Problems For Ordinary Differential Equations Research Articles

A Variable Step Size Multi-Block Backward Differentiation Formula for Solving Stiff Initial Value Problem of Ordinary Differential Equations

A variable step size multi-block backward differentiation formula for solving stiff initial value problems of ordinary differential equations with a variable step size strategy was derived. The proposed method (VSSMBBDF) computes two approximate solution values at a time per integration step. The stability properties are achieved by varying the step size ratio in the formula to generate more zero stable schemes. The proposed method is also found to be an A-Stable scheme across different choices of the step size. The method is capable of solving stiff IVPs of ODEs. Approximates result from the system of stiff ODE problems considered are found to favorably validate the performance of the new method in terms of accuracy of the scale error and less executional time in respect to the two methods compared in the study. Hence, the proposed method can be an alternative solver for stiff IVPs of ODEs.

Unique Solvability of the Initial-Value Problem for Fractional Functional Differential Equations—Pantograph-Type Model

Contrary to the initial-value problem for ordinary differential equations, where the classical theory of establishing the exact unique solvability conditions exists, the situation with the initial-value problem for linear functional differential equations of the fractional order is usually non-trivial. Here we establish the unique solvability conditions for the initial-value problem for linear functional differential equations of the fractional order. The advantage is the lack of the calculation of fractional derivatives, which is a complicated task. The unique solution is represented by the Neumann series. In addition, as examples, the model with a discrete memory effect and a pantograph-type model from electrodynamics are studied.

On solution of pseudohyperbolic equation with constant coefficients

Thepaperproposesthemethodofformingtheexactsolutionofthefirstinitialboundaryvalueproblemforone-dimensionallinearpseudohyperbolicequation with constant coefficients. Toobtainthesolutiontype, themodificationofpartitionmethod (Fourier method) is used, when the type of one of the solution functional factors is consideredto be known. Atthesametime, theinitialproblemisreducedto parameterized family of Cauchy problems for ordinary differential equations. Thepaperpresentsexplicitly calculated formulas, whichspecify the solution. The qualitative research of the solution properties has been conducted. Theconditionsforcoefficientsintheformofinequalitieshave been obtained that is indicative ofboundedness and variability of the solutions. Several examples confirming the results obtained have been considered.

Solution of the Cauchy Problem for Ordinary Differential Equations Using the Collocation and Least Squares Method with the Pade Approximation

Solution of the Cauchy Problem for Ordinary Differential Equations Using the Collocation and Least Squares Method with the Pade Approximation

A 14‐Order Hybrid Block Method in Variable Step‐Size Mode for Solving Second‐Order Initial Value Problems

The search for efficient higher order methods is a constant goal in numerical analysis. In this paper, a higher order two‐step hybrid block method is presented to directly solve second‐order initial value problems in ordinary differential equations. In addition to the higher order, the proposed method has been formulated in variable step‐size mode to extract its best performance. Comparisons with other methods in the literature show the good accuracy it can provide. Theoretical aspects such as linear stability and convergence analysis are also discussed.

Algorithms of algebraic order nine for numerically solving second-order boundary and initial value problems in ordinary differential equations

Algorithms of algebraic order nine for numerically solving second-order boundary and initial value problems in ordinary differential equations

HERMITE WAVELET METHOD FOR APPROXIMATE SOLUTION OF HIGHER ORDER BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

In this paper, Hermite wavelet method (HWM) is considered for numerical solution of 12- and 13-order boundary value problems (BVPs) of ordinary differential equations (ODEs). The proposed algorithm for HWM developed in Maple software converts the ODEs into an algebraic systems of equations. These algebraic equations are then solved by evaluating the unknown constants present in the system of equations and the approximate solution of the problem is obtained. Test problems are considered and their solutions are investigated using HWM-based algorithm. The obtained results from the test problems are compared with exact solution, and with other numerical methods solution in the existing literature. Results comparison are presented both graphically and in tabular form showing close agreement with exact solution, and greater accuracy than homotopy perturbation method (HPM) and differential transform method (DTM).

The stability of the parametric Cauchy problem of initial-value ordinary differential equations revisited

The stability of the parametric Cauchy problem of initial-value ordinary differential equations revisited

Legendre-Petrov-Galerkin Chebyshev spectral collocation method for second-order nonlinear differential equations

<p style='text-indent:20px;'>We propose a Legendre-Petrov-Galerkin Chebyshev spectral collocation method for initial value problems (IVPs) of second-order nonlinear ordinary differential equations (ODEs). The Legendre-Petrov-Galerkin method is applied to time discretization and the nonlinear term is dealt with Chebyshev spectral collocation method. The scheme results in a simple algebraic system by choosing appropriate basis functions. Optimal error estimates in <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm for the single and multiple interval methods are given. As an application of the method, we construct the space-time spectral schemes for solving some nonlinear time-dependent partial differential equations (PDEs). Numerical experiments suggest the efficiency of the methods.</p>

Naimark Problem for a Fractional Ordinary Differential Equation

Рассматривается задача для обыкновенного дифференциального уравнения дробного порядка, где краевые условия задаются в форме линейных функционалов. Это позволяет охватить достаточно широкий класс линейных локальных и нелокальных условий. Дробная производная понимается в смысле Герасимова-Капуто. Получено необходимое и достаточное условие однозначной разрешимости задачи. В терминах специальных функций найдено представление решения исследуемой задачи. Доказана теорема существования и единственности решения. Библиография: 27 названий.

Двумерное течение Пуазейля ньютоновской жидкости с вязкостью, зависящей от температуры

The problem of a two-dimensional Poiseuille flow of a Newtonian fluid with a viscosity exponen-tially dependent on temperature under the influence of longitudinal pressure and temperature gra-dients is reduced to a two-parameter boundary value problem for a third-order ordinary differential equation with respect to a dimensionless unknown characterizing the change in temperature across the channel. In the range of negative values of the dimensionless longitudinal temperature gradient, the solution has two branches. By closing, the branches form a boundary on the parameter plane beyond which Poiseuille-type flows do not exist. The solutions belonging to different branches dif-fer in the flow rate and intensity of heat transfer by the liquid through solid boundaries. The latter circumstance entails deceleration of the flow in the near-wall areas. In the region of positive values of the dimensionless longitudinal temperature gradient, the solution is unique. With an increase in the mentioned parameter, the heat flow to the liquid from the walls increases, which entails the formation of a more bulged velocity profile compared to the isothermal case. At sufficiently large values of the longitudinal temperature gradient, there appears a ‘stick’ flow with a uniform velocity distribution everywhere except for narrow layers near solid boundaries. It is shown that the rate is proportional to the ratio of the densities of the transverse and longitudinal heat flows. Explicit ex-pressions for the pressure and shear stress fields are given. The properties of monotonicity of the velocity and temperature profiles as functions of the transverse coordinate have been studied. It is shown that, regardless of the values of the dimensionless parameters of the problem, the velocity has a positive maximum on the central axis of the channel and monotonically decreases to zero at the solid boundary.

Reduction theorems for stability of compact sets in time-varying systems

Reduction theorems provide a framework for stability analysis that consists in breaking down a complex problem into a hierarchical list of subproblems that are simpler to address. This paper investigates the following reduction problem for time-varying ordinary differential equations on Rn. Let Γ1 be a compact set and Γ2 be a closed set, both positively invariant and such that Γ1⊂Γ2⊂Rn. Suppose that Γ1 is uniformly asymptotically stable relative to Γ2. Find conditions under which Γ1 is uniformly asymptotically stable. We present a reduction theorem for uniform asymptotic stability that completely addresses the local and global version of this problem, as well as two reduction theorems for uniform stability and either local or global uniform attractivity. These theorems generalize well-known equilibrium stability results for cascade-connected systems as well as previous reduction theorems for time-invariant systems. We also present Lyapunov characterizations of the stability properties required in the reduction theorems that to date have not been investigated in the stability theory literature.

An accurate method for dispersion characteristics of surface waves in layered anisotropic semi-infinite spaces

In this paper, we develop a novel method for the dispersion characteristics of surface waves in layered anisotropic semi-infinite spaces. Compared to solving the algebraic transcendental eigenequations directly, this method can find all surface waves accurately. For solving accurately, the first-order state equation is formed by introducing the dual variable. Some properties of the complex Hamiltonian matrix are presented and proved, then based on these properties, the radiation condition for the homogeneous anisotropic half-space is given. And, based on the scaling and squaring algorithm and mixed energy matrix, the precise integration method (PIM) is presented for solving accurately the eigenvalue problem of ordinary differential equations (ODEs) corresponding to surface waves. All eigenfrequencies can be found with certainty by using the concept of the eigenvalue count in the Wittrick-Williams (W-W) algorithm. Numerical examples show that the proposed method is highly accurate and efficient for solving dispersion relations of surface waves in layered anisotropic semi-infinite spaces, which involves the comparison of other previously published methods.

Deep limits of residual neural networks

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The contribution of this paper is to show that, for the residual neural network model, the deep layer limit coincides with a parameter estimation problem for a nonlinear ordinary differential equation. In particular, whilst it is known that the residual neural network model is a discretisation of an ordinary differential equation, we show convergence in a variational sense. This implies that optimal parameters converge in the deep layer limit. This is a stronger statement than saying for a fixed parameter the residual neural network model converges (the latter does not in general imply the former). Our variational analysis provides a discrete-to-continuum Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence result for the objective function of the residual neural network training step to a variational problem constrained by a system of ordinary differential equations; this rigorously connects the discrete setting to a continuum problem.

Modified Adomian Method through Efficient Inverse Integral Operators to Solve Nonlinear Initial-Value Problems for Ordinary Differential Equations

The present manuscript examines different forms of Initial-Value Problems (IVPs) featuring various types of Ordinary Differential Equations (ODEs) by proposing a proficient modification to the famous standard Adomian decomposition method (ADM). The present paper collected different forms of inverse integral operators and further successfully demonstrated their applicability on dissimilar nonlinear singular and nonsingular ODEs. Furthermore, we surveyed most cases in this very new method, and it was found to have a fast convergence rate and, on the other hand, have high precision whenever exact analytical solutions are reachable.

Методи розв’язування початкової задачі для нелінійних інтегро-диференціальних рівнянь з оцінкою локальної похибки

One of the modern scientific methods of researching phenomena and pro-cesses is mathematical modeling, which in many cases allows replacing the real process and makes it possible to obtain both a qualitative and a quantitative pic-ture of the process. Since the exact solutions of such models can be found in very individual cases, it is necessary to use approximate methods. In applied mathematics, fractional-rational approximations, which under appropriate con-ditions give a high rate of convergence of algorithms, bilateral and monotonic approximations have become widely used.In this work, using the technique of constructing one-step methods for solv-ing the initial problem for ordinary differential equations and developing the sought solution into a finite continued fraction, a numerical method for solving the Cauchy problem for nonlinear integro-differential equations of the Volterra type is proposed. The values of the parameters at which the nonlinear method of the first and second order of accuracy is obtained are found.Computational formulas are proposed, which at each integration step allow obtaining an upper and lower approximation to the exact solution without additional references to the right-hand side of the integro-differential equation. Calculation formulas, in which the main terms of the local error differ only in sign, form a two-sided method. We take the half-sum of bilateral approximations to the exact solution as the approximate solution at the given integration point, and the absolute value of the half-difference determines the error of the obtained result.The modular nature of the proposed algorithms makes it possible to ob-tain several approximations to the exact solution of the initial problem for the nonlinear integro-differential equation at each point of integration. The comparison of these approximations gives useful information in the matter of choosing the integration step or in assessing the accuracy of the result

A Single-Step Modified Block Hybrid Method for General Second-Order Ordinary Differential Equations

A multistep collocation approach is used to derive a single-step modified block hybrid method (MBHM) of order five for solving general second-order initial-value problems (IVPs) of ordinary differential equations (ODEs). The new method's basic convergence property is established, and its numerical accuracy is demonstrated using numerical examples from the literature. The new method outperforms similar methods in terms of accuracy, earning it a recommendation as a likely candidate for solving general second-order ODEs.

Generalization of a class of uniformly optimized [formula omitted]-step hybrid block method for solving two-point boundary value problems

This study aims to develop, analyze and implement an efficient method for approximating two-point boundary value problems of ordinary differential equations. The method contains six and twelve implicit formulas, respectively, for the one-step and two-step schemes. The continuous approximations, using the shifted Chebyshev polynomial as the basis function, were obtained via evaluations at three different points on the selected one-step method, including two optimized hybrid points. Evaluations were carried out on six different points on the selected two-step method, including four generalized optimized hybrid points. Qualitative analysis of the method proves the proposed methods are consistent, zero-stable, convergent and have a larger region of absolute stability. The quantitative analysis shows that the methods compared favorably well and established some superiority strength with the existing methods.

An improved dynamic model and numerical simulation of yarn in the ring spinning system

The nonlinear dynamic behavior of ballooning yarn in the ring spinning system is investigated. An improved dynamic model of ballooning yarn with axial conveying speed is established. Based on Hamilton’s law of variation principle, the three-dimensional dynamic differential equations are derived. Then, the shooting method is employed to solve the two-point boundary value problem of ordinary differential equations. The numerical results of balloon profiles and distribution tension curves of ballooning yarns are discussed. The numerical results imply that with the increase of Ω, the yarn conveying speed increases, the minimum tension at the guide eye increases, and the helical loops of the balloon profile decrease. It is also found that as Ω increases, the spatial area of the guide eye tension [Formula: see text] that forms the dynamic balloon decreases. When the yarn conveying speed is medium or high, under different tensions at the guide eye, the yarn tensions along the height direction of the balloon change in the same trend, and the tensions of the ballooning yarn at the same height are very close. In addition, with the increase of Ω, the distribution tension of the ballooning yarn changes significantly under the same guide eye tension.

Fifth Order Improved Runge-Kutta Nystrom Method Using Trigonometrically-Fitting for Solving Oscillatory Problems

In this paper, the Trigonometrically Fitted Improved Runge-KuttaNystrom method is proposed as a novel method with four stages and fifth order for solving oscillatory problems. This method is intended to integrate second-order initial value problems using the trigonometrically fitting approach. To increase the method'saccuracy, the principal frequency of the problem푤∈ℝ, is used. It is discovered that the new method is more precise when compared with the other existing Runge-Kutta Nystrom and IRKN5 methods. To show how well the TFIRKN5 method works, test problems for second-order ordinary differential equations (ODEs) are solved. The numerical outcomes show that the novel approach outperforms methods that have already been published.