The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to show the accuracy of the mathematical methods which is based on the floating point arithmetic, we apply the stochastic arithmetic and new condition to study the efficiency of the method which is based on two successive approximations. Thus the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are employed. Finding the optimal iteration of the method, optimal approximation and the optimal error are some of advantages of the stochastic arithmetic, the CESTAC method and the CADNA library in comparison with the floating point arithmetic and usual packages. The theorems are proved to show the convergence analysis of the Adomian decomposition method for solving the mentioned problem. Also, the main theorem of the CESTAC method is presented which shows the equality between the number of common significant digits between exact and approximate solutions and two successive approximations.This makes in possible to apply the new termination criterion instead of absolute error. Several examples in both linear and nonlinear cases are solved and the numerical results for the stochastic arithmetic and the floating-point arithmetic are compared to demonstrate the accuracy of the novel method.

This paper solves the problem of eliminating smearing, defocusing, and noise of aerospace images (pictures) of the earth's surface obtained from remote probing. The type of distortion (smearing or defocusing) is determined by a modified spectral method (with the values of the distortion parameters determined from original derived formulas). The accuracy of the image reconstruction is enhanced by solving integral equations (an ill-posed problem) to determine the type of distortion and to estimate the distortion parameters. A new noise model-multipolar pulse noise-which is more adequate than bipolar pulse noise is proposed, along with a filter for filtering it out. It is shown that the image-reconstruction error can depend on the sequence in which the smear/defocusing and noise are eliminated. Results are presented for processing distorted pictures of a certain section of the earth's surface. (C) 2018 Optical Society of America.

The inverse problem of spectroscopy—reconstruction of continuous spectra by solving the Fredholm integral equation of the first kind (an ill-posed problem)—is considered. The equation is solved by the Tikhonov regularization method using the method of computational experiments, according to which, along with initial example P, where experimental spectrum u is specified and true spectrum z is sought for, “similar” model example spectrum Q (or several examples) with specified z and modeled u is processed with allowance for additional information on true spectrum z in example P. This approach makes it possible to choose regularization parameter α and estimate the error in reconstructing spectrum z in example P. Some numerical illustrations are presented. Different response functions of spectrometers are considered: slotlike, triangular, diffraction, Gaussian, dispersion, and exponential; identical widths a(λ) at a level of 0.5 and identical integral widths W(λ) are used.

Several methods of separation of continuous overlapping spectral lines are compared. In the majority of methods, the profile of each line is modeled by a Gaussian or Lorentzian, and total measured...