A novel quaternion linear matrix equation solver through zeroing neural networks with applications to acoustic source tracking

Due to its significance in science and engineering, time-varying linear matrix equation (LME) problems have received a lot of attention from scholars. It is for this reason that the issue of finding the minimum-norm least-squares solution of the time-varying quaternion LME (ML-TQ-LME) is addressed in this study. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. In light of that, two new ZNN models are introduced to solve the ML-TQ-LME problem for time-varying quaternion matrices of arbitrary dimension. Two simulation experiments and two practical acoustic source tracking applications show that the models function superbly

Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images

The importance of quaternions in a variety of fields, such as physics, engineering and computer science, renders the effective solution of the time-varying quaternion matrix linear equation (TV-QLME) an equally important and interesting task. Zeroing neural networks (ZNN) have seen great success in solving TV problems in the real and complex domains, while quaternions and matrices of quaternions may be readily represented as either a complex or a real matrix, of magnified size. On that account, three new ZNN models are developed and the TV-QLME is solved directly in the quaternion domain as well as indirectly in the complex and real domains for matrices of arbitrary dimension. The models perform admirably in four simulation experiments and two practical applications concerning color restoration of images

Runge–Kutta pairs suited for SIR-type epidemic models

Modeling the infectious diseases concludes in systems of ordinary differential equations (ODEs). The compartments in these equations (e.g., the numbers of susceptible, infectious, or immunized individuals) change in time. The ODEs arriving in these models are quadratic. Thus, we may apply special type of Runge–Kutta (RK) pairs for solving them. Here, we construct a new RK pair of orders five and four that is special tuned for this type of ODEs. Its superiority over standard RK pairs from the literature is illustrated when applied to various epidemic models, valid in measuring COVID-19 spread

Mathematical modeling and the study of exchange processes in disperse boundary layer control actions

A significant interest of researchers is attracted to the effective management and forecasting of exchange processes in the boundary layer, which are key for the implementation of effective and reliable equipment. Modeling of exchange processes occurring in a high-speed dispersed boundary layer with external influences is a very difficult task. Mathematical modeling allows us to develop reliable devices and engines for the fields of aircraft, energy, shipbuilding with minimal costs for its creation. Despite the interest of numerous groups of researchers around the scientific projects and a large number of works, the current theory of the boundary layer is imperfect. This may be due to several circumstances: firstly, the theory of single-phase turbulent flows of continuous media is far from being completed, secondly, turbulent flows with dispersed impurities in the form of particles greatly complicate the already intricate flow pattern. Interest in dispersed flows is particularly relevant due to the fact that almost all gas-dynamic flows contain a certain concentration of particles, and their impact can provoke significant changes in the structure of the boundary layer and affect the intensity of exchange processes. The article proposes a two-fluid mathematical model describing the motion of a high-speed dispersed boundary layer on a surface with hemispherical damping cavities. The use of hemispherical damping cavities allows to reduce turbulent exchange in the boundary layer, which makes it possible to control the intensity of metabolic processes. The possibility of a significant reduction of turbulent heat transfer and friction in the dispersed boundary layer is established. The proposed method of impact on the turbulent transport in the boundary layer will improve the equipment and installations, including GTU and GTE used in various industries of our country, such as energy, aircraft, shipbuilding

Runge–Kutta Embedded Methods of Orders 8(7) for Use in Quadruple Precision Computations

High algebraic order Runge–Kutta embedded methods are commonly used when stringent tolerances are demanded. Traditionally, various criteria are satisfied while constructing these methods for application in double precision arithmetic. Firstly we try to keep the magnitude of the coefficients low, otherwise we may experience loss of accuracy; however, when working in quadruple precision we may admit larger coefficients. Then we are able to construct embedded methods of orders eight and seven (i.e., pairs of methods) with even smaller truncation errors. A new derived pair, as expected, is performing better than state-of-the-art pairs in a set of relevant problems

Real-Time Estimation of R0 for COVID-19 Spread

We propose a real-time approximation of R0 in an SIR-type model that applies to the COVID-19 epidemic outbreak. A very useful direct formula expressing R0 is found. Then, various type of models are considered, namely, finite differences, cubic splines, Piecewise Cubic Hermite interpolation and linear least squares approximation. Preserving the monotonicity of the formula under consideration proves to be of crucial importance. This latter property is preferred over accuracy, since it maintains positive R0. Only the Linear Least Squares technique guarantees this, and is finally proposed here. Tests on real COVID-19 data confirm the usefulness of our approach

Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems

Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits

The family of Numerov-type methods that effectively uses seven stages per step is considered. All the coefficients of the methods belonging to this family can be expressed analytically with respect to four free parameters. These coefficients are trained through a differential evolution technique in order to perform best in a wide range of Keplerian-type orbits. Then it is observed with extended numerical tests that a certain method behaves extremely well in a variety of orbits (e.g., Kepler, perturbed Kepler, Arenstorf, Pleiades) for various steplengths used by the methods and for various intervals of integration

Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems

Numerical Method for Solving of the Anomalous Diffusion Equation Based on a Local Estimate of the Monte Carlo Method

This paper considers a method of stochastic solution to the anomalous diffusion equation with a fractional derivative with respect to both time and coordinates. To this end, the process of a random walk of a particle is considered, and a master equation describing the distribution of particles is obtained. It has been shown that in the asymptotics of large times, this process is described by the equation of anomalous diffusion, with a fractional derivative in both time and coordinates. The method has been proposed for local estimation of the solution to the anomalous diffusion equation based on the simulation of random walk trajectories of a particle. The advantage of the proposed method is the opportunity to estimate the solution directly at a given point. This excludes the systematic component of the error from the calculation results and allows constructing the solution as a smooth function of the coordinate