoids nonlinear equations. Numerical examples of the application of the technique, both to synthetic and experimental spectra, demonstrate the efficacy of the proposed approach in enabling an effective enhancement of the spectrometer's resolution., Comment: submitted to Applied Spectroscopy Journal

The direct gravimetry problem is solved by dividing each deposit body into a set of vertical adjoining bars, whereas in the inverse problem, each deposit body is modelled by a homogeneous ellipsoid of revolution (spheroid). Well-known formulae for the z-component of gravitational intensity for a spheroid are transformed to a convenient form. Parameters of a spheroid are determined by minimizing the Tikhonov smoothing functional with constraints on the parameters, which makes the ill-posed inverse problem by unique and stable. The Bulakh algorithm for initial estimating the depth and mass of a deposit is modified. The proposed technique is illustrated by numerical model examples of deposits in the form of two and five bodies. The inverse gravimetry problem is interpreted as a gravitational tomography problem or, in other words, as "introscopy" of Earth's crust and mantle., Comment: 24 pages, 12 figures. arXiv admin note: text overlap with arXiv:1508.04410

We propose the generalized quadrature methods for numerical solution of singular integral equation of Abel type. We overcome the singularity using the analytical calculation of the singular integral expression. The problem of solution of singular integral equation is reduced to nonsingular system of linear algebraic equations without shift meshes techniques employment. We also propose generalized quadrature method for solution of Abel equation using the singular integral. Relaxed errors bounds are derived. In order to improve the accuracy we use Tikhonov regularization method. We demonstrate the efficiency of proposed techniques on infrared tomography problem. Numerical experiments show that it make sense to apply regularization in case of highly noisy sources only. That is due to the fact that singular integral equations enjoy selfregularization property.

The direct gravimetry problem is solved using the subdivision of each body of a deposit into a set of vertical adjoining bars, and in the inverse problem each body of a deposit is modeled by a uniform ellipsoid of revolution (spheroid). Well-known formulas for z-component of gravitational intensity of a spheroid are transformed to a convenient form. Parameters of a spheroid are determined by minimizing the Tikhonov smoothing functional using constraints on the parameters. This makes the ill-posed inverse problem by unique and stable. The Bulakh algorithm for initial estimating the depth and mass of a deposit is modified. The technique is illustrated by numerical model examples of deposits in the form of two and five bodies. The inverse gravimetry problem is interpreted as a gravitational tomography problem or the intravision of the Earth's crust and mantle., Comment: 20 pages, 12 figures