Exact Differential Equation Research Articles

Investigation of accelerated moving load on dynamic response of FG Timoshenko nanobeam in thermal environment based on nonlocal strain gradient theory

For the first time, this paper investigates the forced vibrations of a functionally graded (FG) nanobeam with an accelerating moving load in a thermal environment. There is no exact coupling solution for the vibrations of nanobeam with an accelerated moving load, so this paper aims to provide a method to obtain an accurate solution for nanoscale structures with an accelerated moving load. The equations of motion are derived using Timoshenko's beam theory and the nonlocal strain gradient theory (NSGT). The Laplace method has been used to solve the coupling and exact differential equations. Then, by inverting Laplace from the coupled equations, an exact solution of the temporal response for FG nanobeam with constant acceleration and initial velocity in a thermal environment was obtained. The natural frequency was compared with previous works for validity and had acceptable results. Finally, the effect of parameters such as changes in acceleration and velocity of moving force, negative acceleration, power law index of FG material, different temperatures, nonlocal parameter, and longitudinal scale parameter on the maximum dynamic displacement of nanobeam is investigated. These results can be used better to design FG nanostructures with accelerated moving loads. Considering the sensitivity of data analysis in nano dimensions, it is necessary to provide an analytical solution method in these dimensions to reduce the error percentage in nano dimensions to zero. which is presented in this work for the first time.

Insights into the thermal characteristics and dynamics of stagnant blood conveying titanium oxide, alumina, and silver nanoparticles subject to Lorentz force and internal heating over a curved surface

Abstract It is very significant and practical to explore a triple hybrid nanofluid flow across the stuck zone of a stretching/shrinking curved surface with impacts from stuck and Lorentz force factors. The combination (Ag–TiO2–Al2O3/blood) hybrid nanofluid is studied herein as it moves across a stagnation zone of a stretching/shrinking surface that curves under the impact of pressure and Lorentz force. Exact unsolvable nonlinear partial differential equations can be transformed into ordinary differential equations that can be solved numerically by similarity transformation. It was discovered that predominant heat transfers and movement characteristics of quaternary hybrid nanofluids are dramatically affected. Numerous data were collected from this study to illustrate how parameters of flow affect the temperature, velocity, heat transmission, and skin friction characteristics. The axial and radial velocities for both fluids (Newtonian and ternary hybrid nanofluid) are increased due to the increasing function of the curvature parameter, magnetic field, and suction parameter. Additionally, the direct relationship between the temperature and heat transfer decreases the heat transfer rate by the curvature parameter, magnetic field, suction parameter, Prandtl number, and heat source/sink. The higher the values of the curvature parameter, the higher the shear stress and velocity.

Aplanatic freeform-mirror-based optical systems.

The exact partial differential equation to design aplanatic freeform-mirror-based optical systems is presented. The partial differential equation is not limited by the number of freeform surfaces or their orientations. The solutions of this partial differential equation can be useful as initial setups that can be optimized to meet higher criteria. One of these solutions is tested as an example of the initial setup, and the results are as expected by the theory.

Optimal projection filters with information geometry

We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo et al. (IEEE Trans Autom Control 43(2):247–252, 1998, Bernoulli 5(3):495–534, 1999), where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (Math Control Signals Syst 28(1):1–33, 2016) where a direct L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong et al. (Proc Lond Math Soc 119(1):176–213, 2019, Projection of SDEs onto submanifolds. “Information Geometry”, 2023 special issue on half a century of information geometry, 2018), based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges.

Cauchy Characteristics and Some Fluid Flow Problems

The nature of the differential equation for steady compressible flow is different for subsonic and supersonic flows. For subsonic flow the equation, the equation is of hyperbolic type. In this case the simple wave equation linearised theory. The exact differential equation for the steady flow of a compressible fluid as the same nature as that of linearised theory for steady subsonic flow, the differential equation of the elliptic type and for the steady supersonic flow, the differential equation is of the hyperbolic type. The general theory of the method of characteristics for the case of two independent variables is particularly easy to visualize and computation methods for this case need to be studied extensively. In this paper we discussed some basic properties of irrational flows and boundary conditions that occur in aeronautical engineering problems.

Hybrid discrete-continuous truncated Wigner approximation for driven, dissipative spin systems

We present a systematic approach for the semiclassical treatment of many-body dynamics of interacting, open spin systems. Our approach overcomes some of the shortcomings of the recently developed discrete truncated Wigner approximation (DTWA) based on Monte-Carlo sampling in a discrete phase space that improves the classical treatment by accounting for lowest-order quantum fluctuations. We provide a rigorous derivation of the DTWA by embedding it in a continuous phase space, thereby introducing a hybrid discrete-continuous truncated Wigner approximation (DCTWA). We derive a set of operator-differential mappings that yield an exact equation of motion (EOM) for the continuous SU(2) Wigner function of spins. The standard DTWA is then recovered by a systematic neglection of specific terms in this exact EOM. The hybrid approach permits us to determine the validity conditions and to gain detailed understanding of the quality of the approximation, paving the way for systematic improvements. Furthermore, we show that the continuous embedding allows for a straightforward extension of the method to open spin systems subject to dephasing, losses and incoherent drive, while preserving the key advantages of the discrete approach, such as a positive definite Wigner distribution of typical initial states. We derive exact stochastic differential equations for processes which cannot be described by the standard DTWA due to the presence of non-classical noise. We illustrate our approach by applying it to the dissipative dynamics of Rydberg excitation of one-dimensional arrays of laser-driven atoms and compare it to exact results for small systems.

On the Exact Computation of Linear Frequency Principle Dynamics and Its Generalization

Recent works show an intriguing phenomenon of Frequency Principle (F-Principle) that deep neural networks (DNNs) fit the target function from low to high frequency during the training, which provides insight into the training and generalization behavior of DNNs in complex tasks. In this paper, through analysis of an infinite-width two-layer NN in the neural tangent kernel (NTK) regime, we derive the exact differential equation, namely Linear Frequency-Principle (LFP) model, governing the evolution of NN output function in the frequency domain during the training. Our exact computation applies for general activation functions with no assumption on size and distribution of training data. This LFP model unravels that higher frequencies evolve polynomially or exponentially slower than lower frequencies depending on the smoothness/regularity of the activation function. We further bridge the gap between training dynamics and generalization by proving that LFP model implicitly minimizes a Frequency-Principle norm (FP-norm) of the learned function, by which higher frequencies are more severely penalized depending on the inverse of their evolution rate. Finally, we derive an \textit{a priori} generalization error bound controlled by the FP-norm of the target function, which provides a theoretical justification for the empirical results that DNNs often generalize well for low frequency functions.

On The Geometric Approach of Differential Equations

Expressing physical facts around us possesses to use mathematical models in both theory and application. Since these models usually involves differential equations, we present a novel geometric approach to these models using developable ruled surfaces and line congruences. In this paper, a tangent developable ruled surface according to first order differential equation and its solution function is presented. Moreover, we expressed a line congrunce based on the solution of exact differential equation and its solution function. Also, some examples of tangent developable ruled surfaces and line congruences are given.

Advancing engineering students’ conceptual understanding through puzzle-based learning: a case study with exact differential equations

AbstractCurrent views on the teaching of differential equations (DEs) are shifting towards the use of graphical and numerical methods. Motivated by recent research suggesting that puzzle-based learning (PzBL) can improve the teaching and learning of STEM subjects and by the lack of relevant studies for DEs, we designed two tasks—sophism and paradox—to explore undergraduate engineering students’ conceptual understanding of a classical topic—exact DEs—and to analyse the process of meaning-making during collaborative learning in small groups. One hundred and thirty-five undergraduate engineering students from a public university in Iran participated. In response to recent research signalling the tendency of the students to procedural learning of DEs, we analyse how the students in our study engaged in small group work on puzzle tasks, gaining a more conceptual understanding of exact DEs and acknowledging the efficiency of PzBL in their responses to a questionnaire and in interviews.

Exact equations to design aplanatic sequential optical systems

We present the exact differential equations to design an aplanatic sequential optical system, a system that is free of spherical aberration and linear coma. We get the exact set of equations from the Fermat principle and the Abbe sine condition. We solve the mentioned set of equations by implementing the Runge-Kutta algorithm. We test the solutions using commercial ray-tracing software and confirm the expected behavior of the optical system.

Temporal smoothing - A step forward for time-spectral methods

Time-spectral methods may feature substantial advantages over time-stepping solvers for solution of initial-value ODEs and PDEs, but their efficiency depends on the smoothness of the solution. We present two methods to overcome this problem. The first involves transforming the differential equation to an equation for a new variable, related to the time-integrated solution, before applying the solution algorithm. In the second method, a procedure for transformation to exact differential equations of a running average is outlined. Examples of solution of stiff problems and problems with multiple time scales are presented, employing the time-spectral Generalized Weighted Residual Method (GWRM). It is found that the smoothing algorithms have a significant positive effect on convergence.

Exact local fractional differential equations

Here the theory of local fractional differential equations is extended to a more general class of equations akin to usual exact differential equations. In the process, a new notion has been introduced which is termed as α-exact local fractional differential equation. The theory of such equations parallels that for the first order ordinary differential equations. A criterion to check the α-exactness emerges naturally and also a method to find general solutions of such equations. This development completes the basic theory of the local fractional differential equations. The solved examples demonstrate how complex functions arise as solutions which will be useful in understanding the processes taking place on fractals.

Liouville’s equations for random systems

Given a random system, a Liouville’s equation is an exact partial differential equation that describes the evolution of the probability density function of the solution. In this article, we derive Liouville’s equations for the first-order homogeneous semilinear random partial differential equation. This is done for all finite-dimensional distributions of the random field solution, starting with dimension one, then dimension two, and finally generalizing to any dimension. Several examples, including the linear advection equation with random coefficients, are treated. As a corollary, we deduce Liouville’s equations for path-wise stochastic integrals and nonlinear random ordinary differential equations.

An Analytical and Experimental Evaluation of a Heat Sink Under Constant Heat Flow and Forced Convection Heat Transfer

In this study, the exact transient differential equation was used to calculate the convection heat loss in a heat sink with a rectangular cross section fin. The result of the analytic solution was compared to the result from experiments conducted on a standard heat sink. The experiments were performed at a constant heat flow of 9000 W/m2 and a low airflow rate ranging from 12 to 100 cm3/s in seven steps. The comparative results showed that while there was good agreement between the experimental result and the exact solution, the average error ratio increased with an increase of the airflow rate. However, the maximum average error ratio between the experimental result and the exact solution did not exceed 6.4%. The maximum temperature distribution in the heat sink was obtained at a time of 900 s in all experiments.

Analysis of the Movement Characteristics of the Pump Valve of the Mine Emulsion Pump Based on the Internet of Things and Cellular Automata

Studying the movement characteristics of the coalmine emulsion pump valve is of great significance for optimizing the dynamic response characteristics of the pump valve, reducing the hysteresis effect, and improving the volumetric efficiency. This article combines the Internet of Things (IoT) and cellular automata techniques to investigate the movement characteristics of the valve of the emulsion pump. Based on Adolf’s exact differential equation and Runge–Kutta iterative method, the movement displacement and movement of the pump valve spool speed curve are computed using Scilab software. We employ cellular automata and AMESim to establish the hydraulic system model of emulsion pump and analyze the movement characteristics of pump valve movement displacement, speed, stability, and closing hysteresis through simulation. Finally, the IoT techniques and a test device are used to evaluate the movement displacement of the pump valve. The experimental results verify the feasibility of using the proposed method to study the pump valve motion characteristics, greatly reduce the cost of testing and parameterized design, and contribute to the development of highly reliable and efficient emulsion pump valves.

Some of the methods used to solve complete and incomplete differential equations

This paper studies the methods used to solve complete and in complete differential equations and types of first order and second order and Exact differential equation to solve integration general in This equation Fur there more, and the Special cases to find the integration factor use solve those types of equations is use as well,supported by a relevant variety of examples.

Exploring the exact differential equations with GeoGebra software

In this paper, we will show the solutions of certain exact differential equations that are obtained through the interactive GeoGebra software. GeoGebra software has been of great motivational support in the processes of teaching and modeling in mathematics in various universities and colleges, especially in the engineering careers of the University of Antofagasta in the courses of differential equations and calculation of several variables

恰当方程概念的导入及齐次函数欧拉定理必要性的另一种证法和它的应用

恰当方程概念的导入及齐次函数欧拉定理必要性的另一种证法和它的应用

Boundary tracking control of flexible beams for transferring motions

Although flexible beams for transmitting both translational and rotational large motions are used in practice such as ocean drill pipes, their control has not been considered. This paper develops boundary feedback controllers to stabilise these beams at their reference configurations. Exact nonlinear partial differential equations governing motion of the beams in three-dimensional space are derived and used in the control design. The designed controllers guarantee globally practically asymptotically stability of the beam motions at the reference states (i.e. positions and rotations of a straight beam moving axially with a desired velocity and rotating around its axial axis with a desired velocity). In the control design and analysis of well-posedness and stability, we utilise different transformations between the earth-fixed and body-fixed coordinates, Sobolev embeddings, and a Lyapunov-type theorem developed for a class of evolution systems in Hilbert space. Simulation results are also included to illustrate the effectiveness of the proposed control design.

Three-dimensional time-optimal path planning in the ocean

Autonomous underwater vehicles (AUVs) operate in the three-dimensional and time-dependent marine environment with strong and dynamic currents. Our goal is to predict the time history of the optimal three-dimensional headings of these vehicles such that they reach the given destination location in the least amount of time, starting from a known initial position. We employ the exact differential equations for time-optimal path planning and develop theory and numerical schemes to accurately predict three-dimensional optimal paths for several classes of marine vehicles, respecting their specific propulsion constraints. We further show that the three-dimensional path planning problem can be reduced to a two-dimensional one if the motion of the vehicle is partially known, e.g. if the vertical component of the motion is forced. This reduces the computational cost. We then apply the developed theory in three-dimensional analytically known flow fields to verify the schemes, benchmark the accuracy, and demonstrate capabilities. Finally, we showcase time-optimal path planning in realistic data-assimilative ocean simulations for the Middle Atlantic Bight region, integrating the primitive-equation of the Multidisciplinary Simulation Estimation and Assimilation System (MSEAS) with the three-dimensional path planning equations for three common marine vehicles, namely propelled AUVs (with unrestricted motion), floats (that only propel vertically), and gliders (that often perform sinusoidal yo-yo motions in vertical planes). These results highlight the effects of dynamic three-dimensional multiscale ocean currents on the optimal paths, including the Gulf Stream, shelfbreak front jet, upper-layer jets, eddies, and wind-driven and tidal currents. They also showcase the need to utilize data-assimilative ocean forecasts for planning efficient autonomous missions, from optimal deployment and pick-up, to monitoring and adaptive data collection.