Abel’s Differential Equations
نویسندگان
چکیده
This paper is the text of a talk given by the second author at the Chern Conference. Intended for a general audience, it is based on joint work in progress with Mark Green. The purposes of the talk were (i) to discuss Abel’s differential equations (DE’s) for algebraic curves in both classical and modern form, (ii) to explain in a special case the extension of Abel’s DE’s to general algebraic varieties that Mark and I have developed, and (iii) to discuss the integration of Abel’s DE’s in this special case. The emphasis throughout is on the geometric content of the differential equations.
منابع مشابه
Yang-Laplace transform method Volterra and Abel's integro-differential equations of fractional order
This study outlines the local fractional integro-differential equations carried out by the local fractional calculus. The analytical solutions within local fractional Volterra and Abel’s integral equations via the Yang-Laplace transform are discussed. Some illustrative examples will be discussed. The obtained results show the simplicity and efficiency of the present technique with application t...
متن کاملFractional Calculus for Solving Abel’s Integral Equations Using Chebyshev Polynomials
Abstract In this paper, the numerical method for solving Abel’s integral equations is presented. This method is based on fractional calculus. Also, Chebyshev polynomials are utilized to apply fractional properties for solving Abel’s integral equations of the first and second kind. The fractional operator is considered in the sense of RiemannLiouville. Although Abel’s integral equations as singu...
متن کاملConstruction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)
We provide a new mathematical technique leading to the construction of the exact parametric or closed form solutions of the classes of Abel’s nonlinear differential equations ODEs of the first kind. These solutions are given implicitly in terms of Bessel functions of the first and the second kind Neumann functions , as well as of the free member of the considered ODE; the parameter ν being intr...
متن کاملLegendre Wavelets for Systems of Fredholm Integral Equations of the Second Kind
Orthogonal functions and polynomials have been used by many authors for solving various problems. The main idea of using orthogonal basis is that a problem reduces to solving a system of linear or nonlinear algebraic equations by truncated series of orthogonal basis functions for solution of problem and using the operational matrices. Here we use Legendre wavelets basis on interval [0, 1]. Some...
متن کاملUsing Abel's Theorem to Explain Repeated Roots of the Characteristic Equation
This document describes how one can derive the solutions to a linear constant coefficient homogeneous differential equation with repeated roots in the characteristic equation with Abel’s Theorem.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002