This research presents a comparative analysis of numerical methods for solving first-kind Fredholm integral equations using the Bubnov–Galerkin method with various wavelet and orthogonal polynomial bases. The bases considered are constructed from Legendre, Laguerre, Chebyshev, and Hermite wavelets, as well as Alpert multiwavelets and CAS wavelets. The effectiveness of these bases is evaluated by measuring errors relative to known analytical solutions at different discretization levels. Results show that global orthogonal systems—particularly the Chebyshev and Hermite—achieve the lowest error norms for smooth target functions. CAS wavelets, due to their localized and oscillatory nature, produce higher errors, though their accuracy improves with finer discretization. The analysis has been extended to incorporate perturbations in the form of additive noise, enabling a rigorous assessment of the method’s stability with respect to different wavelet bases. This approach provides insight into the robustness of the numerical scheme under data uncertainty and highlights the sensitivity of each basis to noise-induced errors.

This article addresses a two-dimensional inverse problem of magnetotelluric sounding under the assumption of E-polarized electromagnetic fields. The main focus is on the construction of a forward numerical model based on the Helmholtz equation with a complex coefficient, and the recovery of electrical conductivity from boundary measurements. The second-order finite difference method is employed for numerical simulation, providing stable approximations of both the direct and the conjugate problems. The inverse problem is formulated as a minimization of a data misfit functional, and solved using Nesterov’s accelerated gradient descent method, which ensures fast convergence and robustness to noise. Numerical experiments are presented for a synthetic model featuring a smooth background conductivity and a localized anomaly. Comparison between the exact and reconstructed solutions demonstrates the high accuracy and efficiency of the proposed algorithm. The developed approach can serve as a foundation for constructing practical inversion schemes in geophysical exploration problems.

This paper investigates a modification of the fictitious domain method with continuation in the lower-order coefficients for the unsteady Navier-Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded 2D or 3D domain. The modification enables {a solution-dependent} choice of the critical parameter. Global-in-time existence and convergence of a weak solution to the auxiliary problem are proved, and local-in-time existence and convergence of a unique strong solution are established. For the strong solution, a new higher-order convergence rate estimate in the penalization parameter is obtained. The introduced framework allows us to apply a pointwise divergence free finite element method as a discretization technique, leading to strongly mass conservative discrete fictitious domain method. A numerical example illustrates the performance of the method.

In this research, surface Au concentration measurements are considered as a spatially correlated geochemical field associated with deep occurrences of disturbing masses using real geological exploration data from the Novo-Khairuzovsky gold deposit in East Kazakhstan. The approach is based on the relationship between potential-field continuation problems and reconstruction of subsurface geological anomalies from surface observations. The considered approaches include Tikhonov and Lavrentiev regularization, SVD, and TSVD. Special attention is given to regularization parameter selection using the L-curve method, Morozov discrepancy principle, and GCV. Comparative computational analysis is performed to evaluate the accuracy, stability, and efficiency of these methods in solving first-kind Fredholm integral equations. Results are assessed using error metrics and spatial visualization of reconstructed fields within a Geographic Information System (ArcGIS), enabling consistent geospatial interpretation. Results show that Lavrentiev regularization with L-curve criterion provides the most stable and reliable reconstruction across all depths, achieving high correlations ($R=0.8876$ at 100 m and $R=0.8049$ at 200 m) with low reconstruction errors. Tikhonov regularization performs acceptably at 100 m but becomes less stable at greater depths. Among spectral methods, TSVD improves stability compared with classical SVD, while standard SVD shows weak correlations and larger reconstruction errors due to high noise sensitivity.