Numerical Solution Of Differential Equations Research Articles

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.

Analysis of opioid transmission crisis model in resilient cities based on numerical solutions of differential equations

The spread of synthetic and non synthetic opioids is a common phenomenon in real life. With the rapid development of information media, it provides a hotbed for the spread of synthetic and non synthetic opioids. The transmission process of synthetic and non synthetic opiates is described by using the differential equation method based on the traditional prevalence model. The propagation location of synthetic and non synthetic opioids can be obtained. Heroin is mainly distributed in Franklin, Kentucky, Claremont, Ohio, Lehigh, Pennsylvania, Frederick, Virginia and Putnam, West Virginia. Fentanyl is mainly distributed in Britt, Kentucky, green, Ohio and lanat, Pennsylvania. Buprenorphine distribution table: buprenorphine is mainly distributed in jasmine, Kentucky, snow, Ohio and chetherfield, Virginia. Select the index with high correlation coefficient according to the situation. Then, establish relevant models to study the effects of heroin and fentanyl on drug addiction. According to the epidemic model and grey correlation model, the state to be predicted is obtained from the prediction table, and the grey prediction value is corrected by Markov correction factor to obtain any important parameter limit on which the success of new identification depends. Finally, it is necessary to raise the awareness of the audience, which helps to reduce the probability of transmission of synthetic and non synthetic opioids in the population from the perspective of event fuzziness. In addition, the government and the media should pay attention to the power of interpersonal communication, pay attention to cultivating attention to the communication of synthetic and non synthetic opioids, and guide interpersonal communication with good credibility. At the same time, the early warning mechanism of synthetic and non synthetic opioids and the early publication of information can effectively solve the drug use crisis.

Algebraic Systems with Positive Coefficients and Positive Solutions

The paper is devoted to the existence, uniqueness and nonuniqueness of positive solutions to nonlinear algebraic systems of equations with positive coefficients. Such systems appear in large numbers of applications, such as steady-state equations in continuous and discrete dynamical models, Dirichlet problems, difference equations, boundary value problems, periodic solutions and numerical solutions for differential equations. We apply Brouwer’s fixed point theorem, Krasnoselskii’s fixed point theorem and monotone iterative methods in order to extend some known results and to obtain new results. We relax some hypotheses used in the literature concerning the strict monotonicity of the involved functions. We show that, in some cases, the unique positive solution can be obtained by a monotone increasing iterative method or by a monotone decreasing iterative method. As a consequence of one of our results, we recover the existence of a non-negative solution of the Leontief system and describe a monotone iterative method to find it.

An Applied N C Differentiation Interpolation technique for improved random Anomalous values in Data Mining

In data mining, the word “interpolation” refers to interpolating some anonymous information from a given set of known information. The method of interpolation is extensively used as a valuable tool in science and engineering. The predicament is a classical one and dates back to the time of Newton, who needed to solve such a problem in analyzing data on the numerical computations. Numerical applications of interpolation include derivation of computational techniques for numerical differentiation, numerical integration and numerical solutions of differential equations. In this paper a closest fit Application of the formula to numerical data for recovering haphazard Anomalous values in Data Mining has been shown in the case of representing the data on the dataset global carbon dioxide emissions from fossil fuel burning by Fuel Type corresponding as a method of time. The formula is suitable in the situation where the values of the argument are at equal interval.

On RHF and Bernoulli Polynomial for the numerical solution of differential equations

Approximation of the solution of the differential equations is done by Bernoulli polynomial. Bernoulli polynomial and operational matrix of differentiation were used in reducing differential equations into algebraic equations. The method and its application is demonstrated through illustrative examples and found that the method is computationally attractive. The Bernoulli polynomial method has been applied to compare the numerical solution of differential equations with the existing method of Rationalized Haar Function

First Redefined Zagreb Index of Generalized Transformation Graph

Approximation of the solution of the differential equations is done by Bernoulli polynomial. Bernoulli polynomial and operational matrix of differentiation were used in reducing differential equations into algebraic equations. The method and its application is demonstrated through illustrative examples and found that the method is computationally attractive. The Bernoulli polynomial method has been applied to compare the numerical solution of differential equations with the existing method of Rationalized Haar Function.

Stochastic rounding: implementation, error analysis and applications.

Stochastic rounding (SR) randomly maps a real number x to one of the two nearest values in a finite precision number system. The probability of choosing either of these two numbers is 1 minus their relative distance to x. This rounding mode was first proposed for use in computer arithmetic in the 1950s and it is currently experiencing a resurgence of interest. If used to compute the inner product of two vectors of length n in floating-point arithmetic, it yields an error bound with constant with high probability, where u is the unit round-off. This is not necessarily the case for round to nearest (RN), for which the worst-case error bound has constant nu. A particular attraction of SR is that, unlike RN, it is immune to the phenomenon of stagnation, whereby a sequence of tiny updates to a relatively large quantity is lost. We survey SR by discussing its mathematical properties and probabilistic error analysis, its implementation, and its use in applications, with a focus on machine learning and the numerical solution of differential equations.

Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge–Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general format of so-called modified two-derivative diagonally implicit Runge–Kutta methods (TDDIRK) is proposed. Their order conditions up to order six are derived by introducing a set of bi-colored rooted trees and deriving new elementary weights. Next, we build exponential fitting conditions in order for these modified TDDIRK methods to treat oscillatory solutions, leading to EFTDDIRK methods. In particular, a family of 2-stage fourth-order, a fifth-order, and a 3-stage sixth-order EFTDDIRK schemes are derived. These can be considered as superconvergent methods. The stability and phase-lag analysis of the new methods are also investigated, leading to optimized fourth-order schemes, which turn out to be much more accurate and efficient than their non-optimized versions. Finally, we carry out numerical experiments on some oscillatory test problems. Our numerical results clearly demonstrate the accuracy and efficiency of the newly derived methods when compared with existing trigonometically/exponentially fitted implicit Runge–Kutta methods and two-derivative Runge–Kutta methods of the same order in the literature.

A New Algorithm Based on Bernstein Polynomials Multiwavelets for the Solution of Differential Equations Governing AC Circuits

We extend the application of multiwavelet-based Bernstein polynomials for the numerical solution of differential equations governing AC circuits (LCR and LC). The operational matrix of integration is obtained from the orthonormal Bernstein polynomial wavelet bases, which diminishes differential equations into the system of linear algebraic equations for easy computation. It appeared that fewer wavelet bases gave better results. The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution. The error function was calculated and illustrated graphically for the reliability and accuracy of the proposed method. The proposed method examined several physical issues that lead to differential equations. HIGHLIGHTS Differential equations governing AC circuits are converted into the system of linear algebraic equations using Bernstein polynomial multiwavelets operational matrix of integration for easy computation The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution The error function is calculated and shown graphically GRAPHICAL ABSTRACT

Numerical solution of Hermite differential equation using the spline method of order 1 with GeoGebra

In this paper we will show the visualization of the approximations that can be obtained by means of the order 1 spline method for Hermite differential equations with well-interactive examples of GeoGebra applets.Here, we will dedicate ourselves to publisize, the great benefit that can be obtained, in the process of generating new mathematical knowledge for learning and teaching the numerical solutions of differential equations.

CREATION AND EVALUATION OF THE STRUCTURES GRID IN CURVILINEAR AREAS

The article concerns methods of a structural curvilinear grid constructing in areas of geometrically complex shape and its evaluation from the quality point of view. Equidistribution methods based on differential equations were used to construct the grid at the boundary and inside the region. The numerical solution of differential equations was realized by the finite difference method. For the problems of uniform arrangement of grid nodes on the boundary and for the problems of constructing curved grids inside the region, implicit difference schemes were constructed and methods of scalar sweep and alternating directions were used. The results of numerical calculations are obtained and graphs of curved grids are presented for different numbers of grid nodes. The quality of the grid was studied according to four criteria such as orthogonality, elongation, convexity and adaptability, which corresponds to the division of the considered area into equal subdomains, i.e. cells.

Bayesian numerical methods for nonlinear partial differential equations

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of PDEs is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.

PARALLEL IMPLEMENTATION OF THE YANENKO METHOD FOR SOLVING THE HEAT EQUATION

In this article, we will consider the parallel implementation of the Yanenko algorithm for the two-dimensional heat equation, and the sweep method was used to numerically solve the heat equation. The implementation of the sequential program is carried out simply in two-part steps by the longitudinal-transverse run, however, parallelization of two fractional steps with an indefinite scheme is difficult due to the creation of inter-process communication of data. In the course of the study, a parallel data distribution with one-dimensional decompositions is shown in the application of the Yanenko method for calculating heat conductivity. The results of parallelization of this task using the 1D decomposition were obtained and acceleration and efficiency images were analyzed in order to evaluate the parallel program. Currently, modeling of processes by numerical solution of differential equations is widely used in every field of Science, the most common methods bring the differential problem to a system of linear algebraic equations, methods that solve such systems include various startup options. The emergence and development of computing systems using Multi-Core processors and graphics accelerators make the problem of startup parallelization relevant; the results of the study are used for teaching in research institutes and universities.

ЗАСТОСУВАННЯ СИСТЕМ КОМП’ЮТЕРНОЇ МАТЕМАТИКИ ДЛЯ РОЗВ’ЯЗАННЯ ПРИКЛАДНИХ ЗАДАЧ

Purpose. To show the possibilities of using mathematics software in solving problems arising in the construction of mathematical models of various processes and, thus, to reveal the importance of realizing the professional orientation of the mathematical training of students of natural and engineering specialties. Methodology. A number of mathematical models that are presented in the form of a Cauchy problem for a common first-order differential equation are considered in this paper. Mathematical models considered in this paper describe chemical and ecological processes. Euler's numerical method is used to solve the problems that describe the proposed models. This method is implemented in special mathematical software (Mathcad, Matlab). Findings. It is shown how the use of special functions designed to solve the Cauchy problem solves the problems proposed in mathematical models. Graphs are built in Mathcad and Matlab, including graphs that can be used to compare analytical and numerical solutions of problems obtained by Euler's method. Originality. The paper concludes that mathematical models of many technological and physical processes in different industries can be represented by the Cauchy problem for a common differential equation. Practical value. The considered mathematical models show the importance of realization of professional orientation of mathematical training of students of natural and engineering specialties. Conclusions. Mathematical modeling allows you to study and evaluate various processes (physical, economic, environmental), and therefore knowledge of the basics of modeling and special mathematical software is very important in the training of modern specialists in various specialties. The problems presented in the article are used in teaching the courses of numerical methods and computer mathematics in the study of the topic «Numerical Solution of Differential Equations». References 10, tables 2, figures 9.

Determination of load distribution in a given medium according to the values of the loads at certain points

Taking into account force, temperature and other loads, the stress and strain state calculations methods of spatial structures involve determining the distribution of the loads in the three-dimensional body of the structure [1, 2].
 In many cases the output data for this distribution can be the values of loadings in separate points of the structure. The problem of load distribution in the body of the structure can be solved by three-dimensional discrete interpolation in four-dimensional space based on the method of finite differences, which has been widely used in solving various engineering problems in different fields. A discrete conception of the load distribution at points in the body or in the environment is also required for solving problems by the finite elements method [3-7].
 From a geometrical point of view, the result of three-dimensional interpolation is a multivariate of the four-dimensional space [8], where the three dimensions are the coordinates of a three-dimensional body point, and the fourth is the loading at this point. Such interpolation provides for setting of the three coordinates of the point and determining the load at that point. The simplest three-dimensional grid in the three-dimensional space is the grid based on a single sided hypercube. The coordinates of the nodes of such a grid correspond to the numbering of nodes along the coordinate axes.
 Discrete interpolation of points by the finite difference method is directly related to the numerical solutions of differential equations with given boundary conditions and also requires the setting of boundary conditions.
 If we consider a three-dimensional grid included into a parallelepiped, the boundary conditions are divided into three types: 1) zero-dimensional (loads at points), where the three edges of the grid converge; 2) one-dimensional (loads at points of lines), where the four edges of the grid converge; 3) two-dimensional (loads at the points of faces), where the five edges of the grid converge. The zero-dimensional conditions are boundary conditions for one-dimensional interpolation of the one-dimensional conditions, which, in turn, are boundary conditions for two-dimensional conditions, and the two-dimensional conditions are boundary conditions for determining the load on the inner points of the grid.
 If a load is specified only at certain points of boundary conditions, then the interpolation problem is divided into three stages: one-dimensional load interpolation onto the line nodes, two-dimensional load interpolation onto the surface nodes and three-dimensional load interpolation onto internal grid nodes.
 The proposed method of discrete three-dimensional interpolation allows, according to the specified values of force, temperature or other loads at individual points of the three-dimensional body, to interpolate such loads on all nodes of a given regular three-dimensional grid with cubic cells.
 As a result of interpolation, a discrete point framework of the multivariate is obtained, which is a geometric model of the distribution of physical characteristics in a given medium according to the values of these characteristics at individual points.

Preface

We are pleased to introduce the Proceedings of the V International Scientific and Technical Conference “Mechanical Science and Technology Update” (MSTU-2021) organized by the Omsk State Technical University and held on March 16-17, 2021.The conference materials cover the generalized issues of achievements in applied mechanics, engineering and materials science, quality management, mathematical modeling of processes, numerical solution of differential equations and development of micro-mechanical systems.The conference addresses topical issues of development and research on the problems of applied mechanics, mathematical modelling, and material science.List of International Program Committee, Organizing Committee, Organizing Institutions are available in this pdf.

Numerical solution of fluid queueing models for communication networks

Discrete stochastic models of queues are commonly employed in communication networks for their analysis and control. Fluid models, largely stochastic, are being used recently due to their simple character. However, asymptotic analysis of queueing models leads to deterministic fluid models and efficient methods for computing their solution are needed. This paper develops procedures for numerical solution of fluid queueing models whose arrival processes with rates that are piecewise constant. Solutions for fluid queue with single input, queue with two classes under priority and a model with two fluid queues in tandem are developed and implemented. The piecewise linear nature of the dynamics are employed to develop these efficient solution procedures by determining the solution at time instances of rate change of the input. This aviods using numerical solution of differential equations and also greatly reduces the computational complexity. Computer programs are developed implementing the numerical solutions. Motivated by their application to communication network sources, ON/OFF fluid inputs are considered for analysis. The developed numerical solution procedures are applied to the ON/OFF sources as special cases of the piecewise constant rate inputs.

Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics

Markov-type inequalities are often used in numerical solutions of differential equations, and their constants improve error bounds. In this paper, the upper approximation of the constant in a Markov-type inequality on a simplex is considered. To determine the constant, the minimal polynomial and pluripotential theories were employed. They include a complex equilibrium measure that solves the extreme problem by minimizing the energy integral. Consequently, examples of polynomials of the second degree are introduced. Then, a challenging bilevel optimization problem that uses the polynomials for the approximation was formulated. Finally, three popular meta-heuristics were applied to the problem, and their results were investigated.