Home – Home – Jornals – Numerical Analysis and Applications 2026 number 2

Delaunay triangulation and Voronoi partitioning are used to construct computational grids in numerical methods such as the finite element method and the finite volume method. A two-grid technique is considered that utilizes both the nodes of a Delaunay triangulation and the vertices of a Voronoi partitioning. This approach makes it possible to construct operator-difference approximations of the vector calculus operators (gradient, divergence, and curl) on a merged MVD (merged Voronoi-Delaunay) grid which consists of orthodiagonal quadrilaterals. The paper investigates an application of MVD grids for finite element analysis of two-dimensional boundary value problems using a Dirichlet problem for an elliptic equation in an anisotropic medium as an example. Two approaches are examined: Delaunay triangulation with additional Voronoi vertices as extra nodes and direct application of MVD grids. The results of computations on a sequence of progressively refined meshes employing different types of finite elements are presented.

In this paper, we consider problems of obtaining upper bounds for the components of root-mean-square errors for computational constructions of approximation of an unknown probability density for a given sample. Examples are the computer functional kernel and projection algorithms as well as their important special case - the multidimensional analog of the frequency polygon. These bounds are then used in choosing such versions of kernel and projection algorithms that provide a given level of error in a density approximation.

This study propounded a numerical approach for solving a mathematical model of pollutant spread through forest resources using shifted orthogonal Bernoulli polynomials (OBPs). The model is based on a system of ordinary differential equations, which is transformed into an algebraic system using the collocation approach based on shifted OBPs. Newton's method is employed to obtain numerical solutions, and the results are compared with those obtained using the Runge-Kutta method of fourth-order (RK4) to demonstrate the efficiency of the proposed method. The results demonstrate good agreement with the RK4 method, indicating the proposed method's acceptability for modeling pollutant spread through forest resources.

The article studies a problem of optimal control of heating for a homogeneous half-line. The heating problem is set for a heat conduction equation defined on a half-line where the temperature tends to zero at infinity. At the origin of the spatial coordinate a heat flux, i. e. a non-homogeneous boundary condition of the second kind, is given. The heat flux is modeled using a heating function which is a continuous broken line. This choice of the function is explained by the properties of the technical device under study. The article proves the existence of a solution to such a problem and the uniqueness of its classical solution with a certain error. The optimality of the heating control in this paper means that at the boundary x=0 the temperature at any time is maximum permissible (or, in the first time interval, maximum possible), and at the same time does not exceed a certain critical value, which is chosen to be equal to 1. For the optimal control, a recurrence formula is found at different times, it is proven that this is exactly the optimal solution. That is, at large values of the heat flux the critical temperature at the boundary will be exceeded at some point in time, and at the smaller values the temperature will be lower than that allowed by the material. It is also proven that the heat flux found is the exact upper bound of all admissible heat fluxes for a given discrete control and that such a flux is unique.

The real life problems related to engineering and physical systems are theoretically studied using mathematical models and are generally formulated using linear and non-linear differential equations. This work proposes a numerical technique to find approximate solutions of differential equations utilizing polynomials as base approximation functions and metaheuristic optimization algorithms like Differential Evolution (DE) and Particle Swarm Optimization (PSO) for obtaining the optimal values of coefficients of the polynomials in order to get the desired approximate solution. The algorithms for the proposed method have been executed using MATLAB for computer programming. The effectiveness of the approach suggested in this paper is found to be better than or at least comparable to other numerical methods suggested earlier for solving differential equations.

Numerous higher-order iterative methods have been proposed in the literature for locating roots of nonlinear equations. Among these, methods with optimal order are of particular interest due to their superior efficiency. However, not all of them exhibit consistent performance across all scenarios. Some offer low accuracy, while others suffer from slow convergence, yet there are methods that fail to maintain the desired convergence order in certain applications. This paper aims to address these shortcomings. Consequently, we introduce a novel three-point iterative scheme, whose formulation is based on the widely used two-point King's fourth-order method. This scheme attains eighth-order convergence at the cost of four function evaluations per step. As such, it is optimal according to the Kung-Traub conjecture and boasts an efficiency index of 1.682, which surpasses that of Newton's method and many other higher-order techniques. To assess the methods' performance and validate its theoretical properties, we present several numerical examples. Furthermore, we provide a detailed analysis of the complex dynamics through graphical representations of the basins of convergence, comparing our method with those of other established techniques. The computational results and convergence visualizations confirm that our scheme outperforms existing methods in the literature.

In this article, we are interested in the complex version of Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation for grayscale image inpainting as well as the multi-component Cahn-Hilliard systems for image inpainting, that is an extension approach for color image inpainting. We have studied well-posedness of the steady state problem associated to the complex Bertozzi-Esedoglu-Gillete-Cahn-Hilliard equation as well as to Bertozzi-Esedoglu-Gillete-Cahn Hilliard systems. Then, Backward Euler discretization on time has been considered in both models mentioned above. We were able to prove the stability of the backward Euler scheme. Finally, we do some numerical simulations that confirm the theoretical results and show the efficiency of the scheme. The simulations were done using FreeFem++.

A method is proposed for restoring two coefficients in a mixed boundary value problem for an inhomogeneous hyperbolic partial differential equation of the second order based on additional information about the solution of the boundary value problem for a given fixed value of the spatial argument of the solution. The problem simulates the propagation of small transverse vibrations of a finite string with an end affected by the gravity of a body with varying mass. The proposed algorithm provides a sequential restoration of the multiplier in the heterogeneity of the oscillation equation and the coefficient in the nonclassical boundary condition based on the values of one additional function of one argument.