scholarly journals Algebraic solutions of differential equations over ℙ1 −{0,1,∞}

2018 ◽  
Vol 14 (05) ◽  
pp. 1427-1457
Author(s):  
Yunqing Tang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Elliptic Curve ◽  
Convergence Condition ◽  
Differential Galois Group ◽  
Reduction Modulo ◽  
Local Monodromy ◽  
Monodromy Groups ◽  
Almost All ◽  
Algebraic Solutions

The Grothendieck–Katz [Formula: see text]-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo [Formula: see text] has vanishing [Formula: see text]-curvatures for almost all [Formula: see text] has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on [Formula: see text] We prove a variant of this conjecture for [Formula: see text] which asserts that if the equation satisfies a certain convergence condition for all [Formula: see text] then its monodromy is trivial. For those [Formula: see text] for which the [Formula: see text]-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of [Formula: see text]-curvatures and certain local monodromy groups. We also prove similar variants of the [Formula: see text]-curvature conjecture for an elliptic curve with [Formula: see text]-invariant [Formula: see text] minus its identity and for [Formula: see text].

On the Local Structure of Mahler Systems

2020 ◽  
Author(s):  
Julien Roques
Keyword(s):  
Differential Equations ◽  
Local Structure ◽  
Alternative Proof ◽  
Reduction Modulo ◽  
Modulo P ◽  
Almost All ◽  
The Given ◽  
Algebraic Solutions

Abstract This paper is a 1st step in the direction of a better understanding of the structure of the so-called Mahler systems: we classify these systems over the field $\mathscr{H}$ of Hahn series over $\overline{{\mathbb{Q}}}$ and with value group ${\mathbb{Q}}$. As an application of (a variant of) our main result, we give an alternative proof of the following fact: if, for almost all primes $p$, the reduction modulo $p$ of a given Mahler equation with coefficients in ${\mathbb{Q}}(z)$ has a full set of algebraic solutions over $\mathbb{F}_{p}(z)$, then the given equation has a full set of solutions in $\overline{{\mathbb{Q}}}(z)$ (this is analogous to Grothendieck’s conjecture for differential equations).

scholarly journals Calculating Galois groups of third-order linear differential equations with parameters

2018 ◽  
Vol 20 (04) ◽  
pp. 1750038
Author(s):  
Andrei Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Galois Group ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Unipotent Radical ◽  
Third Order ◽  
Differential Galois Group ◽  
Bessel Differential Equation ◽  
Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

scholarly journals Lyapunov Spectrum of Nonautonomous Linear Young Differential Equations

2019 ◽  
Vol 32 (4) ◽  
pp. 1749-1777 ◽  
Author(s):  
Nguyen Dinh Cong ◽  
Luu Hoang Duc ◽  
Phan Thanh Hong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Lyapunov Exponents ◽  
Integrability Condition ◽  
Lyapunov Spectrum ◽  
Triangular Systems ◽  
Almost All ◽  
Stochastic Setting ◽  
Two Parameter ◽  
Finite Moments

Abstract We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is independent of the driving path for triangular systems which are regular in the sense of Lyapunov. In the stochastic setting, the system generates a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are random variables of finite moments. Finally, we prove a Millionshchikov theorem stating that almost all, in a sense of an invariant measure, linear nonautonomous Young differential equations are Lyapunov regular.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

scholarly journals Oscillation and non-oscillation of some neutral differential equations of odd order

1992 ◽  
Vol 15 (3) ◽  
pp. 509-515 ◽  
Author(s):  
B. S. Lalli ◽  
B. G. Zhang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solution ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
Neutral Term ◽  
Odd Order ◽  
Existence Criterion ◽  
Oscillation Of Solutions

An existence criterion for nonoscillatory solution for an odd order neutral differential equation is provided. Some sufficient conditions are also given for the oscillation of solutions of somenth order equations with nonlinearity in the neutral term.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

scholarly journals New Bi-modular Material Approach to Buckling Problem of Reinforced Concrete Columns

2016 ◽  
Vol 6 (1) ◽  
pp. 19 ◽  
Author(s):  
Ahmad Salah Edeen Nassef ◽  
Mohammed A. Dahim
Keyword(s):  
Differential Equation ◽  
Reinforced Concrete ◽  
Differential Equations ◽  
Concrete Columns ◽  
Neutral Axis ◽  
Reinforced Concrete Columns ◽  
New Approach ◽  
Codes Of Practice ◽  
Governing Differential Equation ◽  
Transverse Deflection

<p class="1Body">This paper was investigating the buckling problem of reinforced concrete columns considering the reinforced concrete as bi – modular material. Governing differential equations was driven. The relation between the non-dimensional transverse deflection and non-dimensional distance between centroid axis and the neutral axis "eccentricity" was drawn to enable the solution of the governing differential equation. The new approach was verified with different experimental results and different codes of practice.<strong></strong></p>

Sign in / Sign up

Export Citation Format

Share Document