Journal of Optimization, Differential Equations and their Applications (JODEA - Dnipro National Universuty) / Журнал з оптимізації, диференціальних рівнянь та їх застосувань
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Galerkin Approximation of a Shielded Double-Sided Microstrip Line with Boundary Singularities
Get PDFWe 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
The Influence of Non-Isothermal Conditions on Pressure Head Jumps in the Nonlinear Consolidation Problem in the Presence of Geobarriers
Get PDFA 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
On Systems of Neural ODEs with Generalized Power Activation Functions
Get PDFWhen constructing neural network-based models, it is common practice to use time-tested activation functions such as the hyperbolic tangent, the sigmoid or the ReLU functions. These choices, however, may be suboptimal. The hyperbolic tangent and the sigmoid functions are differentiable but bounded, which can lead to vanishing gradient problem. The ReLU is not bounded but is not differentiable in the point 0, which may lead to suboptimal training in some optimizers. One can attempt to use sigmoid-like functions like the cubic root, but it is also not differentiable in the point 0. One activation function that is often overlooked is the identity function. Even though it doesn’t induce nonlinear behavior in the model by itself, it can help build more explainable models more quickly due to non-existent cost of its evaluation, while the non-linearities can be provided by the model’s evaluation rule. In this article, we explore the use of specially-designed unbounded differentiable generalized power activation function, the identity function, and their combinations for approximating univariate time series data with neural ordinary differential equations. Examples are given
On PDE Formulation of the Relaxed Version of Generalized Active Contour Model in Anisotropic Sobolev Space
Get PDFMostly motivated by the image segmentation problem and its applications to the extraction of agricultural crop fields from satellite data, we propose a new PDE (partial differential equation) formulation for the piecewise constant approximation of satellite images. Our PDE approach is aimed at obtaining a relaxed version of active contour model with sufficiently fast segmentation process and with a flexible initialization option
Generalized Active Contour Model in an Anisotropic Variable Exponent Sobolev Space
Get PDFMostly motivated by the practical applications especially in the field of satellite remote sensing of agricultural territories, we develop a novel approach to the domain decomposition basing on the anisotropic version of the Chan-Vese active contour model. With that in mind we propose a new statement of this problem in variable Sobolev spaces, provide a rigorous mathematical analysis of the proposed optimization problem, establish sufficient conditions of its consistency and solvability, show that the objective functional is Gateaux differentiable, and derive the corresponding optimality conditions. To illustrate the validity of the obtained results, we give some examples of numerical simulations with the real satellite images
Pulse and Point Optimal Control of Linear Hyperbolic Integro-Differential Systems with Mixed Boundary Conditions
Get PDFWe investigate an optimal control problem for a linear hyperbolic integrodifferential equation, subject to Robin-type boundary conditions. The control operates as an operator on the equation’s right-hand side, integrating Dirac delta functions concerning temporal or spatial variables. A priori inequalities in negative norms are established for the boundary value problem (BVP) operator. From these findings, we derive theorems on the problem’s well-posedness and the existence of optimal control across multiple scenarios, subject to specific assumptions regarding the admissible control set and the quality criterion
Study of the Dynamics of Product Sales Process with the Help of Zolotas Model
Get PDFA new mathematical model describing the dynamics of sales of goods on the market has been proposed. This model takes into account the following characteristics: the average welfare of buyers, the maximum welfare of buyers, the number of buyers who know about the incoming product and the level of market saturation with this product. Examples demonstrating the features of the constructed model are given
On Uniqueness and Continuous Dependence of the Weak Solutions of Neutral Stochastic Functional Differential Equations in Infinite Dimensional Spaces
Get PDFIn this paper we establish uniqueness of the weak solutions of nonlinear stochastic functional differential equations of neutral type in Hilbert spaces. We also study the continuous dependence of such solutions on the initial dat
Regional Feedback Stabilization of Distributed Bilinear Systems with Time Delay
Get PDFThe current study focuses on the regional stabilization of infinite dimensional bilinear systems with time delays, evolving in a spatial domain Ω. It consists of studying the asymptotic behaviour of such a system in a subregion ω of Ω. Then, we demonstrate regional weak stabilization under weak observability conditions, while regional strong stabilization is achievable under exact observability conditions. Illustrative examples and simulations are included to validate the theoretical results
Influence of Dispersal Asymmetry on Total Biomass in Two-Patch Environment with Generalized Growth Rate
Get PDFIn this paper, we consider a two-patch model coupled by migration terms, where each patch follows a generalized logistic law. In the first consideration, we assume that the two patches are sources and in the second, we assume that one patch is source and the second is sink. Two complete classifications for two models (source-source and source-sink) are given: one concerning when dispersal asymmetry causes smaller or larger total biomass than no dispersal, and the other addressing how the total biomass changes with dispersal asymmetry. For two models, we find that the total biomass is either strictly decreasing or initially strictly increasing then strictly decreasing with respect to dispersal asymmetry. Populations are persist in both patches for source-source case and going to extinction in both patches for source-sink case