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The problem of determining the parameters of a large system of non-autonomous differential equations consisting of subsystems connected in an arbitrary order by non-local boundary conditions is solved. Unknown parameters participate both in differential equations and in boundary conditions. The problem under study is reduced to a parametric optimal control problem with a mean-square residual criterion estimating the degree of non-fulfillment of additionally specified boundary conditions. To apply first-order numerical methods, analytical formulas for the components of the gradient of the objective functional in the space of optimized parameters are obtained. The analysis of the obtained results of computer experiments is made using a test problem as an example.

This paper aims to provide an efficient numerical method based on the second Chebyshev wavelets for solving the fractional Langevin equation. Applying this operational matrix of fractional-order integration of second Chebyshev wavelets converts the original problem into a system of algebraic equations, which could be solved by the Newton method. After analyzing the method, the error bound is estimated. Moreover, the method's efficiency through a few numerical examples is evaluated.

We consider nonlinear dynamical systems as a model of interaction of components of a gene network which regulates early stages of an embryonic stem cells state. A parametric analysis of these dynamical systems is performed in order to describe the (non)uniqueness and stability of their equilibriums. We have obtained a criterion of existence of periodic trajectories near these points and localized these oscillations on the phase portraits of dynamical systems which describe these processes. A special software for cloud numerical experiments with these systems has been elaborated.

This article explores the computational intricacies of H. Rutishauser's quotient-difference (Q-D) algorithm and C programming code, a revolutionary advancement in polynomial analysis. Our specific focus is on cubic polynomials featuring absolute, distinct non-zero real roots, emphasizing the algorithm's distinctive capability to simultaneously approximate all zeros independently of external data. Notably, it proves invaluable in diverse domains, such as determining continuous fraction representations for meromorphic functions and serving as a powerful tool in complex analysis for the direct localization of poles and zeros. To bring this innovation into practice, the article introduces a meticulously crafted C language program, complete with a comprehensive algorithm and flowchart. Supported by illustrative examples, this implementation underscores the algorithm's robustness and effectiveness across various real-world scenarios.

In this paper, a linear second-order finite difference scheme is proposed for the Allen-Cahn equation with a general positive mobility. The Crank-Nicolson scheme and Taylor's formula are used for temporal discretization, and the central finite difference method is used for spatial approximation. The discrete maximum bound principle (MBP), the discrete energy stability and L^∞-norm error estimation are discussed, respectively. Finally, some numerical examples are presented to verify our theoretical results.

We present in this work a convergence analysis of a Finite Difference method for solving on quadrilateral meshes 2D-flow problems in homogeneous porous media with a full permeability tensor. We start with the derivation of the discrete problem by using our finite difference formula for a mixed derivative of second order. A result of existence and uniqueness of the solution for that problem is given via the positive definiteness of its associated matrix. Their theoretical properties, namely, stability on the one hand (with the associated discrete energy norm) and error estimates (with L²-norm, relative L²-norm and L^∞-norm ) are investigated. Numerical simulations are shown.

The manuscript presents the results of an application of a numerical method to solve one-dimensional hyperbolic equations. These equations simulate the dynamics of a liquid in a pipe with varying cross-sections. The equations are written in terms of pressure-head and discharge. Radial-basis functions and least-squares optimization are used for the numerical simulation. This numerical method is specialized for working with arbitrary nodal distribution in the problem domain. The basics of the application of the numerical method were introduced in our previous work. In the current work, we updated previously applied methods by means of getting rid of the time-marching approach and applying another adaptive refinement technique. Three cases of the simulations of the reservoir-pipe-valve system are described, indicating that the sharp time-gradient phenomenon is reproduced by the model.

The paper presents a Symmetric Hyperbolic Thermodynamically Compatible model of a saturated porous medium for the case of finite deformations and its linearisation for the description of small amplitude seismic wave fields in porous media saturated with fluid. The model allows us to describe wave processes for different phase states of the saturating fluid during its transition from solid to liquid state, for example during thawing of permafrost and decomposition of gas hydrates under the influence of temperature. To numerically solve the governing equations of the model, a finite difference method on staggered grids has been developed. It was used to perform test calculations for a model of the medium containing a layer of gas hydrate in a homogeneous elastic medium. The study showed that the characteristics of the wave fields in saturated porous media depend significantly on the porosity, which varies with temperature.

This work provides a high accuracy analysis of the reduced Adini Stokes element method developed in [7] for the Brinkman model. We show that this method is uniformly convergent for the velocity with convergence order O(h ²) in a mesh- and parameter-dependent norm over general quasi-uniform rectangular meshes. A proper postprocessing technique is also proposed to improve the precision of the pressure. Numerical examples confirm our theory.

The article investigates the application of hyperbolization for parabolic equations to the material and thermal balances' equations for a mathematical model of oxidative regeneration of a spherical catalyst grain with detailed kinetics. The initial spherical grain model is constructed using a diffusion approach. It is a nonlinear system of differential equations in a spherical coordinate system. The material balance of the gas phase is described by diffusion-convection-reaction equations with source terms compiled for concentrations of substances of the gas phase; the balance of the solid phase is represented by nonlinear ordinary differential equations. The thermal balance equation of the catalyst grain is the thermal conductivity equation with an inhomogeneous term corresponding to the grain heating during a chemical reaction. Slow processes of heat and mass transfer in combination with fast chemical reactions lead to significant difficulties in the development of a computational algorithm. Hyperbolization of the parabolic equations is applied to avoid the computational complication. It consists in the introduction of a second time derivative multiplied by a small parameter, in order to expand the stability area of the computational algorithm. An explicit three-layer difference scheme is constructed for the modified model. It is implemented in the form of a software module. The convergence analysis of the developed algorithm is presented. A comparative analysis of the new computational algorithm with the previously constructed one is carried out. The advantage of the new algorithm while maintaining the order of accuracy is shown. The result of the implemented new algorithm is the profiles of the distribution of temperature and substances along the radius of the catalyst grain.