Ordinary differential equations and Easter Island: a survey of recent research developments on the relationship between humans, trees, and rats

2018 ◽  
Vol 5 (3) ◽  
pp. 929-936 ◽  
Author(s):  
Lorelei Koss
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Easter Island ◽  
The Relationship

The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 170-184
Author(s):  
Richard Atkins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shortest Paths ◽  
Equivalence Problem ◽  
Second Order ◽  
System Of Differential Equations ◽  
Lagrange Equations ◽  
Invariant Functions ◽  
Underlying Geometry ◽  
The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

EQUIVALENCE OF THE TWO METHODS IN CALCULATING THE EQUILIBRIUM SHAPES OF CYLINDRICAL VESICLES

1999 ◽  
Vol 13 (16) ◽  
pp. 547-553
Author(s):  
SHAOGUANG ZHANG ◽  
ZHONGCAN OUYANG ◽  
JIXING LIU
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
The Other ◽  
General Shape ◽  
Calculus Of Variation ◽  
And Topology ◽  
Shape Equation ◽  
Equilibrium Shapes ◽  
Comparison Of The Results ◽  
The Relationship

So far, two methods are often used in solving the equilibrium shapes of vesicles. One method is by starting with the general shape equation and restricting it to the shapes with particular symmetry. The other method is by assuming the symmetry and topology of the vesicle first and treating it with the calculus of variation to get a set of ordinary differential equations. The relationship between these two methods in the case of cylindrical vesicles, and a comparison of the results are given.

scholarly journals Solution of Higher-Order ODEs Using Backward Difference Method

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Mohamed Bin Suleiman ◽  
Zarina Bibi Binti Ibrahim ◽  
Ahmad Fadly Nurullah Bin Rasedee
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Method ◽  
Numerical Technique ◽  
Higher Order ◽  
First Order ◽  
The Relationship

The current numerical technique for solving a system of higher-order ordinary differential equations (ODEs) is to reduce it to a system of first-order equations then solving it using first-order ODE methods. Here, we propose a method to solve higher-order ODEs directly. The formulae will be derived in terms of backward difference in a constant stepsize formulation. The method developed will be validated by solving some higher-order ODEs directly with constant stepsize. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The result presented confirmed our hypothesis.

scholarly journals On Spectral Properties of Doubly Stochastic Matrices

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 369
Author(s):  
Mutti-Ur Rehman ◽  
Jehad Alzabut ◽  
Javed Hussain Brohi ◽  
Arfan Hyder
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Quadratic Forms ◽  
Spectral Properties ◽  
Singular Values ◽  
Low Rank ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
The Relationship

The relationship among eigenvalues, singular values, and quadratic forms associated with linear transforms of doubly stochastic matrices has remained an important topic since 1949. The main objective of this article is to present some useful theorems, concerning the spectral properties of doubly stochastic matrices. The computation of the bounds of structured singular values for a family of doubly stochastic matrices is presented by using low-rank ordinary differential equations-based techniques. The numerical computations illustrating the behavior of the method and the spectrum of doubly stochastic matrices is then numerically analyzed.

MARKOV ATTRACTORS: A PROBABILISTIC APPROACH TO MULTIVALUED FLOWS

2008 ◽  
Vol 08 (01) ◽  
pp. 59-75 ◽  
Author(s):  
FRANCO FLANDOLI ◽  
JOSÉ A. LANGA
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Probabilistic Approach ◽  
Global Attractors ◽  
Markov Kernel ◽  
Markov Selections ◽  
The Relationship

Deterministic ordinary differential equations without uniqueness may be studied in the framework of Markov selections. This motivates a concept of Markov attractor, namely a set which attracts, in a suitable probabilistic sense, a Markov kernel. The relationship with respect to global attractors for multivalued flows in the sense of Ball [4] or Melnik and Valero [18] is also established.

scholarly journals Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complete Classification ◽  
First Integrals ◽  
Second Order ◽  
Point Transformations ◽  
Maximal Algebra ◽  
The Relationship ◽  
Linear Odes

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

scholarly journals An Exponential Spline Difference Scheme for Solving a Class of Boundary Value Problems of Second-Order Ordinary Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Dunqian Cao
Keyword(s):  
Differential Equations ◽  
Difference Scheme ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Spline Function ◽  
Boundary Value ◽  
Second Order ◽  
Smooth Functions ◽  
Exponential Spline ◽  
The Relationship

In this paper, we mainly study an exponential spline function space, construct a basis with local supports, and present the relationship between the function value and the first and the second derivative at the nodes. Using these relations, we construct an exponential spline-based difference scheme for solving a class of boundary value problems of second-order ordinary differential equations (ODEs) and analyze the error and the convergence of this method. The results show that the algorithm is high accurate and conditionally convergent, and an accuracy of 1/240h6 was achieved with smooth functions.

scholarly journals Solving Nonstiff Higher Order Odes Using Variable Order Step Size Backward Difference Directly

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohamed bin Suleiman ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Variable Order ◽  
Numerical Techniques ◽  
Step Size ◽  
The Relationship

The current numerical techniques for solving a system of higher order ordinary differential equations (ODEs) directly calculate the integration coefficients at every step. Here, we propose a method to solve higher order ODEs directly by calculating the integration coefficients only once at the beginning of the integration and if required once more at the end. The formulae will be derived in terms of backward difference in a constant step size formulation. The method developed will be validated by solving some higher order ODEs directly using variable order step size. To simplify the evaluations of the integration coefficients, we find the relationship between various orders. The results presented confirmed our hypothesis.

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