Numerical Solution Of Differential Equations Research Articles

On the learning of high order polynomial reconstructions for essentially non-oscillatory schemes

Approximation accuracy and convergence behavior are essential required properties for the computed numerical solution of differential equations. These requirements restrict the application of deep learning networks in the domain of scientific computing. Moreover, the recipe to create suitable synthetic data which can be used to train a good model is also not very clear. This study focuses on learning of third order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstructions using classification neural networks with small data sets. In particular, this work (i) proposes a novel way to obtain a third order WENO reconstruction which can be posed as classification problem, (ii) gives simple and novel approach to sample data sets which are small but rich enough to inherit the latent feature of inter-spatial regularity information in the constructed data, (iii) it is established that sampling of train data sets impacts quantitatively as well as qualitatively the required accuracy and non-oscillatory properties of resulting ENO3 and WENO3 schemes, (iv) proposes to use a limiter based multi model to retain desired accuracy as well as non-oscillatory properties of the resulting numerical schemes. Computational results are given which established that learned networks perform well and retain the features of the reconstruction methods.

Причины возникновения и развития дефектов рельсов в кривых на участках обращения поездов повышенной массы и длины

Purpose: determine the causes of defects in rails in areas of circulation of trains of increased weight and length through modeling the operation of rails in curved sections of the railway track using digital twins. Methods: t study the influence of various factors on rail wear in curved sections of the track, software systems were used ased on the numerical solution of differential equations for the movement of rolling stock and the operation of the railway track under the train. To determine the stresses in the rails in order to predict the development of contact-fatigue defects, dynamic calculations using the finite element method were used. Results: it was revealed that in order to reduce rail wear in a curved section of the track in sections with predominantly freight traffic, the elevation of the outer rail must be calculated based on the actually realized speeds of freight trains. It has been established that at the moment the influence of the amount of rail inclination on rail wear has not been sufficiently studied. As an analysis of the position of the rails at various inclines has shown, for the curve under consideration it is most advisable to use a 1/30 inclination within the limits of a circular curve and with a 1/20 to 1/30 inclination within the transition curve. To reduce the likelihood of contact-fatigue defects occurring in such areas, it is necessary to provide for the implementation of measures to lubricate the side working edge of the outer rail thread and the rolling surface of the inner rail thread within the entire curve. Practical importance: the research results indicate the need to revise the approach to calculating the elevation of the outer rails in curves with heavy freight train traffic. The possibility of extending the service life of rails by revising the system for planning and implementing work on lubricating and grinding rails in curves in areas where trains of increased mass and length operate

Review of approximation formulas for fractional derivatives of variable order

Fractional differential equations of variable order, depending on a time or space variable, are successfully used to study time and/or spatial dynamics. The purpose of this review is to consider the latest research and results related to the basic definitions and approximation formulas for fractional derivatives of variable order. The review begins with an examination of existing definitions based on various physical and practical knowledge. The following are formulas for discretizing fractional derivatives, since they play a key role in numerical modeling, providing an effective means of describing complex systems with long-term memory, heterogeneous parameters and non-local interactions. Their use not only improves the accuracy of models, but also facilitates the numerical solution of differential equations, which is essential for analyzing and predicting the behavior of real processes in various fields of science and engineering. This review is intended to help readers select appropriate definitions and discretization formulas to effectively solve specific physical and engineering problems.

Hyperparameter optimization of orthogonal functions in the numerical solution of differential equations

Numerical methods for solving differential equations often rely on the expansion of the approximate solution using basis functions. The choice of an appropriate basis function plays a crucial role in enhancing the accuracy of the solution. In this study, our aim is to develop algorithms that can identify an optimal basis function for any given differential equation. To achieve this, we explore fractional rational Jacobi functions as a versatile basis, incorporating hyperparameters related to rational mappings, Jacobi polynomial parameters, and fractional components. Our research develops hyperparameter optimization algorithms, including parallel grid search, parallel random search, Bayesian optimization, and parallel genetic algorithms. To evaluate the impact of each hyperparameter on the accuracy of the solution, we analyze two benchmark problems on a semi‐infinite domain: Volterra's population model and Kidder's equation. We achieve improved convergence and accuracy by judiciously constraining the ranges of the hyperparameters through a combination of random search and genetic algorithms. Notably, our findings demonstrate that the genetic algorithm consistently outperforms other approaches, yielding superior hyperparameter values that significantly enhance the quality of the solution, surpassing state‐of‐the‐art results.

Retraction Note: Data simulation of optimal model for numerical solution of differential equations based on deep learning and genetic algorithm

Retraction Note: Data simulation of optimal model for numerical solution of differential equations based on deep learning and genetic algorithm

FİNİTE ELEMENT ANALYSİS OF THE OUTER TUBE PART OF THE ENDOSCOPE

Even after cleaning and disinfecting the endoscope, the possibility of contamination is high and can adversely affect the health of patients. The US Centers for Disease Control report that there are approximately two million infected victims each year, and approximately 20,000 people succumb to the infection. In recent news, several deaths at UCLA Ronald Reagan Medical Center have been linked to bacterial contamination of endoscopes. Hence, there is a need to design an endoscope that will reduce or eliminate the chances of these deadly bacteria entering the body. In this article, we will look at the finite element analysis of the outer tube part of a newly designed disposable endoscope. The finite element method (SEM) is a popular method for the numerical solution of differential equations arising in engineering and mathematical modeling. Traditional areas of structural analysis include typical problem areas related to heat transfer, fluid flow, mass transport, and electromagnetic potential. Keywords: endoscope, outer tube, finite element method, mathematical modeling, analysis.

Numerical Solution of Differential Equations

Numerical Solution of Differential Equations

Several different ways to increase the accuracy of the numerical solution of the first order wave equation

The article presents several different ways to increase the accuracy of the numerical solution of differential equations. The comparison of schemes with different accuracy for the first order wave equation problem is presented. The schemes of the first order of upwind scheme, the second order of accuracy of McCormack, the third order of accuracy of Warming–Cutler–Lomax, and the scheme of the fourth order of accuracy of Abarbanel–Gotlieb–Turkel were applied. The condensed computational grids for the McCormack scheme are used, and the results using an adaptive grid for the McCormack scheme were compared.

A study of strong convergence of differential equations based on Euler’s algorithm

Abstract Differential equations have important applications in many fields, such as chemistry, biology, epidemiology, and finance. Most analytic solutions of differential equations are difficult to obtain. Therefore numerical solutions of differential equations become an important tool. The truncated Euler method is proposed in this paper, and we investigate how the truncated EM solution of the derived SDDE converges strongly under the local Lipschitz condition and the one-sided linear growth condition after relaxation. The basic strong convergence theorem is set up and the new notation X(t,x;s) is introduced as an analytic solution of the stochastic differential equation. Establish the assumption that the coefficients of the drift term and the coefficients of the diffusion term of the stochastic differential equation satisfy a contractionary monotonicity condition in order to prove, by induction, that the exact solution of this stochastic differential equation is bounded for a long time. Different examples are given to compare the simulated deviations between the numerical and analytical solutions of Euler’s algorithm and the modified Euler’s algorithm for the initial value problem of fractional order differential equations and to analyze the convergence of the two numerical solutions. The truncated Euler method is applied to highly nonlinear time-transformed stochastic differential equations by means of the dyadic principle, and it is proved that the order of convergence of the strong convergence of the truncated Euler-Maruyama method for time-transformed stochastic differential equations is min ( α , γ , 1 2 − ɛ ) \min \left( {\alpha ,\gamma ,{1 \over 2} - \varepsilon } \right) .

Алгоритмы управления карьерным самосвалом при проскальзывании колес

Mine dump trucks with a carrying capacity of 90 tons or more are equipped with an electromechanical transmission that controls the traction motors of the driving wheels. Due to unfavorable operation conditions of dump trucks, there are strict requirements to the driving safety along unpaved roads of open pits. The results of BELAZ dump trucks tests and authors maintenance experience have shown that the active security system is especially needed and gives essential assistance to a driver. Loss of control of а loaded truck, for example, during sharp braking on a slippery icy road, can lead to а serious accident and additional costs for repairing mining equipment. Thus, the development of traffic safety systems of mine dump trucks with electromechanical transmission is an important task. The methods of automatic control, mathematical modeling and numerical solution of differential equations are used to develop control algorithms for a mine dump truck. To test the proposed solutions, the method of full-scale experiment is used in real-life driving conditions along the dirt and icy roads. Control algorithms that use data of additional front (driven) wheels sensor are proposed. The results of tests of a BELAZ dump truck with a carrying capacity of 90 tons are presented. The results confirm the effectiveness of the developed control system in comparison with the one currently used in stock-produced items. The developed ABS system with speed sensors of the front wheels of a dump truck has proved the braking effectiveness along a slippery road. It makes it possible to improve the controllability of the dump truck significantly under conditions of lack of road adhesion, preventing the dump truck from skidding and wheels blocking. And it makes it possible to reduce mechanical effects on the transmission and anti-slip system sensitivity to different traffic conditions. The adaptive system remains operational if one of the front wheel sensors fails and if compared to the standard system it works equally effectively in various traffic conditions. The implementation of the obtained results will improve traffic safety and reduce the accident rate of expensive mining equipment.

Analysis of the technical level of hydraulic fluid power with motor-wheels

The analysis of the use of hydraulic fluid power of rotation of hydraulic motor-wheels for tracked and wheeled machines was carried out. The hydraulic principle schemes for the use of high-speed axial-piston hydraulic motors with planetary gearboxes and high-torque low-speed radial-piston hydraulic motors are considered, in particular, hydraulic devices for ensuring reliable movement in the event of obstacles from the road surface. The features of static and dynamic calculation methods using a mathematical model for the numerical solution of differential equations when calculating pressure fluctuations and the frequency of rotation of hydraulic motors depending on the tasks of changing the intensity of the pump supply and the working volume of the hydraulic motor, taking into account its efficiency, are shown. A comparative analysis of the technical level of hydraulic motor-wheels was carried out and a significant increase in the output parameters of high-torque radial piston hydraulic motors with respect to pressure, rotation frequency and mass-to-torque ratio was revealed. The article may be useful for engineers, scientists and master's students.

Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies

A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.

Numerical Solutions of High-Order Differential Equations with Polynomial Coefficients Using a Bernstein Polynomial Basis

The paper presents a novel method that allows one to establish numerical solutions of linear and nonlinear ordinary differential equations—with polynomial coefficients—that contain any finite products of the unknown functions and/or their general derivatives. The presented algorithm provides numerical solutions of these differential equations subject to initial or boundary conditions. This algorithm proposes the desired solution in terms of B-polynomials (Bernstein polynomial basis) and then uses the orthonormal relation of B-polynomials with its weighted dual basis with respect to the Jacobi weight function to construct a linear/nonlinear system in the unknown expansion coefficients which can be solved using a suitable solver. The properties of B-polynomials provide greater flexibility in which to impose the initial or boundary conditions at the end points of the interval [0, R] and enable us to obtain exactly and explicitly some of the unknown expansion coefficients in the form of a suggested numerical solution. Consequently, the presented algorithm leads to a linear or nonlinear algebraic system in the unknown expansion coefficients that has a simpler form than that was obtained by the other algorithms. So that, this procedure is a powerful tool that we may utilize to overcome the difficulties associated with boundary and initial value problems with less computational effort than the other techniques. An accepted agreement is obtained between the exact and approximate solutions for the given examples. The error analysis was also studied, and the obtained numerical results clarified the validity of the theoretical results.

Unsteady mixed convective flow of hybrid nanofluid past a rotating sphere with heat generation/absorption: an impact of shape factor

Purpose This study aims to examine the flow of unsteady mixed convective hybrid nanofluid over a rotating sphere with heat generation/absorption. The hybrid nanofluid contains different shapes of nanoparticles (copper [Cu] and aluminium oxide [Al2O3]) in the base fluid (water [H2O]). The influence of different shapes (sphere, brick, cylinder, platelets and blades) of nanoparticle in water-based hybrid nanofluid is also investigated. Design/methodology/approach To analyse the nanomaterial, the flow model is established, and in doing so, the Prandtl’s boundary layer theory is incorporated into the present model. The bvp4c approach, i.e. finite difference method, is used to find the numerical solution of differential equations that is controlling the fluid flow. The effect of relevant flow parameters on nanofluid temperature and velocity profile is demonstrated in detailed explanations using graphs and bar charts, whereas numerical results for Nusselt number and the skin’s coefficient for various form parameters are presented in tabular form. Findings The rate of heat transfer is least for spherical-shaped nanoparticle because of its smoothness, symmetricity and isotropic behaviour. The rate of heat transfer is highest for blade-shaped nanoparticles as compared to other shapes (brick, cylindrical and platelet) of nanoparticles because the blade-shaped nanoparticles causes comparatively more turbulence flow in the nanofluid than other shapes of nanoparticle. Heat generation affects the temperature distribution and, hence, the particle deposition rate. The absorption of heat extracts heat and reduce the temperature across the rotating sphere. The heat generation/absorption parameter plays an important role in establishing and maintaining the temperature around the rotating sphere. Research limitations/implications The numerical study is valid with the exception of the fluctuation in density that results in the buoyancy force and the functional axisymmetric nanofluid transport has constant thermophysical characteristics. In addition, this investigation is also constrained by the assumptions that there is no viscosity dissipation, no surface slippage and no chemically activated species. The hybrid nanofluid Al2O3–Cu/H2O is an incompressible and diluted suspension. The single-phase hybrid nanofluid model is considered in which the relative velocity of water (H2O) and hybrid nanoparticles (Al2O3–Cu) is the same and they are in a state of thermal equilibrium. Practical implications Study on convective flow across a revolving sphere has its applications found in electrolysis management, polymer deposition, medication transfer, cooling of spinning machinery segments, spin-stabilized missiles and other industrial and technical applications. Originality/value The originality of the study is to investigate the effect of shape factor on the flow of electrically conducting hybrid nanofluid past a rotating sphere with heat generation/absorption and magnetic field. The results are validated and provide extremely positive balance with the recognised articles. The results of the study provide many appealing applications that merit further study of the problem.

RETRACTED ARTICLE: Data simulation of optimal model for numerical solution of differential equations based on deep learning and genetic algorithm

Calculus equation is an important tool for mathematical research and plays an important role in most natural science research. Since the beginning of the eighteenth century, people have gradually used differential and integral equations to solve physical problems. In general, several different aspects of differential equations in the field of mathematics are concerned and studied by most scholars. However, this paper studies and establishes the optimal model for numerical solution of differential equations through deep learning and genetic algorithm. In this paper, the solution of ordinary differential equations is solved through the use of polynomial function space, while the linear combination of simple function x and its product nx can obtain multinomial function space. The space function form of polynomial is very simple, and the operation ability is very strong. Almost all functions can be approximated, and the function space can be transformed by a simple function. Through data simulation test results, it can be found that the oscillation of neural network output is stronger and stronger with the increase in depth, that is to say, the deeper depth endows the neural network with stronger oscillation properties, so for the oscillation function, the depth neural network fitting effect is better than the shallow neural network. Therefore, by combining deep learning and genetic algorithm, this paper studies and establishes the optimal model for numerical solution of differential equations, and finds that the deep neural network can largely complete data simulation.

Development of a Platform for Distributed Energy Resources Management on the Basis of a Digital Twin

The paper discusses the development of a platform for distributed energy resources management based on digital twins. The platform use cases include demand response, electric vehicle charging, peer-to-peer energy trading, storage scheduling, virtual power plant, and so on. Thanks to the digital twin, the platform can perform the use cases controlling either real operation-stage equipment or virtual design-stage simulation models. The platform offers mass distributed energy resources owners and operators to improve the power supply quality (including stability), reduce costs (including transaction overhead), and gain emerging market opportunities (including participation in various aggregators' programs). Software and equipment vendors are interested in the platform's capability to quickly assemble distributed energy management systems almost without programming. The digital twin and the platform are designed with the viewpoint-based approach established by the international systems engineering standard ISO/IEC/IEEE 42010. The typical power system digital twin architecture is described. The major kinds of mathematical models as part of digital twins are presented: physical models based on numerical solutions of differential equations and optimization problems, machine learning models, knowledge-based models. The interoperability of such heterogeneous models is ensured on the basis of the ontological model of distributed energy. The platform architecture is represented from three key viewpoints: functional, information, and software. To formalize and ultimately automate the integration of heterogeneous models, we propose novel mathematical methods of model-based system engineering based on category theory, including universal constructions and the multicomma. The multicomma category is shown to be constructed using standard product, exponent, and pushout constructions, which makes it possible to establish a number of its practically significant properties.

A Quantum-Computing-Based Method for Solving Quantum Confinement Problem in Semiconductor

Technology computer-aided design (TCAD) of semiconductor devices relies on the numerical solution of differential equations in devices. Recent advances in quantum computing provide a new opportunity for TCAD simulations to be performed on a quantum computer. Based on a variational quantum algorithm, we develop a quantum-computing-based method to solve quantum confinement problems in semiconductor nanostructures. As the number of numerical discretization grid points for solving the Schrödinger equation increases, the number of qubits needed scales only logarithmically, ~ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${O}[\log(N)]$ </tex-math></inline-formula> . The method is applied to solve quantum confinement problems at all dimensions, which are related to confinement in a quantum well, semiconductor nanowire, and semiconductor quantum dot structures. The method can treat an anisotropic band structure and electrostatic potential in semiconductor nanostructures. We further show that the design of ansatz plays an important role in the performance of the method in terms of solution accuracy. The quantum-computing-based method can compute the energies and wave functions of both the ground and excited states with high accuracy.

Fractional-step Runge–Kutta methods: Representation and linear stability analysis

Fractional-step methods are a popular and powerful divide-and-conquer approach for the numerical solution of differential equations. When the integrators of the fractional steps are Runge–Kutta methods, such methods can be written as generalized additive Runge–Kutta (GARK) methods, and thus the representation and analysis of such methods can be done through the GARK framework. We show how the general Butcher tableau representation and linear stability of such methods are related to the coefficients of the splitting method, the individual sub-integrators, and the order in which they are applied. We use this framework to explain some observations in the literature about fractional-step methods such as the choice of sub-integrators, the order in which they are applied, and the role played by negative splitting coefficients in the stability of the method.

Numerical solutions of differential equations having cubic nonlinearity using Boole collocation method

The aim of the study is to develop a numerical method for the solution of cubic nonlinear differential equations in which the numerical solution is based on Boole polynomials. That solution is in the form of the truncated series and gives approximate solution for nonlinear equations of cubic type. In this method, firstly, the matrix form of the serial solution is set and the nonlinear differential equation is converted into a matrix equation system. By adding the effect of both the conditions of the problem and the collocation points to this system of equations, we obtain the new system of equations. The coefficients of Boole-based serial solution are obtained from the solution of the resulting system of equations. The theoretical part is reinforced by considering three test problems. Numerical data for Boole solutions of test problems and absolute error functions are given in tables and figures.

Nonlinear Dynamics of the Primary Oscillation Circuit of a Mems Gyroscope Under the Action of Phase-locked Loop and Automatic Gain Control Systems

In this work, we study the dynamics of primary oscillations of a high-Q micromechanical resonator - a sensitive element of an RR-type MEMS gyroscope - under the action of various implementations of a phase-locked loop system operating in conjunction with an automatic gain control system for an electrostatic drive. The study of the dynamics of the object is carried out both numerically and analytically - using the averaging method. Conditions for the stability of a stationary regime in a linear approximation are obtained. The questions of accuracy of various methods of numerical solution of differential equations of the circuit of primary oscillations are considered. The influence of the mechanical nonlinearity of the resonator on the dynamics of the resonator and the control system has been studied. An implementation of a low-order PLL circuit that does not contain a double-frequency spurious signal at the output of the phase detector is proposed. The output characteristics of control systems (speed, capture bandwidth, etc.) are analyzed and qualitative conclusions are drawn about the features of the interaction between the dynamics of a mechanical oscillatory link and the PLL-ARC circuit.