Methods For Differential Equations Research Articles

The Riccati-Bernoulli sub-optimal differential equation method for analyzing the fractional Dullin-Gottwald-Holm equation and modeling nonlinear waves in fluid mediums

<abstract><p>The present study investigates the fractional Dullin-Gottwald-Holm equation by using the Riccati-Bernoulli sub-optimal differential equation method with the Bäcklund transformation. By employing a well-established criterion, the present study reveals novel cusp soliton solutions that resemble peakons and offers valuable insights into their dynamic behaviors and mysterious phenomena. The solution family encompasses various analytical solutions, such as peakons, periodic, and kink-wave solutions. Furthermore, the impact of both the time- and space-fractional parameters on all derived solutions' profiles is examined. This investigation's significance lies in its contribution to understanding intricate dynamics inside physical systems, offering valuable insights into various domains like fluid mechanics and nonlinear phenomena across different physical models. The computational technique's straightforward, effective, and concise nature is demonstrated through introduction of some graphical representations in two- and three-dimensional plots generated by adjusting the related parameters. The findings underscore the versatility of this methodology and demonstrate its applicability as a tool to solve more complicated nonlinear problems as well as its ability to explain many mysterious phenomena.</p></abstract>

A SHORT SOLUTION OF THE LOCAL FRACTIONAL (2+1)-DIMENSIONAL DISPERSIVE LONG WATER WAVE SYSTEM

In this paper, a local fractional Riccati differential equation method is applied. A new travelling wave solution to the nonlinear local fractional [Formula: see text]-dimensional dispersive long water wave system is investigated. After travelling wave transformation, the governing model studied is converted into nonlinear ordinary differential equation. Some properties with the strain conditions are also reported.

Numerical Mean-square and Asymptotic Stability Analysis for the Weak Simpson Method

A weak Simpson method has order of weak convergence one in general and has order of weak convergence three under certain additional assumptions. The proposed method has the potential to overcome some of the numerical instabilities that are often experienced when using explicit Euler methods. This work aims to determine the mean-square stability region of the weak Simpson method for linear stochastic differential equations with multiplicative noises. In this work, a mean-square stability region of the weak Simpson scheme is identified, and step-sizes for the numerical method where errors propagation are under control in a well-defined sense are given. The main results are illustrated with numerical examples.

The traveling wave solutions to a variant of the Boussinesq equation

In this study, we study a variant of the Boussinesq equation called as \(B( n+1,\,1,\,n )\) equation, and construct some traveling wave solutions by using an effective approach called the extended trial equation method. Thus, the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions, which show the existence of various mathematical and physical structures and events in the fundamental equation considered, have been constructured. In order to make a more detailed examination of the physical behavior of these solutions, two- and three-dimensional graphs of some solution functions were drawn with the help of the Mathematica package program. In the section of Discussion, we suggest a more general version of the trial equation method for nonlinear differential equations.

NEW IMPLICIT MULTISTEP METHOD FOR ODE’S

A new class of multistep methods for stiff ordinary differential equations is presented. The method is based on the transformation of the arguments of the original system into purely algebraic combinations of the solutions of previous steps. The scheme differs from the classical multistep methods in that the state variables, instead of functions of them, are approximated by means of linear combination of previous steps. A family of coefficients is deduced from a combined analysis of convergence order and stability. Numerical results are presented for three test problems.

A super-localized generalized finite element method

AbstractThis paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method’s basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method’s applicability for challenging high-contrast channeled coefficients.

An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

ФОРМУВАННЯ УМІНЬ МАТЕМАТИЧНОГО МОДЕЛЮВАННЯ ПРИКЛАДНИХ ЗАДАЧ МЕТОДАМИ ДИФЕРЕНЦІАЛЬНИХ РІВНЯНЬ

The article focuses attention on the need to learn how to solve applied problems, substantiates the need to master the skills of mathematical modelling as a universal method of solving applied and practical problems, presents the experience of students of higher education in forming the skills of mathematical modelling of applied problems using the methods of differential equations, reveals interdisciplinary connections, examples of solving geometric and physical problems from the topic “Differential Equations” are given. Problem solving was carried out in three stages, namely: building a mathematical model, studying the model and interpreting the result.

A New Approximation Method for Multi-Pantograph Type Delay Differential Equations Using Boubaker Polynomials

In this paper, a new approaching technique is offered to unravel multi-pantograph-type delay differential equations. The suggested new method is a collocation method based on integration and Boubaker polynomials. As the main idea of the method, the process starts by approaching the first derivative function in the equation in the form of truncated Boubaker series. Then this approximating form is composed in the matrix form. The unknown function is then obtained by integrating the approximate derivative function and expressing it as a matrix. Using the approximation, the matrix forms for the proportionally delayed terms in the equation are derived. In addition, operational matrix forms are constructed for convenience in the method. By using these matrix forms and matrix operations, the problem is reduced to a system of algebraic linear equations. The method is illustrated through numerical implementations and compared with existing techniques in the literature. The results demonstrate the effectiveness and reliability of the proposed approach, highlighting its superiority over other methods.

Threshold factorization of the Drell-Yan quark-gluon channel and two-loop soft function at next-to-leading power

We present a factorization theorem of the partonic Drell-Yan off-diagonal processes gq¯qg→γ∗+X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ g\\overline{q}(qg)\ o {\\gamma}^{\\ast }+X $$\\end{document} in the kinematic threshold regime z=Q2/ŝ→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ z={Q}^2/\\hat{s}\ o 1 $$\\end{document} at general subleading powers in the (1 − z) expansion. Focusing on the first order of the expansion (next-to-leading power accuracy with respect to the leading power qq¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q\\overline{q} $$\\end{document} channel), we validate the bare factorization formula up to Oαs2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({\\alpha}_s^2\\right) $$\\end{document}. This is achieved by carrying out an explicit calculation of the generalized soft function in d-dimensions using the reduction to master integrals and the differential equations method. The collinear function is a universal object which we compute from an operator matching equation at one-loop level. Next, we integrate the soft and collinear functions over the convolution variables and remove the remaining initial state collinear singularities through PDF renormalization. The resulting expression agrees with the known cross section in the literature.

İNTEGRO-DİFERANSİYEL DENKLEMLERİN SAYISAL ÇÖZÜMÜNE UYGULANAN KOLLOKASYON YÖNTEMİ

Integro-differential equations are encountered in such fields of study as mechanics, physics, chemistry, biophysics, astronomy, economic theory, and population dynamics. In rare cases the solution methods for differential and/or integral equations can be generalized to integro-differential equations; but in general, numerical methods have to be applied. Recent years have seen the development of a large number of methods applicable to integro-differential equations. The present study aims to compare these newer methods with the classical method of point collocation, which is one of the weighted residual methods. The method was applied to test problems chosen from the literature, both linear and nonlinear integro-differential equations, and was seen to give good results.

Convergence and stability of modified partially truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments

ABSTRACT This paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of the underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.

Numerical oscillation and non-oscillation analysis of the mixed type impulsive differential equation with piecewise constant arguments

The purpose of this paper is to study oscillation and non-oscillation of Runge–Kutta methods for linear mixed type impulsive differential equations with piecewise constant arguments. The conditions for oscillation and non-oscillation of numerical solutions are obtained. Also conditions under which Runge–Kutta methods can preserve the oscillation and non-oscillation of linear mixed type impulsive differential equations with piecewise constant arguments are obtained. Moreover, the interpolation function of numerical solutions is introduced and the properties of the interpolation function are discussed. It turns out that the zeros of the interpolation function converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge–Kutta method. To confirm the theoretical results, the numerical examples are given.

Three loop QCD corrections to the heavy-light form factors in the color-planar limit

We present the analytic expressions for the color-planar contributions to the heavy-light form factors at three loops in perturbative QCD. These form factors play an important role in the precision predictions of various observables in top quark and flavour physics. We compute the master integrals using the method of differential equations. We perform the ultraviolet renormalization for all the appearing fields and parameters. The analytic results for the renormalized form factors are expressed in terms of generalized harmonic polylogarithms. We also study the Sudakov behaviour of these form factors in the asymptotic limit, which enables us to obtain the complete logarithmic three-loop and partial four-loop contributions.

Numerical simulation of an external ballistic problem using analytical approach and atmosphere flow visualization by finite element method

Purpose. Analysis of the dynamic characteristics of the cargo in the presence of the speed of its carrier, taking into account the influence of the external environment and the determination of factors that affect the operational characteristics and parameters for the dynamic system control. Methodology. An approximate analytical approach to solving the nonlinear problem of external ballistics of a system with time-dependent parameters, which is implemented using the asymptotic perturbation method, and a numerical algorithm for modeling a dynamic process using a 3D software complex and the finite element method, which allows to visualize the nature of the flow around the object under wind load conditions. Findings. An analytical approach to solving the nonlinear problem of external ballistics of a system with time-varying parameters is proposed, as well as a numerical model, solution and visualization of a dynamic process that can be applied in problems of mathematical physics and engineering calculations. Originality. The use of an approximate analytical approach to solving the nonlinear problem of external ballistics, which includes a system of time-varying parameters, is a significant innovative step. For the first time, a three-dimensional distribution of the free fall parameters under atmospheric conditions was obtained depending on the time of free fall under the condition of an initial velocity different from zero. Obtaining the characteristics of the distribution, taking into account the aerodynamic quality of the object, made it possible to adapt the methods of analytical mechanics and differential equations with variable coefficients to the solution of the applied problem. The obtained characteristic three-dimensional surface allows to perform a study of the aerodynamic quality of the object under study using numerical methods and to visualize dynamic processes in three-dimensional space in order to obtain a qualitative picture of the perturbation of the object by the air flow. This integrated approach to the study of ballistic characteristics of cargo forms the scientific principle to perform engineering calculations to solve the problem of controlling dynamic systems. Practical value. An approximate analytical approach and a calculation model of the dynamic process of a mechatronic unmanned system make it possible to increase its efficiency in the presence of an initial speed and ensure the accuracy of the application of ballistic cargo delivery systems in real operating conditions.

Preface

The 2023 7th International Conference on Mechanics, Mathematics and Applied Physics (ICMMAP 2023) took place in Dalian, China during 8th to 10th September 2023. The conferences in this series started in 2017 as an academic conference and gradually expanded to include wider topic fields as well. Now the International Conference on Mechanics, Mathematics and Applied Physics has been developed into an annual international conference based in China with topics encompassing the whole area of mechanics, mathematics and applied physics.The aim of the Conference was to provide a platform for the exploration of both fundamental topics and new applications of research fields related to mechanics, mathematics and applied physics, bring different scientific communities together, and thus facilitate the contacts between science, technology and industry. All the attendees were welcomed to give comments and raise questions to the speeches and presentations during the Conference.More than 100 participants attended the Conference which consisted of ten keynote speeches, as well as plenty of oral and poster contributions. Among them, Prof. Zhenxin Liu (Dalian University of Technology, China), who is interested in the researches on Stochastic Dynamical Systems and Stochastic Differential Equations, delivered a keynote speech on The Averaging Principle for SDEs with Monotone Coefficients. And Prof. Chongjun Li (Dalian University of Technology, China) made a report on Revised SBFEM on Arbitrary Polygons and Faceted Polyhedrons with the Second Order Completeness by Elimination of Bubble Functions. He focuses his research on Numerical Approximation, Computational Geometry, Multivariate Spline Theory, and Spline Finite Elements. And the other speakers performed their speeches one by one, sharing research experience and professional insights on specific topics. It is really a great opportunity to promote academic communication and collaboration worldwide.This volume gathers papers from regular contributions of ICMMAP 2023; each contribution submitted for publication has been peer reviewed and the Editors are very grateful to the referees for their careful support in improving the original manuscripts. The manuscripts accepted for publication cover different topics of mechanics, mathematics and applied physics: Computational Mechanics, Multiphase Fluid Mechanics, Partial Differential Equation, Simulation and Monte-Carlo Methods, Nuclear Electronics, etc.It is a great honor to present this volume of Journal of Physics: Conference Series and we deeply thank the authors for their enthusiastic and high-grade contribution.Finally, we would like to thank the Conference Chair, the Publication Chair, the members of the Technical Program Committee as well as Organizing Committee, the conference secretariat and the support from Dalian Jiaotong University that allowed the success of ICMMAP 2023.The Committee of ICMMAP 2023List of Committee Member are avaible in this pdf.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

A high-order explicit time-domain FEM using 15-node tetrahedral elements for room acoustics modeling: Basic performance

Time-domain room acoustic modeling using mass-lumped higher-order tetrahedral finite elements with an explicit time-marching scheme is highly attractive because of its excellent geometrical flexibility with unstructured meshing and applicability into massively parallel computing. However, the standard mass-lumped tetrahedral elements yield an unstable scheme, and only implicit time-marching schemes are available. This study proposes a novel wave-based room acoustics solver based on a high-order explicit time-domain FEM. The present solver uses mass-lumped 15-node tetrahedral elements for spatial discretization and a dissipation-free two-stage partitioned Runge-Kutta time integration for time discretization. The 15-node tetrahedral elements had been developed to apply mass-lumping methods to 10-node tetrahedral elements, which are the most common finite element in engineering applications, but its applicability in room acoustics problems is unknown. Higher accuracy and efficiency of the proposed method over the standard method using 10-node tetrahedral elements is presented through an eigenvalue analysis of a long duct model and acoustic simulations in a small room. Additionally, frequency-dependent sound absorbing boundary is implemented by revising an auxiliary differential equation method.

SDE method for cosmic rays modulation in the heliosphere statistical error and solution uniqueness

The cosmic ray modulation in the heliosphere models comes to a new phase of their evolution, where the still higher number of models will be public with published source code. At the dawn of this period, we want to address two topics, the statistical error of their results and the uniqueness of their solutions.We present a method for the evaluation of statistical error for the numerical stochastic differential equation method, which is probably the most used method to numerically solve Parker's equation. We defined a limit of statistical error, for which we present a method to estimate the number of particles needed to be simulated to reach this limit. The estimation of statistical error from a scan of parametric space of two currently available models with public code, for the SOLARPROP model and Geliosphere 2D model, is presented.We present a test of the uniqueness of the solution of Parker's equation for 1D and 2D models of heliospheric modulation. Namely for the 1D B-p model and Geliosphere 2D model. The dependence of solution uniqueness on the selected model's input parameters is presented and discussed. Program summaryProgram Title: CudaHelioCommanderLightCPC Library link to program files:https://doi.org/10.17632/cptpkwphn4.1Developer's repository link:https://github.com/gladiatus55/CudaHelioCommanderLightLicensing provisions: GPLv3Programming language: C#, .NET 6.0Nature of problem: Despite the decades-long history of cosmic ray modulation in the heliosphere models, the topics of the statistical error of their results, and the uniqueness of their solutions were in published articles addressed only in a very limited way. A precise description of the method for statistical error evaluation, with examples applied to available models with published source code, is needed. The uniqueness of the solution of Parker's equation was not addressed in published literature.Solution method: The estimation of statistical error was evaluated for two currently available models with public code, the SOLARPROP model and Geliosphere 2D model, by a scan of their parametric space. The dependence of solution uniqueness on the selected model's input parameters was also done by scanning their parametric space.

Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.