The paper examines a difference equation, analogous to the Helmholtz equation, which describes the point vortex dynamics on a plane under control. The dependence of the asymptotic behaviour of the moment of vorticity of a system of point vortices on the control parameters is established. In the presence of random perturbations, for a system of point vortices the parameter estimation problem is solved. An optimal control allowing the maintenance of two point vortices on a specified rotational orbit under minor changes in the initial control parameters has been found.
We study minimax estimation of functionals of solutions to parabolic problems under nonlinear observations with rapidly oscillating coefficients. An averaging approach is developed, the existence of a minimax estimator is proved, and the convergence of the estimation error to that of the averaged problem is established.
To model highly oscillatory processes (for example, processes determined by the rhythms of electroencephalograms (EEG) measured at certain points in the cerebral cortex), a special system of neural singular ordinary differential equations (NSODE), the trajectories of which exhibit chaotic behavior for certain parameter values, has been designed. Further, a special algorithm for minimizing the loss function was used to adjust the parameters of the NSODE system.The presented algorithm demonstrates an effective approach to the identification of parameters of nonlinear dynamical systems of the NSODE type in the context of EEG modeling. The success of the approach is due to the synergistic combination of numerical optimization methods, including the regularized Gauss-Newton method with SVD decomposition of the corresponding matrices, adaptive tuning of optimisation parameters and complex control of numerical stability. Also important is the multi-level analysis of the quality of the results, which includes both traditional statistical metrics and modern visualization methods. The obtained modeling results were then used to predict epileptic seizures in patients under observation.
An algorithm for solving a class of linear variable order fractional partial differential equations (FPDEs) is presented in this paper. We utilized a combination of Caputo fractional derivatives with the Haar wavelet collocation method (HWCM) and Eiler–Fouirier method to solve linear variable order FPDEs. Fractional calculus is a more complex form of standard mathematical calculus. FPDEs is an equation that uses a fractional derivative (FD) by t > 0 (time variable) and two derivatives by x ∈ R (speciousness variable). FPDEs are used to model a wide range of phenomena in the fields of science, electric circuits, engineering and mechanical systems. In this paper we study solvability of the Cauchy and boundary value problem is constructed for FPDE. Parabolic pseudodifferential equations this pseudodifferential operator with noon-smooth symbols introduced by Eidel’man S. and Drin’ Ya. for the first time. The Haar wavelets, which are made up of piecewise constant functions, are the mathematically simplest of all the wavelet families. These wavelets have the advantage of being analytically integrated any number of times. In Haar wavelet two functions, sclaling and mother function, play an essential role. We may therefore draw the conclusion that the Haar wavelet collocation method and Eiler–Fouirier method is an effective method for finding the solution of linear VO FPDEs.
We propose a hybrid evolutionary-sparse optimization framework for the automated synthesis of polynomial Lyapunov functions certifying exponential stability of nonlinear autonomous systems on a prescribed compact domain. Candidate Lyapunov functions are encoded as height-bounded genetic programming (GP) expression trees restricted to addition and multiplication, which guarantees polynomial semantics and yields interpretable closed-form certificates. Each tree is expanded into a canonical polynomial representation and mapped to its induced monomial support. Conditioning on this support, we fit the coefficients using a constrained sparse-optimization routine that minimizes an empirical Lyapunov penalty together with a sparsity-promoting term, implemented through a staged proximal scheme based on firm thresholding. The coefficient-fitting step is exposed as a feasibility oracle that returns "feasible" precisely when the sampled quadratic bounds and dissipation inequality hold on the chosen sample set. We establish two complementary guarantees. First, under full-support regrow mutation, we derive an explicit finite-generation (hitting-time) bound for the GP outer loop: if at least one oraclefeasible template exists in the height-bounded search class, then, for any target confidence level, there exists a finite number of generations after which a feasible structure is found with at least that confidence; moreover, the non-discovery probability decreases geometrically in the number of generations, with rate governed by the offspring count, mutation probability, minimal regrow mass, and the number of feasible templates. Second, we provide deterministic sampling certificates on a compact set, showing that sufficiently small empirical violation implies satisfaction of the Lyapunov inequalities on the full domain. Numerical experiments on benchmark systems demonstrate that the method produces sparse, human-readable Lyapunov polynomials.
This paper serves as a continuation of previous research on approximating solutions for reaction-diffusion equations with multivalued interaction functions. While previous work established a Machine Learning framework using Physics-Informed Neural Networks (PINNs) and Pasch-Hausdorffregularization, this study investigates the underlying Monte Carlo integration techniques required to estimate the loss functionals for such non-unique, discontinuous problems. We analyze the convergence properties of Monte Carlo estimators applied to the regularized interaction terms and demonstrate that as the regularization parameter k → ∞, the variance of the standard Monte Carlo estimator grows unboundedly, necessitating advanced sampling strategies. We propose an Adaptive Importance Sampling approach to improve the accuracy of stochastic optimization in the presence of multivalued terms and provide rigorous error bounds for the proposed method.
We conducted computer modeling and finite element analysis of stress concentration near an elliptical hole in thin plates made of a functionally graded material (FGM) under uniaxial tension. We investigated the influence of the direction and law of plate material heterogeneity on the magnitude of the stress concentration coefficient. The article provides recommendations for a method for reducing stress concentration near the hole. Rational parameters of FGM plate heterogeneity for the considered problems were found, allowing the stress concentration coefficient to be reduced to ∼ 70%. A mechanical effect was also observed: using a FGM for a plate with the proposed law of elastic modulus variation leads to a reduction in both stress and strain intensity near the hole. The results of this study can be useful in the design and optimization of plate elements with holes in new equipment as an effective practical method for reducing stress concentration through the use of functionally graded materials with specific mechanical properties.
This paper presents a mathematical analysis of a deterministic model that explores the interactions between terrorism and counter-terrorism strategies. The model includes a ratio-dependent incidence rate, which enables tracking of changes as the proportion of terrorists relative to susceptibles rises. The dynamics of terrorism are described and investigated using a system of ordinary differential equations. The mathematical analysis of the model reveals two equilibrium states: a terror-free state and a terrorpersistent state, with stability assessed via the basic reproduction number R0. Both local and global stability behaviours of these equilibria are thoroughly examined. The findings reveal that the terror-free equilibrium is both locally and globally asymptotically stable when R0 < 1, indicating the conditions under which terrorism can be eradicated. Conversely, the terror-persistent equilibrium is locally asymptotically stable when R0 > 1, signifying the potential for sustained terrorism under certain conditions. Additionally, the terror-persistent state is globally asymptotically stable under specific scenarios, depending on model parameters. Finally, numerical simulations are conducted to validate the theoretical findings
This paper studies contractibility of m-efficient and weak m-efficient solution sets in the context of semi-infinite set optimization with respect to a partial set order relation. Sufficient conditions for starshapedness of the image and upper semicontinuity of the constraint set map have been obtained and further employed to examine the aforementioned contractibility by way of a nonlinear scalarization function. In the end, an application to uncertain vector optimization is provided to exemplify the utility of our results.
This paper presents a two-dimensional search algorithm for quasi-Newton methods applied to ill-conditioned and degenerate unconstrained optimization problems. At each iteration, the space is decomposed into an orthogonal sum of two subspaces based on the spectral decomposition of the approximate Hessian (updated via the BFGS or SR1 formula) and a regularization parameter. The search direction in one subspace is computed using a quasi-Newton scheme, while an alternative optimization method (e.g., gradient descent or conjugate gradient) is employed in the complementary subspace. The next iterate is obtained by minimizing a fourth-order local model of the objective function in two dimensions with respect to the step-size parameters along these directions. The proposed approach enables efficient handling of spectral degeneracy by combining curvature-aware and gradient-based updates within a unified framework. The efficiency of the proposed method is demonstrated through numerical experiments on standard test problems from unconstrained optimization and machine learning. The results are compared with implementations from widely used software environments, including R, Scilab, Python, and PyTorch.
In this paper, we propose a new version of a projection algorithm for the generalized variational inequalities problem (GVIP) involving a multi-valued mapping. In this work, we introduce a new combination of the main algorithmic steps proposed in projection methods that have been developed previously for classical or generalized variational inequalities problem with the aim of significantly reducing the computational cost. For the convergence, the assumptions required for the underlying mapping are continuity and a condition that is weaker than the pseudo-monotonicity. We show that this method is well-defined and its corresponding algorithm is globally convergent. Preliminary comparative experiments on several test problems to validate the effectiveness of the proposed method.
We investigate the long-term behavior of an epidemic using a stochastic SIRV (Susceptible-Infectious-Recovered-Vaccinated) model that incorporates time dependent parameters and random perturbations. The model is motivated by tuberculosis (TB) transmission dynamics and includes vaccination of newborns as a key intervention strategy. Random environmental fluctuations are modeled via Wiener processes, and the systems stochastic stability is analyzed. We derive conditions for the existence of a unique global solution and examine the extinction and persistence of the disease under stochastic influences. Numerical simulations are provided to illustrate the theoretical results and to explore the impact of parameter variability and noise intensity on the epidemic’s evolution. This work contributes to the understanding of how random disturbances affect long-term epidemic dynamics and can inform public health strategies under uncertainty. This project has received funding through the MSCA4Ukraine project (AvH ID:1233636), which is funded by the European Union.
This paper proposes an extension of Zoutendijk’s Method of Feasible Directions (MFD) for solving linearly constrained multi-objective optimization problems. The approach integrates a Hit-and-Run sampling technique to generate a diverse set of feasible points and uses an iterative two-step process. This process involves determining an improving feasible direction by solving a quadratic subproblem and adjusting the step size using the multi-objective Armijo rule. As a result, the method generates a sequence of feasible points that converge to a critical Pareto point, and under convexity assumptions, to a Pareto-optimal point. The process is applied to all points in the generated population, enabling an adequate approximation of the Pareto front. The convergence of the method is proven. A comparative study, conducted in MATLAB on 25 linearly constrained multi-objective optimization benchmark problems, evaluates our MFD against the NSGA-II algorithm and another variant of MFD that uses a different direction search subproblem and sampling procedure. The quality of the approximated Pareto front is assessed using four metrics: purity, spacing, hypervolume, and generalized spread. The results show that our MFD outperforms the other methods in terms of both convergence speed and the quality of the Pareto front for the tested problems
The method of reflected waves was extended to the finite vibrating string, subject to concentrated (at both its ends) and distributed (along its length) forces. To this end, the periodic odd continuation along the infinite space-time strip was implemented to all functions, influencing the above vibrations: 1) the initial, 2) the boundary (converted into the double layers) and 3) the right-hand side of the 1D non-homogeneous wave equation, resulting to a weak formulation of the IBVP, instead of the original strong one. The weak solution to the IBVP as the sum of regular and singular distributions was obtained by convolving the fundamental solution of the 1D wave operator and the above continued functions, similarly to obtaining the weak solution to the Cauchy problem for the 1D non-homogeneous wave equation. The weak solution to the IBVP was shown can be transformed (through some intermediate representations) into the well known strong one, obtained by the method of separation of variables. The opposite is also true, that is the strong solution to the IBVP can be transformed into a representation of the weak one by two successive integrations by parts, provided that the convergence of some Fourier series is interpreted in the weak sense (as distributions).
A new type of chaotic attractors, whose basin of attraction is the entire phase space, is considered. The main difference between these attractors and the known ones is that any trajectory starting from the basin of attraction first enters a unique transport channel (which is a straight line), and then the trajectory reaches the attractor itself along this channel. For any quadratic dynamic system generating the mentioned attractor, a new concept of a uniquely defined degenerate autonomous quadratic dynamic system with exactly one real double equilibrium point is introduced. It is shown that if the degenerate system exhibits chaotic behavior, then the original (non-degenerate) system also exhibits similar chaotic behavior.
In this paper, we introduce and study a modified inertial subgradient extragradient iterative algorithm for solving bilevel split quasimonotone variational inequality problems with a fixed point constraint of demimetric mappings in the framework of real Hilbert spaces. The method involves a strongly monotone operator at the upper-level problem and quasimonotone mapping at the lower level. We obtain a strong convergence result of the proposed method under some mild conditions on the algorithm parameters without the prior knowledge of the operator norm or the coefficient of the underlying operator in the scope of infinite dimensional real Hilbert spaces. Finally, some numerical demonstrations are given to illustrate the gains of this method. Our results generalize and improve some well-known results in the literature.
A nonlinear boundary value problem is considered for a system of parabolic equations in an inhomogeneous region, that requires conjugation conditions. The boundary value problem serves as a mathematical model of the nonlinear non-isothermal filtrationconsolidation process in a heterogeneous soil mass. A distinctive feature at the physical level is the application of a nonlinear Darcy’s law incorporating the threshold gradient, where this gradient depends on the thermal state of the porous medium. These features are also reflected in the conjugation conditions. The finite element method is applied to obtain approximate discontinuous solutions of the corresponding system of quasilinear parabolic equations. The existence and uniqueness of an approximate generalized solution are proven. Estimates for the accuracy of the finite element solutions are also established in terms of total approximation. A model test example is used to analyze the differences in pressure head and temperature distributions between the case studied in this article and the classical formulation.
We provide a mathematical and numerical analysis for solving an eigenvalue boundary value problems (Dirichlet and Neumann) for Helmholtz operator, which describes a regular shielded double-sided microstrip transmission line problem. The paper presents a new adapted method for numerical calculation of dispersion characteristics of this problem, and, in particular, balanced microstrip line problem. The boundary value problem for a regular planar transmission line is solved in a rigorous formulation considering the singularity of the behavior of the current density at the edges of the microstrips. The method of discretization by Galerkin approximations of the boundary value problem using an additional basis from Chebyshev polynomials of the 1st and 2nd kind was adapted for the case of a doubly connected domain (two electrodynamically coupled microstrip lines). In an explicit form, the matrix elements for the system of linear algebraic equations were obtained, the solution algorithm of which was implemented in a high-performance computing environment. The dispersion characteristics of a double-sided microstrip and balanced transmission line made on the RO3010 substrate material were calculated using the developed algorithm. As an example of using the obtained dispersion characteristics, a circuit was designed and the scattering characteristics of tapered transition with an exponential profile from a double-sided microstrip with a characteristic impedance Z0 = 50 Ohm to a balanced transmission line with a impedance of Z0 = 100 Ohm were obtained. According to the results of the analysis of exponential transitions with exponents n = 0.707 and n = 1.0 in the function of the strip width versus the longitudinal coordinate, the reflection coefficient is below |Γ| < 0.1.
A combined approach to identifying the coordinates of point impulse sources, using observation data at certain time intervals at characteristic points, has been transferred to the case of anisotropy. The corresponding mathematical model is constructed based on the assumption that the movement of particles in the domain is quasiideal and satisfies the generalization of Darcy’s law; at the same time, the diffusion component of the movement of pollutants is neglected. The initial problem is divided into two smaller ones: it is assumed to construct a hydrodynamic mesh and, based on this, to identify the coordinates of pollution sources. In the first case, numerical methods of quasiconformal mappings are applied. In the second case, based on the use of the method of characteristics with respect to the convection equation, an integral equation is constructed. Numerical experiments were conducted, confirming the effectiveness of the previously developed approach in the case of anisotropy. The most significant residuals between the a priori known and calculated coordinates of pollution sources occur on those streamlines that pass near stagnant zones and intersect areas in the vicinity of so-called “key” points (critical points of quasiconformal mapping violation) at the boundary of the domain. In certain cases of filtration tensor distribution, this can have a significant negative impact on the accuracy of the solution.
