Theory Of Equations Research Articles

REDUCTION OF A LINEAR MATRIX OPERATOR OF DIFFERENTIATION IN MANY VARIABLES TO AN OPERATOR WITH AN ORDINARY DIFFERENTIATION OPERATOR

Abstract. In the theory of equations with a linear partial differential operator of the first order, the question of differentiation and integration of vector functions is associated with considering them alternately along the characteristics of the operator, and the representation of their integrals by characteristics in the form of functions of many variables depended on the characteristic first integrals of the operator. The difficulty in this matter was to manage the characteristics and the first integrals for each of their variables so that they were in their places. Such control was implemented by projectors having pairwise ductile properties. The main results of this study are: 1) the use of the reduction method to reduce the matrix operator in partial derivatives to an operator with ordinary differentiations for one variable and 2) the use of projectors in matters of the composition of functions from many variables and the characteristics of the operator for each of the variables. These novelties of the research have a serious perspective.

Dynamically orthogonal narrow-angle parabolic equations for stochastic underwater sound propagation. Part I: Theory and schemes.

Robust informative acoustic predictions require precise knowledge of ocean physics, bathymetry, seabed, and acoustic parameters. However, in realistic applications, this information is uncertain due to sparse and heterogeneous measurements and complex ocean physics. Efficient techniques are thus needed to quantify these uncertainties and predict the stochastic acoustic wave fields. In this work, we derive and implement new stochastic differential equations that predict the acoustic pressure fields and their probability distributions. We start from the stochastic acoustic parabolic equation (PE) and employ the instantaneously-optimal Dynamically Orthogonal (DO) equations theory. We derive stochastic DO-PEs that dynamically reduce and march the dominant multi-dimensional uncertainties respecting the nonlinear governing equations and non-Gaussian statistics. We develop the dynamical reduced-order DO-PEs theory for the Narrow-Angle parabolic equation and implement numerical schemes for discretizing and integrating the stochastic acoustic fields.

Some new identities for colored partitions with parts in multiples of $ 4 $

<p>To start, we provided the deformations of certain specific modular equations derived from Ramanujan's notebook, Part Ⅲ, by using the complete square formula. Subsequently, we established fifteen identities for partitions with distinct colors that arose from these deformed modular equations in the theory of modular equations. Throughout our investigation, we found that the fifteen identities have parts that were in multiples of $ 4 $.</p>

A limiting result for the Ramsey theory of functional equations

A limiting result for the Ramsey theory of functional equations

Minimax solutions of Hamilton-Jacobi equations in dynamic optimization problems for hereditary systems

A survey of the results concerning the development of the theory of Hamilton-Jacobi equations for hereditary dynamical systems is presented. One feature of these systems is that the rate of change of their state depends not only on the current position, like in the classical case, but also on the full path travelled, that is, the history of the motion. Most of the paper is devoted to dynamical systems whose motion is described by functional differential equations of retarded type. In addition, more general systems described by functional differential equations of neutral type and closely related systems described by differential equations with fractional derivatives are considered. So-called path-dependent Hamilton-Jacobi equations are treated, which play for the above classes of systems a role similar to that of the classical Hamilton-Jacobi equations in dynamic optimization problems for ordinary differential systems. In the context of applications to control problems, the main attention is paid to the minimax approach to the concept of a generalized solution of the Hamilton-Jacobi equations under consideration and also to its relationship with the viscosity approach. Methods for designing optimal feedback control strategies with memory of motion history which are based on the constructions discussed are presented. Bibliography: 183 titles.

Heat and Mass Transfer in Infrared Assisted Heat Pump Drying of Purple Yam

Abstract This study focused on an engineering problem of heat and mass transfer in infrared (IR) assisted heat pump (HP) drying of purple yam including both a drying experiment and theory of heat and mass transfer mathematical equations. The experimental drying of yam by the IR assisted HP drying method was performed to evaluate the effect of IR power on drying rate and heating rate. The input drying parameters included the drying air temperature of 50°C, drying air velocity of 2.5 m·s−1 and IR power of 0, 300 and 350 W, in which, at the IR power of 0 W, the HP-only drying mode was performed. The experimental drying results showed that the IR assisted HP drying method could improve the drying rate and heating rate as compared to HP-only drying. At the IR power of 350 W, the drying time was the shortest (150 minutes), followed by the IR power of 300 W (210 minutes) and HP-only drying (300 minutes). The IR assisted HP drying obtained the high heating rate as the time required for the drying material to reach the drying temperature was about 35 and 25 minutes corresponding to the IR power of 300 and 350 W. While in HP-only drying, it took about 270 minutes for the drying material’s temperature to reach nearly the drying air temperature value. Besides, the comparison between the predicted data by numerically solving the heat and mass transfer equations and experimental drying data was also conducted. The analysis results indicated that the predicted data could be used to predict the experimental data accurately. The theoretical results of heat and mass transfer in the IR assisted HP drying process as well as experimental drying results would be the basis for improving the IR assisted HP drying equipment to achieve the drying efficiency including the drying rate, quality of dried products, and energy consumption.

Nonexistence for fractional differential inequalities and systems in the sense of Erdélyi-Kober

<p>Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities.</p>

Review of the research on the qualitative theory of differential equations at St. Petersburg University. II. Locally qualitative analysis of essentially nonlinear systems

This article is the second one in a series of articles devoted to a review of the results of scientific research that have been carried out at the Department of Differential Equations of St.Petersburg University over the past three decades and continue to be carried out at the present time. The first article talked about studies of stable periodic points of diffeomorphisms with homoclinic points and systems of differential equations with weakly hyperbolic invariant sets. This paper presents the results of locally qualitative analysis of essentially nonlinear systems in the neighbourhood of the zero solution, obtained by employees and graduates of the department. A system is said to be essentially nonlinear if the Taylor series expansion of its right-hand sides does not contain linear terms. The study of such systems, firstly, is complicated by a more complex picture of the behavior of solutions compared to quasilinear systems. Secondly, there are not even theoretical formulas for the general solution of a nonlinear first-approximation system, the presence of which is so helpful in the quasi-linear case. All this complicates analysis and significantly limits technical capabilities. Therefore, almost any new results and any new methods of working with such systems are of great interest. One of the most effective tools for working with essentially nonlinear systems turned out to be the logarithmic Lozinsky norms. In a sense, they are an analogue of the characteristic exponents (eigenvalues) used in the theory of quasilinear systems. Research conducted at the department has demonstrated the wide possibilities of using logarithmic norms in a wide variety of problems.

On the combination of Lebesgue and Riemann integrals in theory of convolution equations

On the combination of Lebesgue and Riemann integrals in theory of convolution equations

Analyzing vegetation pattern formation through a time-ordered fractional vegetation-sand model: A spatiotemporal dynamic approach

<p>This paper contributes to the field by developing a fractional-order vegetation-sand model that incorporates memory effects into the traditional integer-order framework. By studying the spatiotemporal dynamics of a time-order fractional vegetation-sand model, the research aimed to deepen our understanding of the complex interactions between vegetation and sand environments, providing insights for effective management and conservation strategies in arid and semi-arid regions. First, using the linear stability theory of fractional differential equations, we conducted a stability analysis of the spatially homogeneous fractional-order vegetation-sand model and provided the parametric conditions for stability and instability. Next, we performed a stability analysis of the spatiotemporal model, utilizing Turing instability to reveal the effects of diffusion and fractional order on vegetation distribution. Through numerical simulations, we demonstrated the spatiotemporal evolution patterns of the model under different environmental conditions and discussed the implications of these dynamic changes for ecological restoration and land management.</p>

Fourth-order neutral dynamic equations oscillate on timescales with different arguments

<p>The theory of neutral dynamic equations on timescales was based to unify the study of differential and difference equations. The article described several oscillating criteria that will be developed for fourth-order-neutral dynamic equations in the presence of various types of arguments on timescales. The goal was to establish all necessary conditions for the solutions of these models to be oscillatory. To construct observation values, ideas from [Y. Sui and Z. Han, Oscillation of second order neutral dynamic equations with deviating arguments on time scales, <italic>Adv. Differ. Equ.</italic>, 10 (2018)] were used. The research seeked to provide sufficient criteria that ensured the oscillation of solutions to these complex dynamic equations using a technique Riccati transformations generalized, emphasizing their importance in the study of oscillatory processes within various scientific and engineering contexts.</p>

A MULTIPOINT BOUNDARY VALUE PROBLEM IN TIME FOR A $2B$-PARABOLIC EQUATION WITH DEGENERACY

One of the most important issues in the general theory of differential equations with partial derivatives is establishing the solvability of boundary value problems. Among the boundary value problems for equations with partial derivatives, problems with nonlocal boundary conditions occupy an important place. Such interest in such problems is caused both by their rich practical application (the process of diffusion, moisture distortion in soils, plasma physics, etc.), and by the needs of the general theory of boundary value problems. A general multipoint boundary value problem for nonuniformly $2b$-parabolic equations with degeneracy is studied. The coefficients of parabolic equations and boundary conditions allow power degeneracy of arbitrary order in terms of time variable and spatial variables at some set of points. To solve the given multipoint boundary value problem, solutions of problems with smooth coefficients in Hölder spaces with the appropriate norm are studied. With the help of interpolation inequalities and a priori estimates, estimates of the solution of auxiliary problems and their derivatives in special Gelder spaces are established. Using the theorems of Ross and Archel, a convergent sequence is distinguished from the compact sequence of solutions of the auxiliary problems, the limiting value of which is the solution of the multipoint boundary value problem in time for the $2b$-parabolic equation with degeneracy. Estimates of the solution of the given problem are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of features of the coefficients of the equations and the boundary conditions. With certain restrictions on the right-hand side of the equation and boundary conditions, an integral image of the given problem is obtained.

The dynamics of a delayed predator-prey model with square root functional response and stage structure

<abstract> <p>In recent years, one of the most prevalent matters in population ecology has been the study of predator-prey relationships. In this context, this paper investigated the dynamic behavior of a delayed predator-prey model considering square root type functional response and stage structure for predators. First, we obtained positivity and boundedness of the solutions and existence of equilibrium points. Second, by applying the stability theory of delay differential equations and the Hopf bifurcation theorem, we discussed the system's local stability and the existence of a Hopf bifurcation at the positive equilibrium point. Moreover, the properties of the Hopf bifurcation were deduced by using the central manifold theorem and normal form method. Analytical results showed that when the time delay was less than the critical value, the two populations will coexist, otherwise the ecological balance will be disrupted. Finally, some numerical simulations were also included to verify the theoretical results.</p> </abstract>

Approximate Damped Oscillatory Traveling Wave in Viscous Compressible Fluid With Capillary Effect

We convert the viscous compressible fluid equations with capillary effect into the equivalent planar dynamical system and illustrate the global qualitative phase diagrams corresponding to its bounded orbits by the qualitative theory of ordinary differential equations. From the evolutionary relationship of bounded orbits in global qualitative phase diagrams and the correspondence between the bounded orbit and traveling wave, we discover the evolutionary rule of the profile of the traveling wave as the viscous coefficient varying and the forming reason for the damped oscillatory traveling wave. Further, we construct the analytical approximate explicit expression of the damped oscillatory traveling waves in accordance with the forming reason and prove its error decreases in exponential form. Finally, we compare the approximate damped oscillatory traveling wave with the numerical solution. The result shows they fit well.

A THEORY OF MULTIDIMENSIONAL FREDHOLM INTEGRAL EQUATIONS HAVING SEPARABLE KERNELS: SOLVABLE IN A REGION SURROUNDING BY THE HYPERPLANES

In this article, we present a theory of multidimensional Fredholm integral equations, having separable kernels, are solvable in a region surrounded by hyperplanes. In derivation of their solutions, we employ the generalized Hilbert-Schmidt theory involving eigenvalues and corresponding normalized eigen functions obtained by separable kernels in a region surrounded by the hyperplanes. Finally, we apply two variables Gegenbauer polynomials and derive a result on the inequality of the solution for the double symmetric Fredholm integral equation.

Study on effect of charged sand/dust atmosphere on the performance of microwave quantum illumination radar

This paper aims to study the effects of charged sand/dust atmosphere on the performances of microwave quantum illumination (QI) radar. Based on Mie particle scattering theory, using a Monte Carlo method for simulating the physical process which photon is scattered multiple times by discrete random distributed particles, the specific attenuation (dB/km) of microwave propagating in sand/dust atmosphere are analyzed under the conditions of varying atmospheric visibility and sand/dust particles with different charged quantities. It is indicated that the specific attenuation obtained by multiple scattering is smaller than those obtained based on Mie theory, for microwave propagating in charged sand/dust atmosphere. The smaller the atmospheric visibility, the greater the difference, while the difference decreases gradually as the atmospheric visibility increases. Then, it is more reasonable to consider multiple scattering attenuation at lower atmospheric visibility. When sand/dust particle is charged, the specific attenuation is increased, however, this increase is not linear. According to quantum illumination radar theory, a beam splitter-based optical link model is used to simulate the sand/dust atmospheric channel. The effects of charged sand/dust atmosphere with different visibility on the detection error probability, signal-to-noise ratio, and maximum detection range for microwave quantum illumination radar are studied by using quantum radar equation and quantum detection error probability theory. The performances of QI and classical two mode noise (TMN) radar are compared and analyzed. These results show that the performances of quantum illumination radar are improved with increasing sand/dust atmospheric visibility. When sand/dust particle is charged, the performances for QI radar are degraded due to attenuation increase. The change in the performances is nonlinear as the change in sand/dust carrying charge quantity. Increasing the signal frequency can enhance the performance of quantum illumination radar when visibility is high, but when visibility is low, the gain from frequency increase is outweighed by the performance degradation caused by increased attenuation, making it inadvisable to raise the frequency in such cases. The comparison with classical radar reveals that QI radar performs better under the condition of lower atmospheric visibility and lower average photon emission, but this advantage diminishes as the number of photons increases. In a word, these results show that the performances of QI radar are more significant at lower atmospheric visibility. Under higher visibility conditions, the QI system SNR can be improved by increasing frequency. The maximum detection range of the QI radar is significantly better than the classical TMN radar.

Ultrafast symmetry-breaking charge separation in Perylenemonoimide-embedded multichromophores: impact of regioisomerism.

Symmetry-breaking charge separation (SB-CS) has recently evolved as an emerging concept offering its potential to the latest generation of organic photovoltaics. However there are several concerns that need to be addressed to reach the state-of-the-art in SB-CS chemistry, for instance, the desirable molecular geometry, interchromophoric distance and extent of electronic coupling. To shed light on those features, it is reported herein, that ortho-functionalized perylene monoimide (PMI) constituted regioisomeric dimer and trimer derivatives with varied molecular twisting and electronic conjugation have been synthesized. In steady-state photophysical studies, all the dimers and trimer derivatives exhibit a larger bathochromic shift in the emission spectra and a significant reduction of fluorescence quantum yield in polar DMF. Among the series of multichromophores, ortho- and self-coupled dimers display the strikingly different optical feature of SB-CS with a very fast charge separation rate (τCS = 80.2 ps) upon photoexcitation in DMF, which is unveiled by femtosecond transient absorption (fs-TA) studies. The SB-CS for two dimers is well-supported by the formation of PMI˙+ and PMI˙- bands in the fs-TA spectra. Further analysis of fs-TA data revealed that, among the other multichromophores the trimer also exhibits a clear charge separation, whereas SB-CS signatures are less prominent, but can not be completely disregarded, for the meta- and para-dimers. Additionally, the charge separation dynamics of those above-mentioned PMI derivatives are devoid of a kinetically favorable excimer or triplet formation. The evidence of a profound charge transfer phenomenon in the ortho-dimer is characterized by density functional theory (DFT) calculations on excited state electronic structures. The excitonic communications in the excited state electronic arrangements unravel the key role of dihedral twisting in SB-CS. The thermodynamic feasibility of CS (ΔGCS) and activation barrier (ΔG≠) of the derivatives in DMF are established from the Rehm-Weller equation and Marcus's theory, respectively. This work is an in-depth study of the effect of mutual orientation of PMIs and regioisomerism in determining sustainable guidelines for using SB-CS.

Performance of copper nanofluid on changing width of stretching sheet using thermal radiation and heat flux

The phenomenon of nanomaterial carrying fluid flow over a non-linear elongated sheet with changing thickness is investigated in this study. The present research focuses at the flow of magnetic nanofluid over non-linear elongated sheets of different thicknesses, with an emphasis on copper nanoparticles distributed in water. The phenomena, which is significant in many industrial applications including paper manufacturing and atomic reactors, is analysed using Prandtl boundary layer theory and Navier-Stoke's equation. The study covers a wide range of physical features. There have been fewer works on the mathematical modelling of stretching sheets with varying widths. The main aim of investigation is to analyse the effect of EMHD, as well as other characteristics like Eckert number, Boit number, radiation effect and absorption factor. Paper manufacturing, dye and filament extrusion, atomic reactors, and metallurgical processes all rely heavily on these sheets of varied widths. The problem employs Navier-Stokes' equation and Prandtl boundary layer theory. The similarity transformation converts partial differential equations into ordinary differential equations. MATLbvp4c programme is used to obtain numerical solutions. The present study helps to achieve desired quality of stretching sheet.

First principle based rate equation (1pRE) for reduction kinetics of Fe2O3 with syngas in chemical looping

Chemical looping combustion (CLC) of solid fuel is a promising technology with inherent CO2 separation and low energy penalty for CO2 capture. Syngas is main intermediate species of solid fuel conversion in CLC, the reduction kinetics of oxygen carriers with syngas play a crucial role in CLC systems. However, the current research obtains the reaction rate constants by fitting the apparent models with the experimental data, and cannot explain the reduction kinetics behavior from a microscopic level. It remains a challenge to compute the reduction kinetics of oxygen carriers with syngas directly from first-principles density functional theory (DFT) without fitting experimental data. This study proposes a first-principle-based rate equation (1pRE) theory and integrates it into the random pore model (RPM) to predict the kinetics of Fe2O3 reduction by syngas in CLC. The developed 1pRE theory utilizes DFT calculations to search for reaction pathways and energy barriers of elementary reactions. Then the DFT data are introduced into the statistical mechanics partition function and transition state theory (TST) to calculate the reaction rate constants. Microkinetic rate equations of elementary reactions occurring at the surface scale are developed to describe the change of surface coverage of different surface species. The 1pRE theory is integrated into the RPM to account for the influence of particle-scale structural changes on the overall conversion rate during the reduction process. The theory can predict the reduction kinetics of oxygen carriers without fitting experimental data and establishes a connection between microscopic insights and macroscopic phenomena. The accuracy was validated by experimental data of Fe2O3 oxygen carriers obtained from the thermogravimetric analyzer (TGA) in the atmosphere of syngas. The developed 1pRE predicts the reduction kinetics of oxygen carriers accurately and can be used to optimize the design of oxygen carrier materials and the scale up of CLC reactors.

Charge and spin thermoelectric transport in benzene-based molecular nano-junctions: a quantum many-body study.

Within the Coulomb blockade regime, our study delves into the charge, spin, and thermoelectric transport characteristics of a benzene-based molecular nano-junction using the Pauli master equation and linear response theory. The charge- and spin-transport studies show strong negative differential conductance features in the current-voltage (I-V) characteristics for the ortho and meta connections of electrodes on either side. Contrarily, the para-connection displays Coulomb staircase behavior. By exploring spin current behavior in the presence of spin-polarized electrodes or an external Zeeman field, we establish a methodology that facilitates precise control over the specific spin flow. Various charge and spin thermoelectric transport coefficients have been studied with varying chemical potentials. We focus on spin-polarized conductance, the Seebeck coefficient, and the figure of merit. By adjusting electrode polarization or employing an external magnetic field, we achieve an impressive peak value for the spin thermoelectric figure of merit, approximately 4.10. This outcome underscores the strategic value of harnessing both spin-polarized electrodes and external magnetic fields within the domain of spin caloritronics.