SimDE App: Simulating and visualizing formal theories using differential equations.

Ordinary differential equations on singular spaces, Z. Bartosiewicz stability in delayed neural networks, J. Belair a condition on multi-existence of periodic solutions for a differential delay equation, Y. Cao control of global economic growth - will the centre hold?, E.N. Chukwu asymptotic behaviour of the Titchmarsh-Weyl coefficient for a coupled second order system, S.L. Clark stability problems for systems of nuclear reactors, C. Cordoneanu comparison theorems for disconjugate linear differential equations, M. Gaudenzi oscillation results for higher order nonlinear neutral delay equations with periodic coefficients, J.R. Graef, M.K. Grammatikopoulos and P.W. Spikes an implicit differential equation related to epidemic models, K.P. Hadeler and R. Shonkwiler the relationship between stability under disturbances and uniform stability in a periodic integrodifferential equation, Y. Hamaya on the asymptotic stability of the equilibrium of the damped oscillator, L. Hatvani on higher order nonlinear differential-difference equations, U. An Der Heiden vector field approximations flow homogencity, H. Hermes hopf bifurcation for a differential-difference equation from climate modeling, G. Hetzer shock layer behaviour for vector boundary value problems, S.J. Kirschvink properties of solutions of nth order equations, W.A.J. Kosmala small solutions to BVP's at resonance with nonhomogeneous nonlinearity, L. Lefton vibrational control of time delay systems, B. Lehman bifurcation set and compound eyes in a perturbed cubic Hamiltonian system, J. Li and Z. Lu gevrey character of formal solutions of nonlinear differential equations, X. Liu finite-difference schemes having the correct linear stability properties for all finite step-sizes, R.E. Mickens evolution of surface functionals and differential equations, Y. Li and J.S. Muldowney some remarks on stability properties in functional differential equations with infinite delay, S. Murakami and T. Yoshizawa periodic orbits of the Froeschle's map, A. Olvera and C. Vargas rotated vector fields, global families of limit cycles and Hilbert's 16th problem, L.M. Perko green's matrices and disconjugacy of a vector difference equation, A. Peterson the poincare manifold for the general case of a planar flow, W. Rivera hopf bifurcation in a class of ODE systems related to climate modeling, P.G. Schmidt on second order two point boundary value problems at resonance, M. Hihnala and S. Seikkala separatrix connections of quadratic gradient vector fields, D.S. Shafer. Part contents.

The article deals with applying mathematics in chemistry and chemistry-technology. Specifically, differential equations are extensively used in various fields of science and technology. That is why the theory of differential equations, as a separate topic in the course of higher mathematics, is of major importance in educational system of future mechanics, physicists, electrical engineers, chemists, mechanical engineers etc. A possibility of using differential equations in solving various chemical problems is demonstrated. Some chemical technology problems are exemplified whose general solution is reduced to separating variables equations, first-order linear differential equations, second-order linear homogeneous differential equations. It is noteworthy that in solving chemical technology problems we deal with all of these types of differential equations. First-order homogeneous differential equations are applied in solving the following problems: chemical compounds chlorination; chemical agent consumption with maximum end product yield in complex reactions. Second-order non-homogeneous differential equations with constant coefficients are used in solving problems of a system of reverse reactions running at constant volume; continuous hydrolysis of solid fat in a spray column. Second-order differential equations which allow reduction of order are utilized for problems such as liquid movement in capillaries. Second-order linear non-homogeneous differential equations with constant coefficients are applied to solve various problems, e.g. to find a law of motion of a particle that falls as a precipitate in a liquid having no initial velocity.

In this paper, we consider two different models of nonlinear ordinary differential equations (ODEs) of second order. We construct two new Lyapunov functions to investigate boundedness of solutions of those nonlinear ODEs of second order. By using the Lyapunov direct or second method and inequality techniques, we prove two new theorems on the boundedness solutions of those ODEs of second order as t to infty . When we compare the conditions of the theorems of this paper with those of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), we can see that our theorems have less restrictive conditions than those in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002) because of the two new suitable Lyapunov functions. Next, in spite of the use of the Lyapunov second method here and in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995; Sun and Meng in Ann. Differ. Equ. 18(1):58–64 2002), the proofs of the results of this paper are proceeded in a very different way from that used in the literature for the qualitative analysis of ODEs of second order. Two examples are given to show the applicability of our results. At the end, we can conclude that the results of this paper generalize and improve the results of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995), Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), and some other that can be found in the literature, and they have less restrictive conditions than those in these references.

Scientific machine learning is a burgeoning discipline which blends scientific computing and machine learning.Traditionally, scientific computing focuses on large-scale mechanistic models, usually differential equations, that are derived from scientific laws that simplified and explained phenomena.On the other hand, machine learning focuses on developing non-mechanistic data-driven models which require minimal knowledge and prior assumptions.The two sides have their pros and cons: differential equation models are great at extrapolating, the terms are explainable, and they can be fit with small data and few parameters.Machine learning models on the other hand require "big data" and lots of parameters but are not biased by the scientists ability to correctly identify valid laws and assumptions.However, the recent trend has been to merge the two disciplines, allowing explainable models that are data-driven, require less data than traditional machine learning, and utilize the knowledge encapsulated in centuries of scientific literature.The promise is to fuse a priori domain knowledge which doesn't fit into a "dataset", allow this knowledge to specify a general structure that prevents overfitting, reduces the number of parameters, and promotes extrapolatability, while still utilizing machine learning techniques to learn specific unknown terms in the model.This has started to be used for outcomes like automated hypothesis generation and accelerated scientific simulation.The purpose of this blog post is to introduce the reader to the tools of scientific machine learning, identify how they come together, and showcase the existing open source tools which can help one get started.We will be focusing on differentiable programming frameworks in the major languages for scientific machine learning: C++, Fortran, Julia, MATLAB, Python, and R.We will be comparing two important aspects: efficiency and composability.Efficiency will be taken in the context of scientific machine learning: by now most tools are well-optimized for the giant neural networks found in traditional machine learning, but, as will be discussed here, that does not necessarily make them efficient when deployed inside of differential equation solvers or when mixed with probabilistic programming tools.Additionally, composability is a key aspect of scientific machine learning since our toolkit is not ML in isolation.Our goal is not to do machine learning as seen in a machine learning conference (classification, NLP, etc.), and it's not to do traditional machine learning as applied to scientific data.Instead, we are putting ML models and techniques into the heart of scientific simulation tools to accelerate and enhance them.Our neural networks need to fully integrate with tools that simulate satellites and robotics simulators.They need to integrate with the packages that we use in our scientific work for verifying numerical accuracy, tracking units, estimating uncertainty, and much more.We need our neural networks to play nicely with existing packages for delay

KANTBP 3.0: New version of a program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel adiabatic approach

59 - Exact Second Order Equations

The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerkin approximation method can be applied to a second-order delay differential equation in two ways: either by converting into a second-order partial differential equation and then into a system of second-order ordinary differential equations (the “second-order Galerkin” method) or by first expressing as two first-order delay differential equations and converting into a system of first-order partial differential equations and then into a first-order ordinary differential equation system (the “first-order Galerkin” method). In this paper, we demonstrate that these subtly different formulation procedures lead to different roots of the characteristic polynomial. In particular, the second-order Galerkin method produces spurious roots near the origin, which must then be identified through substitution into the characteristic polynomial of the original delay differential equation. However, spurious roots do not arise if the first-order Galerkin method is used, which can reduce computation time and simplify stability analyses. We describe these two formulation strategies and present numerical examples to highlight their important differences.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

5 - Ordinary Differential Equations

This thesis concerns the analysis of differential equations with uncertain input parameters, in the form of random variables or stochastic processes with any type of probability distributions. In modeling, the input coefficients are set from experimental data, which often involve uncertainties from measurement errors. Moreover, the behavior of the physical phenomenon under study does not follow strict deterministic laws. It is thus more realistic to consider mathematical models with randomness in their formulation. The solution, considered in the sample-path or the mean square sense, is a smooth stochastic process, whose uncertainty has to be quantified. Uncertainty quantification is usually performed by computing the main statistics (expectation and variance) and, if possible, the probability density function. In this dissertation, we study random linear models, based on ordinary differential equations with and without delay and on partial differential equations. The linear structure of the models makes it possible to seek for certain probabilistic solutions and even approximate their probability density functions, which is a difficult goal in general. A very important part of the dissertation is devoted to random second-order linear differential equations, where the coefficients of the equation are stochastic processes and the initial conditions are random variables. The study of this class of differential equations in the random setting is mainly motivated because of their important role in Mathematical Physics. We start by solving the randomized Legendre differential equation in the mean square sense, which allows the approximation of the expectation and the variance of the stochastic solution. The methodology is extended to general random second-order linear differential equations with analytic (expressible as random power series) coefficients, by means of the so-called Frobenius method. A comparative case study is performed with spectral methods based on polynomial chaos expansions. On the other hand, the Frobenius method together with Monte Carlo simulation are used to approximate the probability density function of the solution. Several variance reduction methods based on quadrature rules and multilevel strategies are proposed to speed up the Monte Carlo procedure. The last part on random second-order linear differential equations is devoted to a random diffusion-reaction Poisson-type problem, where the probability density function is approximated using a finite difference numerical scheme. The thesis also studies random ordinary differential equations with discrete constant delay. We study the linear autonomous case, when the coefficient of the non-delay component and the parameter of the delay term are both random variables while the initial condition is a stochastic process. It is proved that the deterministic solution constructed with the method of steps that involves the delayed exponential function is a probabilistic solution in the Lebesgue sense. Finally, the last chapter is devoted to the linear advection partial differential equation, subject to stochastic velocity field and initial condition. We solve the equation in the mean square sense and provide new expressions for the probability density function of the solution, even in the non-Gaussian velocity case.

5 - Differential equations

Placing human landscape legacies in a dynamic systems framework.

4 - THE APPLICATION OF DIGITAL DIFFERENTIAL ANALYSERS

We analyse nonlinear second-order differential equations in terms of algebraic properties by reducing a nonlinear partial differential equation to a nonlinear second-order ordinary differential equation via the point symmetry f(v)∂v. The eight Lie point symmetries obtained for the second-order ordinary differential equation is of maximal number and a representation of the sl(3,R) algebra. We extend this analysis to a more general nonlinear second-order differential equation and we obtain similar interesting algebraic properties.

Similarity solutions of nonlinear partial differential equations invariant to a family of affine groups