Numerical Solution Of Differential Equations Research Articles

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.

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.

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.

Numerical Analysis of Differential Equation with Type-2 Fuzzy Number as Initial Condition

The primary intention of this article is to study numerical solutions of differential equation with interval Type- 2 Fuzzy Number (T2FN) as the initial condition. The differential equation is first redrafted in the parametric form; then, it is restructured into three systems of linear differential equations. Each system includes two concurrent linear differential equations with respective initial conditions. The classical 4 th order Runge-Kutta method is developed for the above derived systems. The ability of the method is corroborated by illustrating problems.

Evaluation of vertical closed loop system performance by modeling heat transfer in geothermal probes

This study aims to develop and validate the heat transfer models for evaluating the performance parameters of closed loop geothermal systems, considering in detail probe internal geometry, heat exchange between the tubes, surrounding medium, and its depth-dependent temperature. The developed models are based on numerical solutions of differential equations that govern heat transfer in tubes, with refining the thermal resistance coefficients of probe elements. The models allow to calculate performance parameters, first of all, the fluid output temperature and system thermal capacity, for vertical single, double U-shaped and coaxial probes contacting either soil or mine water. We validated the models using the experiments performed in-situ at three sites in Germany and Turkey demonstrating high agreement between calculated and measured parameters. In addition, we proposed and tested the criterion to optimize the geothermal system operation regarding the technology features.

Control theory applications when describing centrifugal concentration processes for mineral raw materials

The article discusses an approach aimed at developing an application methodology for the principles and models used in the theory of process control automation. The problems solved in the article are as follows: to try and represent the gravity concentration process (centrifugal concentration) in the form of a transfer function and to establish its parameters. In this case, the limited amount of data obtained in the course of the experiments is the main limitation for building models in the form of differential equations. An original processing and application methodology is proposed for the averaged data used when designing the model. This approach may be applicable to other ore preparation and processing stages as it involves equipment operation using the black box principle. The study is based on replacement of instantaneous characteristics of the input stream with their averaged values and subsequent integration, allowing the creation of a numerical model that coincides with the actual data. Transfer functions and their parameters for tailings and concentrate were established separately. MatLAB software with the Simulink package was used to verify the calculations. As a result, the transfer function was found for a KC-CVD6 device used at a gold deposit and satisfactory simulation results were obtained. The work uses automatic systems identification, system analysis, and mathematical modeling methods, namely, numerical solutions of differential equations using the MatLAB desktop environment with the Simulink package.

Methodology for applying control theory in modeling the processes of centrifugal enrichment of mineral raw materials

ABSTRACT The article discusses an approach aimed at the formation of a methodology for applying the principles and models used in the automatic control theory in relation to the processes of enrichment of mineral raw materials. The problem solved in the article is posed as follows, try to represent the process of gravitational enrichment, namely centrifugal concentration, in the form of a transfer function and try to find its parameters. The paper proposes an approach to modeling centrifugal concentrators based on transfer functions, using averaged values of output characteristics. Proposal of a mechanism for determining the parameters of transfer functions and their calculation based on experimental data. To check the presented calculations, the MatLAB software tool with the Simulink package was used. As a result, the transfer function of the KC-CVD6 apparatus used in the gold deposit was found and satisfactory results were obtained. The work uses methods of identification of automatic systems, methods of system analysis, as well as methods of mathematical modeling, namely the numerical solution of differential equations, implemented in the MatLAB environment with the Simulink package. A comparison with the classical regression model was made; as a result, the modeling error was reduced by two times. The simulation error for the concentrate yield was less than 1%.

Personalized Algorithm Generation: A Case Study in Learning ODE Integrators

We study the learning of numerical algorithms for scientific computing, which combines mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a departure from the classical approaches in numerical analysis, which typically do not feature such learning-based adaptations. As a case study, we develop a machine learning approach that automatically learns effective solvers for initial value problems in the form of ordinary differential equations (ODEs), based on the Runge--Kutta (RK) integrator architecture. We show that we can learn high-order integrators for targeted families of differential equations without the need for computing integrator coefficients by hand. Moreover, we demonstrate that in certain cases we can obtain superior performance to classical RK methods. This can be attributed to certain properties of the ODE families being identified and exploited by the approach. Overall, this work demonstrates an effective learning-based approach to the design of algorithms for the numerical solution of differential equations. This can be readily extended to other numerical tasks.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ НАПРЯЖЁННО-ДЕФОРМИРОВАННОГО СОСТОЯНИЯ МЕТАЛЛОКОНСТРУКЦИЙ С ПОМОЩЬЮ ГЕОМЕТРИЧЕСКИХ ИНТЕРПОЛЯНТОВ

The work is devoted to carrying out multidimensional interpolation and approximation methods for the numerical solution of differential equations and computer model development of the stress-strain state of metal structures. The core of the work is a fundamental computational algorithm for the numerical solution of differential equations using geometric interpolants on regular and irregular networks. On its basis, computational experiments are carried out on numerical simulation of the stress-strain state of operated reservoirs for storing petroleum products, which form a software package implemented in the Maple interpreter. At the same time, the differential equation for modelling the stress-strain state of an elastic cylindrical shell under axisymmetric loading is improved for the numerical analysis of the stress-strain state of a cylindrical reservoir with geometric imperfections. Also a new approach is proposed to take into consideration the initial conditions of the differential equation, which consists of parallel transfer of the numerical solution to the point, its coordinates correspond to the initial conditions. The advantage of the proposed approach for the numerical solution of differential equations using geometric interpolants is that it eliminates the need to coordinate geometric information in the process of interaction between CAD and FEA systems, by analogy with the isogeometric method.

High Order Multi-block Boundary-value Integration Methods for Stiff ODEs

In this paper, we present a new family of multi-block boundary value integration methods based on the Enright second derivative type-methods for differential equations. We rigorously show that this class of multi-block methods are generally $A_{k_1,k_2}$-stable for all block number by verifying through employing the Wiener-Hopf factorization of a matrix polynomial to determine the root distribution of the stability polynomial. Further more, the correct implementation procedure is as well determine by Wiener-Hopf factorization. Some numerical results are presented and a comparison is made with some existing methods. The new methods which output multi-block of solutions of the ordinary differential equations on application, and are unlike the conventional linear multistep methods which output a solution at a point or the conventional boundary value methods and multi-block methods which output a block of solutions per step. The second derivative multi-block boundary value integration methods are a new approach at obtaining very large scale integration methods for the numerical solution of differential equations.

A Review: Differential transform in numerical study of differential equations

Differential equation plays an important role in understanding various phenomenon of science and engineering. Various methods have been developed by mathematicians to obtain a numerical solution of differential equations with minimum errors, transform methods are one of such numerical methods. Till present no transform methods claims to have zero error in obtaining the numerical or exact solution of the given differential equations. Few of well-known transform methods are Laplace transform, Sumudu transforms, Differential transform, Integral transform method etc. In this paper the review of Laplace transform is presented which is one of the tools used by scientists and researchers in finding the solution to their differential equation. In this work, 18 research papers of applications of Laplace transform have been studied and it is shown how this transform has been used by other researchers to obtain the exact solution of ordinary, partial and fractional differential equations. To present a literature review on application of Laplace transform is the main objective of this work.