Form Of Differential Equations Research Articles

Discrete symmetry analysis of free convection in a cylindrical porous annular microchannel at low-pressure level

Abstract This article studies the natural convection in an annular porous microchannel in case of one wall being heated and another being cooled. For the first time, such a problem was solved using discrete symmetries of the Navier-Stokes equations. Using discrete symmetries, self-similar forms of differential equations were obtained. Solutions of self-similar equations made it possible to obtain velocity and temperature profiles incorporating slip and temperature jump at the channel walls as boundary conditions. The effects of Grashof, Knudsen, Darcy, and Prandtl numbers on the velocity and temperature profiles in the microchannel and Nusselt numbers are demonstrated. At high Grashof numbers, an ascending flow forms near the hot cylinder, while a descending flow develops near the cold cylinder. As the Knudsen number increases, rise of velocity and temperature jump at the walls as well as heat transfer coefficients decrease is observed. An increase in the Darcy number results in higher velocities for both flows. The temperature jump at the heated cylinder increases, remaining unchanged at the cooled cylinder, and the heat transfer coefficient at the heated cylinder drops.

Active Inference and Artificial Spin Ice: Control Processes and State Selection.

Theory and simulations are used to demonstrate implementation of a variational Bayes algorithm called "active inference" in interacting arrays of nanomagnetic elements. The algorithm requires stochastic elements, and a simplified model based on a magnetic artificial spin ice geometry is used to illustrate how nanomagnets can generate the required random dynamics. Examples of tracking and PID control are demonstrated and shown to be consistent with the original stochastic differential equation formulation of active inference. Interestingly, nonlinear response in the form of spikes and spike trains not predicted by the original theory can appear in the nanomagnet system for certain temperature regimes. A theoretical approach using a mean-field approximation for spin systems is proposed, which describes the transition to nonlinear response. Finally, the possibility to create simple magnetic arrays using realistic models is shown with micromagnetic simulations of a simple 17 element array of nanomagnets that include magnetic anisotropies, and exchange and dipolar interactions. Possible applications are simulated to illustrate how nanomagnetic arrays can be used as the stochastic element for feedback control of processes, investigation and control of magnetic state evolution, and as a method to optimize pulsed field magnetic switching protocols.

Forecast modeling of the dynamics of main resources and construction of a simulation system for managing the activities of the regional energy system of the Samara region

Regional energy systems are complex dynamic systems that operate in conditions of constant changes in the external environment. In this regard, the task of developing complex management decisions and conducting a systemic analysis of the efficiency of its activities arises. To solve this problem, it is necessary to improve the mathematical models of the dynamics of capital, labor and fuel resources, since they have the greatest impact on the energy output of the energy system. The paper analyzes existing methods of mathematical description of the dynamics of basic resources and their shortcomings. The paper considers covariance-stationary models of time series in the form of difference equations with a deterministic polynomial trend. The paper presents the results of mathematical modeling of the dynamics of capital, labor and fuel resources based on statistical data on the activities of the energy system of the Samara Region, published in the annual reports of regional ministries and energy companies, and conducts a statistical analysis of them. Based on a comparative analysis of mathematical models, the most effective models of the dynamics of capital, labor and fuel resources with the best predictive qualities were selected. Also, in the work, a single-loop simulation model of the energy system management system was constructed by forming investments for the renewal of capital resources and the structure of the decision support system (DSS) was developed, allowing the formation of mathematically sound management decisions.

S-Contractive Mappings on Vector-Valued White Noise Functional Space and Their Applications

In this paper, we propose a new notion, which we name S-contractive mapping, in a framework of vector-valued white noise functionals Wω⊗N⊂Γ(H)⊗K⊂(Wω⊗N)*. And we give concrete definitions of S-contractive mappings for vector-valued white noise functionals. We establish the fixed-point theorems of S-contractive mappings. As applications, by applying the fixed-point theorems of generalized S-contractive mappings, we prove the existence and uniqueness of a generalized form of differential equations of vector-valued white noise functionals with weak conditions and investigate Wick-type differential equations of vector-valued white noise functionals with generalized conditions.

Existence of Solutions for Generalized Nonlinear Fourth-Order Differential Equations

In this article, we studied the existence of solutions for a more general form of nonlinear fourth-order differential equations by using a new generalization of the Arzelá–Ascoli theorem and Schauder fixed theorem under easier and general conditions. Moreover, we provided some sufficient conditions on the nonlinear function that allowed us to deduce the nonexistence results. Finally, we outlined an example to illustrate our main results.

Optimal control of day-to-day traffic dynamics under user learning with dynamic congestion pricing

Abstract A continuous day-to-day dynamic traffic assignment model is proposed in this study, where several typical day-to-day models are integrated into a more general form. The effects of the past information on the travelers’ routing behavior are characterized by the second-order derivatives of the traffic flow. Thereupon, the day-to-day dynamic takes the form of a second-order non-homogeneous ordinary differential equation with constant coefficients. We prove that the dynamical process is globally asymptotically stable at the user equilibrium with the existence of travelers’ route swapping behavior. Moreover, we prove that there always exists a limited time point, after which the non-negative flows can be guaranteed in the proposed model. The optimal control of the system is realized by utilizing dynamic congestion pricing. Different optimization objectives are investigated, including cost, time, revenue, and combined minimizations. Necessary conditions for optimal congestion prices are analyzed to uncover the bang-bang charging strategy. Numerical examples are provided to illustrate the trajectory of the flow evolution under the optimal control.

Modeling Covid-19 Virus after Lifting Preventive Measures: A Case Study of Kisii County

Purpose: This research is about a new COVID-19 SIR model containing three classes; susceptible S(t), infected I(t), and recovered R(t) with the convex incident rate. Methodology: The NCOVID-19 model was formulated in the following system, the whole population N(t) was divided into three classes S(t), I(t), and R(t), which represented Susceptible, Infected, and Recovered compartments in the form of differential equations. Lyapunov functions were used to validate the stability of the equilibrium of the ordinary differential equations, linearization of the system was also done using Jacobian matrices by finding the derivatives of f(x) for x. Findings: Covid-19 is an infectious disease caused by the novel coronavirus identified as Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2). The people infected by COVID-19 experience mild respiratory problems such as; Fever, dry cough, throat infection, and fatigue. People may also have symptoms such as nasal infection, aches, and sore throat. The pandemic has led to a dramatic loss of human life in Kenya, Africa, and the whole world as it presents an unprecedented challenge to public health, food systems, and the world of work. This case study seeks to model covid-19 virus after lifting preventive measures with a major focus on Kisii County, the subject model was presented in the form of differential equations and the disease-free and endemic equilibrium was calculated for the model. Also, the basic reproduction number R0 = 0.7831 was calculated and the disease-free equilibrium was found to be asymptotically stable meaning that the virus could be eliminated from the population, this showed that the county government of Kisii was in good control of the COVID-19 situation., in addition, The global stability of the model was calculated using the Lyapunov function construction while the Local stability was calculated using the Jacobian matrices. The numerical solutions were calculated using the non-standard finite difference scheme (NFDS) and MATLAB software. Unique Contribution to Theory, Practice and Policy: This study has laid a foundation for future research in the area. In the future, a study that can include the rate of COVID-19 virus mutation and its impacts is recommended.

Analytical and numerical solutions of squeezing nanofluid flow between parallel plates: A comparison of IRPSM and HPM

The nanofluid flow between two parallel plates has many applications across various fields of engineering. The addition of nanoparticles to conventional fluids can improve the rheological properties of nanofluids, enhance the heat transfer rate, and have other beneficial characteristics. Therefore, a squeezing flow problem between two parallel plates has been investigated in this article. The nanofluid flow is composed of alumina (Al2O3) nanoparticles and water, which is used as a base fluid. This study is featured according to the recommendations of a previous study to implement the improved residual power series method (IRPSM). The mathematical model is presented in the form of partial differential equations (PDEs), which are then transformed into ordinary differential equations (ODEs) by means of similarity variables. The considered problem is solved by two analytical methods called IRPSM, the homotopy perturbation method (HPM), and a numerical method (NDSolve). The residual errors of both applied techniques are compared to verify the validation of a new method. The comparison of both applied analytical techniques is conducted at the 17th order of approximations. From the obtained results, we confirmed that both techniques are in great agreement and have a very close relationship. The results of both methods are compared with the numerical method, and it was found that the absolute error between the IRPSM and numerical techniques is much more significant than the absolute error between the HPM and numerical techniques. Also, we have seen that the residuals of IRPSM are quite significant in terms of less residual errors than those of HPM. We confirmed that the IRPSM is a more powerful method than the HPM in terms of solving such types of mathematical problems.

Stability scrutinization of a non‐Newtonian (Williamson) ternary hybrid nanofluid past a stretching/shrinking sheet

AbstractThe thermal superiority of ternary hybrid nanofluids (THNFs) over conventional heat transfer fluid has led to growing interest in their applications. This new type of nanofluid can be customized for cooling systems, heat exchangers, and electronic cooling by carefully selecting nanoparticle types and their volume fraction. Hence, this study seeks to investigate the Heimenz flow in a Williamson THNF over a sheet that stretches or shrinks. The fundamental objective is to assess the effect of the stretching/shrinking parameter, the Weissenberg number, and the nanoparticle volume fraction on the physical quantities and flow profiles. Besides, attention is also given to the occurrences of multiple solutions in this fluid flow situation. By employing a similarity transformation, the governing equations are modified as a simpler form of ordinary differential equations (ODEs). Next, the numerical method is put to use to solve the resulting ODEs system, specifically the bvp4c solver in MATLAB. Significant changes in heat transmission occur due to variations in the Weissenberg number and volume fractions of nanoparticles, particularly when the sheet starts to shrink. The escalating Weissenberg number correlates with growing critical values of the stretching/shrinking parameter, suggesting that both parameters help to hold off the detachment of the boundary layer. These findings emphasize the capacity of THNFs to improve heat transfer performance in numerous applications. This study also reveals that while stretching sheets often have unique solutions, a shrinking sheet has multiple solutions when the shrinking parameter falls within a certain range. By scrutinizing the robustness of these solutions, it was concluded that only one of them maintains stability over an extended period. It is essential to highlight that these present discoveries apply exclusively to the mixture of copper, alumina, and titania. Various mixtures of nanoparticles can demonstrate distinct characteristics of THNFs concerning both flow dynamics and thermal transfer.

АЛГОРИТМИ ПРОГРАМНОЇ РЕАЛІЗАЦІЇ ЦИФРОВИХ СТАТИСТИЧНО ОПТИМАЛЬНИХ РЕГУЛЯТОРІВ АСИНХРОННИХ ЕЛЕКТРОПРИВОДІВ ІЗ СТОХАСТИЧНИМИ НАВАНТАЖЕННЯМИ

The paper is devoted to the development of algorithms for the programm implementation of digital statistically optimal regulators of electric drives, which are under the influence of dynamic loads that change according to stochastic laws. On the basis of continuous transfer functions of statistically optimal regulators for similar loads, which can be approximated by exponential and exponential-cosine correlation functions, the following algorithms were obtained in the form of recurrent difference equations, which can be implemented by software. Recommendations have been developed for the accuracy of determining the parameters of the corresponding digital regulators. Ref. 11, fig. 5.

An improved water strider algorithm for solving the inverse Burgers Huxley equation

In this paper, we introduce an improved water strider algorithm designed to solve the inverse form of the Burgers-Huxley equation, a nonlinear partial differential equation. Additionally, we propose a physics-informed neural network to address the same inverse problem. To demonstrate the effectiveness of the new algorithm and conduct a comparative analysis, we compare the results obtained using the improved water strider algorithm against those derived from the original water strider algorithm, a genetic algorithm, and a physics-informed neural network with three hidden layers. Solving the inverse form of nonlinear partial differential equations is crucial in many scientific and engineering applications, as it allows us to infer unknown parameters or initial conditions from observed data. This process is often challenging due to the complexity and nonlinearity of the equations involved. Meta-heuristic algorithms and neural networks have proven to be effective tools in addressing these challenges. The numerical results affirm the efficiency of our proposed method in solving the inverse form of the Burgers-Huxley equation. The best results were obtained using the improved water strider algorithm and the physics-informed neural network with 10,000 iterations. With this iteration count, the mean absolute error of these algorithms is O(10-4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(10^{-4})$$\\end{document}. Additionally, the improved water strider algorithm is nearly four times faster than the physics-informed neural network. All algorithms were executed on a computing system equipped with an Intel(R) Core(TM) i7-7500U processor and 12.00 GB of RAM, and were implemented in MATLAB.

Numerical study of an electrically conducting nanofluid flow past a vertical stretching Riga plate

The heat and mass transfer through electrically conducting and mixed convective nanofluid flow across an elongating Riga plate is studied. Riga plate is an electromagnetic sheet, in which electrodes are assembled alternatively. The arrangement of Riga plate produces electromagnetic behavior in the fluid flow. In the current analysis, the influence of Arrhenius activation energy, viscous dissipation and heat source/sink over the surface of Riga sheet is considered. The heat and mass transfer are examined by convective boundary conditions and nanoparticles (NPs) zero-mass flux. The Brownian motion and thermophoretic diffusion of fluid molecules are applied via Buongiorno's approach. Von Karman's similarity approach is implemented to reset the modeled equations into the dimensionless form of ordinary differential equations (ODEs). The system of ordinary differential equations is solved numerically via PCM (parametric continuation method). The fluid velocity, temperature field, and concentration gradients are analyzed by varying various flow parameters and graphically portrayed. The numerical results are validated by comparing them with the published studies. It can be resolved form the results that the flow rate decilnes with the effect of power index m, whereas enhances with the consequence of Hartmann number. The significance of thermophoresis term augments the temperature curve of the fluid.

Temperature-dependent density effects on oscillatory mixed convective flow across a non-conducting horizontal circular cylinder embedded in a porous medium

Temperature-dependent density and aligned magnetic field influence on mixed thermal convection heat and current density aspects over a non-conducting porous circular horizontal cylinder is the novelty of this investigation. The magnetic effects over the surface are minimal but maximum away from the surface. The density of the fluid is assumed as an exponential variable of temperature to record the heat and current density rates. The pertinent form of partial differential equations is designed using suitable dimensionless quantities. The computational outputs of fluid velocity, magnetic fields and temperature variations are drafted using fluctuating-stokes quantities. The primitive quantities are utilised to make similar and asymptotic programs in FORTRAN for geometrical and numerical outcomes. The transient and oscillating behaviour of gradient values is sketched by using steady results in the oscillating formula. The finite difference methodology is applied to examine the physical properties with the Gaussian elimination matrix scheme. Effects of buoyant force ( λ ), Prandtl (Pr), magnetic force ( ξ ), density factor ( m ) and MHD Prandtl factor ( γ ) on the thermal behaviour of magneto-hydrodynamic analysis are applied. Results are explored around three angles π / 6 , π / 3 , π of the cylinder. Increasing magnitude in fluid velocity is deduced with high amplitude as porosity, buoyant and density parameters are enhanced.

Rate dependency and fragmentation response of phase field models with micro inertia and micro viscosity terms

We present elliptic, parabolic, and hyperbolic phase field (PF) equations, referred to as EPF, PPF, and HPF, for cohesive zone model (CZM) and Linear Elastic Fracture Mechanics (LEFM) PF models. The micro viscosity and micro inertia terms result in PPF and HPF, that can be solved explicitly in time. The additional advantage of the HPF is implying a finite damage propagation speed and a more favorable time advance limit. The desired micro-viscosity for the HPF is shown to correspond to a damping factor of one. The appropriate length and time scales, nondimensional equations, and (asymptotic) strain-stress solutions are provided for each differential equation. Two sources of rate sensitivity are discussed. First, when the fields are spatially uniform (0D solution) the rate effect arises from the form of differential equation. Asymptotic solutions show that at high loading rates, the energy dissipation has the rates of 2/3 and 1 for the PPF and HPF, respectively, with the former matching the Grady’s rate-sensitivity. The dynamic strength and failure strain show half of this rate. The analysis is extended to damage models with stress-based and bounded driving forces. Second, the rate effect arises from the deviation of 1D from 0D solutions as the damage exceeds a critical value and fragments form. Since this critical value tends to zero for quasi-static loading, this rate-effect is mainly present for low loading rates. This results in lower- and upper-shelf energy limits for the EPF at low and high loading rates, resembling some experimentally observed sigmoidal rate models for energy dissipation.

Statistical analysis of periodic MHD chemically radiative casson flow over an inclined cylinder with entropy optimization

Statistical analysis (SA) stands as an indispensable tool in materials science and engineering, aiding in the understanding and optimization of heat and mass transport processes. This study delves into the interplay between periodic magneto-hydrodynamics (MHD) and chemical reactions concerning entropy generation over an inclined porous cylinder (IPC). We employ a statistical approach, amalgamating the robust 6th order Runge-Kutta (R-K) integration algorithm with the Nachtsheim-Swigert (N-S) shooting iteration method, subsequent to converting the governing equations into Ordinary Differential Equation (ODE) form. The primary objective of this study is to analyze the behavior of entropy generation, unveiling the intricate relationship between entropy dynamics, radiative periodic MHD on IPC, and chemical processes. The numerical findings encapsulate the dynamics of temperature, velocity, and entropy generation, visually portrayed across various dimensionless parameters. The primary findings of this investigation indicate that Casson fluid flow generates 18.99 % more entropy in comparison to Newtonian fluid flow. Furthermore, entropy is 9.87 % larger in an inclined cylinder than in a level cylinder. In contrast, entropy production falls by 3.34 % in the presence of periodic MHD as opposed to non-periodic MHD. Regression study reveals a genuine and significant interaction between variables inside the entropy-generating systems, as well, with a correlation value (R2) of 99.82 % at a 95 % confidence level. The correlation study shows that there is a significant 98.13 % relationship between entropy generation and chemical processes. Understanding the heightened entropy associated with Casson fluid flow benefits tissue engineering, drug delivery systems, and renewable energy. Increased entropy in inclined cylinders aids in optimizing drainage systems, pipeline design, and aerodynamic efficiency in aerospace engineering.

Comparison of Classical and Inverse Calibration Equations in Chemical Analysis.

Chemical analysis adopts a calibration curve to establish the relationship between the measuring technique's response and the target analyte's standard concentration. The calibration equation is established using regression analysis to verify the response of a chemical instrument to the known properties of materials that served as standard values. An adequate calibration equation ensures the performance of these instruments. There are two kinds of calibration equations: classical equations and inverse equations. For the classical equation, the standard values are independent, and the instrument's response is dependent. The inverse equation is the opposite: the instrument's response is the independent value. For the new response value, the calculation of the new measurement by the classical equation must be transformed into a complex form to calculate the measurement values. However, the measurement values of the inverse equation could be computed directly. Different forms of calibration equations besides the linear equation could be used for the inverse calibration equation. This study used measurement data sets from two kinds of humidity sensors and nine data sets from the literature to evaluate the predictive performance of two calibration equations. Four criteria were proposed to evaluate the predictive ability of two calibration equations. The study found that the inverse calibration equation could be an effective tool for complex calibration equations in chemical analysis. The precision of the instrument's response is essential to ensure predictive performance. The inverse calibration equation could be embedded into the measurement device, and then intelligent instruments could be enhanced.

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.

A new convex integration approach for the compressible Euler equations and failure of the local maximal dissipation criterion

Abstract In this paper we establish a new convex integration approach for the barotropic compressible Euler equations in two space dimensions. In contrast to existing literature, our new method generates not only the momentum for given density, but also the energy and the energy flux. This allows for a simple way to construct admissible solutions, i.e. solutions which satisfy the energy inequality. Moreover using the convex integration method developed in this paper, we show that the local maximal dissipation criterion fails in the following sense: There exist wild solutions which beat the self-similar solution of the one-dimensional Riemann problem extended to two dimensions. Hence the local maximal dissipation criterion rules out the self-similar solution. The convex integration machinery itself is carried out in a very general way. Hence this paper provides a universal framework for convex integration which not even specifies the form of the partial differential equations under consideration. Therefore this general framework is applicable in many different situations.

Thermal analysis of mathematical model of heat and mass transfer through bioconvective Carreau nanofluid flow over an inclined stretchable cylinder

In this proceeding the main focus to scrutinize the heat and mass transfer rate increment due to the involvement of motile microorganisms in the Carreau nanofluid flow through an inclined stretchable cylinder under the consequences of activation energy and thermal radiation. This phenomenon has many physical applications in the field of civil, mechanical and seismographic engineering. The research focuses on creating and analyzing new nanocomposites with applications in energy storage, showcasing the transformative impact of nanotechnology on the advancement of high-performance materials. First of all, for the numerical treatment of the above physical model mathematical model is developed in the form of partial differential equations (PDE's) by considering all involving quantities. The resultant model is highly nonlinear and of higher order. Appropriate similarity variables are defined to transform this system of PDE's into first order ordinary differential equations (ODE's). For numerical results we develop a numerical algorithm from nonlinear ODE's and use ‘bvp4c’ built in package of MATLAB. Numerical results are acquired from software in the form of graphical visuals and tabular data as narrated in the results and discussion section. Using this tabular data streamlines are constructed for the better understanding of heat and mass transfer through this phenomenon under the defined constraints. From results, velocity profile increased by the augmentation of values of curvature parameter. Thermal profile is stronger when values of thermophoresis parameter is boosted, Concentration of nanoparticle profile get smoother when temperature coefficient increased, and motile microorganism's density profile is enlarged upon inclining the values of bioconvection lewis number.

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.