Nekrasov matrices, as a subclass of H -matrices, exhibit unique functionalities and hold significant importance in diverse fields. These matrices are characterized by their distinctive structure, which endows them with numerous remarkable properties. Our paper extends the study of Nekrasov matrices by introducing a new subclass of H -matrices, termed DN -matrices, and explores their properties, particularly focusing on the Schur complement.
Two-dimensional multi-term time-space fractional diffusion-wave equations are considered. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. We also prove the numerical stability and convergence of the developed scheme and that the error is O(τ^2 + N^γ - r) , where N, τ, γ, r are the polynomial degree, time step size, Riesz derivative order, and the regularity of the exact solution, respectively.
This paper considers an algorithm for numerical solution of a one-dimensional inverse magnetotelluric sounding problem. The algorithm is based on a special type of algebraic equation which is called a pseudo-quadratic equation. The inverse problem is considered in three variants: 1) for media with fixed geometry; 2) for media with fixed geoelectrical properties; 3) general case. Additionally, an algorithm is proposed for input data processing which provides the existence of a solution to the inverse problem. Numerical experiments realized on test media with different sets of parameters are carried out to study and illustrate the efficiency of the algorithms.
This paper is devoted to modeling the propagation of seismic waves in a chemically inert elastically deformable rock. Only changes in stress and pore pressure are considered, and the chemistry of the saturating pore fluid does not directly affect the deformation of the rock. Chemical effects are taken into account by changing pore pressure and rock deformation in the transport equations. In the numerical solution of the problem under consideration, an algorithm is used to combine a Laguerre integral transform method and a finite difference method. The results of modeling the transport of a dissolved substance through a semi-permeable clay shale are presented.
A continuous-discrete stochastic model describing the dynamics of a spatially heterogeneous population is presented. The individuals of the population are located in a system consisting of two interconnected compartments. The individuals move between the compartments along unidirectional pipes. The duration of individual motion along the pipes is specified by constants or functions depending on time. The individuals located in the second compartment can contact one of the reproduction centers located in this compartment. As a result of contact with a reproduction center, an individual begins the process of fission. The reproduction of the individuals arising due to fission occurs until the number of descendants exceeds a threshold level; otherwise the reproduction of individuals ends. The population formed after the completion of fission contains descendant individuals that are not subject to fission and leave the system over time. The assumptions of the model are formulated, a probabilistic formalization of the model and an algorithm of numerical modeling based on a Monte Carlo method are given. The results of a computational experiment to simulate the dynamics of the population versus the parameters of the model are presented.
A simple computer-efficient grid model for an isotropic random field is developed to provide sufficiently high accuracy of solutions to stochastic transport problems for small correlation lengths. Results of test estimation of the time asymptotics of mean particle flux in a multiplying random medium are presented.
In this paper, we explore a double-step method for solving nonlinear equations containing differentiable and non-differentiable operators. Our approach is based on a combination of three different methods. We have analyzed the local convergence of the suggested method, considering both Lipschitz and L-average conditions, and established the superquadratic ( ≈ 2.414 ) order of convergence. Finally, we have pictured numerical results that are compared with those obtained by using several existing methods.
In this paper we consider a nonlinear fractional differential equations involving the new Caputo–Fabrizio derivative of order γ∈ ] 1pt1,2 1pt [ . We convert the fractional problem to an equivalent nonlinear Volterra integro-differential equation of the second kind, then we investigate the existence and uniqueness of its solution under certain given conditions by using the Schauder fixed point theorem. Finally, we numerically solve the proposed fractional problem by applying the Nyström method, and we provide some suitable examples to support our study.
Conjugate gradient methods represent a powerful class of optimization algorithms known for their efficiency and versatility. In this research, we delve into the optimization of the generalized descent symmetrical Hestenes–Stiefel (GDSHS) algorithm by refining the parameter c , a critical factor in its performance. We employ both analytical and numerical methodologies to estimate an optimal range for c . Through comprehensive numerical experiments, we investigate the impact of different values of c on the algorithm’s convergence behavior and computational efficiency. Comparative analyses are conducted between GDSHS variants with varying c values and established conjugate gradient methods such as Fletcher–Reeves (FR) and Polak–Ribière–Polyak ( PRP^ + ). Our findings underscore the significance of setting c = 1 , which significantly enhances the GDSHS algorithm’s convergence properties and computational performance, positioning it as a competitive choice among state-of-the-art optimization techniques.
The aim of this paper is to derive efficient numerical algorithms for the numerical solution of nonstiff ordinary differential equations by applying the Richardson extrapolation technique to a class of explicit two-derivative Runge–Kutta methods. Theoretical results illustrate that application of this technique has considerable impact on the accuracy and stability properties of the underlying numerical methods. The achieved improvements for the proposed algorithms are also confirmed by the results of some numerical experiments.
An emerging field of study is the application of fractional calculus to iteratively solve the nonlinear equations. Recently, several Newton-type techniques have been proposed that make use of the notion of fractional order derivatives. However, the existence of at least first order derivative is essentially required for the convergence of these methods. On the contrary, we propose a new secant-type method, which is inherently derivative-free, although its construction is based on the idea of conformable fractional derivative of order α∈ (0,1] . The primary objective for the development is to analyze how fractional derivatives have the effect of enlarging the convergence domain. In this regard, the proposed scheme is examined for its convergence characteristics and dynamical features for different values of α in the specified range. Furthermore, the efficacy of method is demonstrated through solving various applied nonlinear problems including the fractional order Burgers’ equation.
In this paper, we will give a new Crank–Nicolson mixed covolume method for parabolic optimal control problems. The state and costate variables are approximated by the lowest order Raviart–Thomas element and the control variable is approximated by piecewise constant function, while Crank–Nicolson scheme is ultilized for temporal discretization. We derive the priori error estimates for the control variable, the state and the costate variables.
A numerical solution by the finite element method of a homogeneous Dirichlet boundary value problem for an elliptic equation is examined (using a Poisson equation as an example) in a two-dimensional convex polygonal domain Ω with a singular right-hand side given by the Dirac delta function. A theorem on the existence and uniqueness of a generalized solution in the fractional Sobolev space H^s(Ω ) , 1/2 < s < 1 , is proved. An approach to discrete analysis of the problem using the finite element method is proposed and investigated. The results of numerical experiments for a model problem, obtained using the FreeFem++ software, are presented. They confirm the error estimate of the difference between the discrete and exact solutions derived in the paper.
This paper presents a two-grid method for solving nonlinear time fractional diffusion equations (TFDEs). First, a fully discrete scheme is constructed by using P-0(2)-P-1 mixed finite elements (MFEs) and L1 formula for spatial and temporal discretization, respectively. Second, the stability and error of the fully discrete scheme are analyzed. Third, a two-grid algorithm (TGA) based on the fully discrete scheme is proposed and its stability and error analysis results are derived. Finally, some numerical examples are provided to support the theoretical results.
Rational techniques for verifying the congruence of complex matrices are discussed. An algorithm is said to be rational if it is finite and uses arithmetical operations only. An important part in verifying the congruence of nonsingular matrices play their cosquares. The verification gets complicated if there are eigenvalues of modulus 1 in the spectrum of cosquares; this is especially true if such eigenvalues are defective. In this direction, the most advanced result is the rational algorithm for matrices A and B whose cosquare is the direct sum J(m)(1) circle plus J(m)(1). Here, this algorithm is extended to the case where the cosquare is the direct sum of two Jordan blocks of distinct orders. This extension is heavily dependent on additional facts concerning the solutions to the matrix equation X - J(m)(inverted perpendicular)(1)XJ(m)(1) = 0, which are found in the present paper.
During dynamic loading, ice demonstrates complex nonlinear behavior which depends on many factors including its strain rate. In practical applications, during the low-speed collision, ice exhibits both viscous and brittle properties. To consider the specifics of local ice failure, a compound model is proposed in this paper, which distinguishes a hydrostatic core and an elastoplastic zone in ice, with the material far from the impact area being in an elastic state. Additionally, volumetric cracking is considered. The model is verified by comparing the results of numerical computations and a laboratory experiment with a spherical indenter. The numerical results demonstrate various phenomena observed in the experiments. The simulations reconstruct nonlinear waves, different destruction patterns, and demonstrate the wave nature of fracturing. The deformation curves calculated confirm the possibility of a qualitative description of ice behavior during the main stage of the collision.
A morphing algorithm included in a three-dimensional structured grid generation technology designed for the numerical solution of differential equations modeling vortex processes in multi-component hydrodynamics is described. The algorithm is intended for the generation of structured grids of a special topology in volumes obtained by deformation of volumes of revolution by bodies formed by surfaces of revolution with parallel axes. The algorithm is developed by using a variational approach for constructing optimal grids and is a non-stationary one: at each iteration the form of the domain and the grid for it are deformed. Then the grid is optimized in accordance with the following optimality criterion: the closeness of the grid to a uniform and orthogonal one. The iterations are continued up to a given degree of deformation. The algorithm allows one to construct grids in domains of very complex geometry, and it is not necessary to describe the boundary of a complex domain, it is sufficient to describe the volume of revolution, the deforming volume, and the parameters of deformation. Examples of grid calculations are given.
Some scheduling problems taking into account energy consumption are considered. Such problems arise in multiprocessor computer systems and take into account resource constraints and parallelization capabilities. For these problems, some algorithms of greedy and list types with guaranteed accuracy estimates in the worst case are known. In this paper, we propose an adaptive genetic algorithm with decoding solutions based on the specifics of the problem statements. A peculiarity is that the crossover operator solves a problem of optimal recombination in full and truncated versions. The call of the crossover operators is implemented adaptively. The categorical and numerical parameters are adjusted adaptively by using modern packages. The results of an experimental study show a statistically significant advantage over the known algorithms on a series of problems of different structure.
Emden–Fowler equations are widely used in mathematical and physical modeling. They describe phenomena in various fields, including astrophysics, quantum mechanics, and nonlinear dynamics. Applications range from modeling stars’ thermal behavior to species’ distribution in a chemical reaction. Researchers continuously seek new methods to solve Emden–Fowler (EF) equations more efficiently and accurately due to their versatility and richness. This article presents a novel approach for solving the generalized EF equations subject to boundary conditions using the Legendre wavelet. First, we convert the problem into equivalent Fredholm integral equations. Next, we use the Legendre wavelet collocation approach and the Newton–Raphson iterative technique to solve the resulting integral equations. The formulation of the proposed algorithm is further supported by its convergence and error analysis. We examine the accuracy of the method by computing the numerical solution and errors for various examples. We compare our numerical outcomes to the exact solution and those achieved by techniques in the literature, such as the Haar wavelet and the optimal homotopy analysis method. The Legendre wavelet collocation method offers superior accuracy with fewer collocation points, making it advantageous.
