DEtools - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : DEtools : Differential Operators : DEtools/newton_polygon

DEtools

  

newton_polygon

  

construct the Newton polygon of a LODE

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

newton_polygon(L, y, u)

newton_polygon(L, y, u, x = x0)

Parameters

L

-

linear homogeneous differential equation

y

-

unknown function to search for

u

-

name

x0

-

(optional) rational or an algebraic number or infinity

Description

• 

The newton_polygon function constructs the Newton polygon of a linear differential operator at the point x0. The linear differential operator L corresponds to the differential equation Ly=0. The equation Ly=0 must be homogeneous and linear in y and its derivatives, and its coefficients must be rational functions in the variable x. The variable u must be a name and specifies the variable for the Newton polynomials. The variable x0 must be a rational or an algebraic number or the symbol infinity. If x0 is not passed as argument, then x0=0 is assumed.

• 

The Newton polygon of a linear differential operator at a point x0 is defined in the following way: for a rational function f, denote v_x0f the valuation at the point x0, that is, the lowest power in the series expansion of f at x0: f=f0xx0v_x0f+terms of higher order.  f0 is called the leading coefficient of f at x0. For real numbers u,v let Qu,v be the subset of RxR defined by {x,y|xuandyv}. Now consider a monomial aididxiyx of L. Define ML as the union of Qi,v_x0aii for i=0..orderL. The Newton polygon is then the lower convex hull of the set ML. The slopes of the Newton polygon are non-negative rational numbers, and with each slope one associates the Newton polynomial Pku. The length lk of a slope k is defined as the length of the projection of this slope onto the x-axis. Pk is a polynomial in the new indeterminate u of degree lkdenomk. Its monomials can be computed from the points which lie exactly on the slope k and the leading coefficients of the corresponding ai's. This Newton polynomial is called "reduced characteristic polynomial" in Barkatou, "Rational Newton Algorithm for computing formal solutions of linear differential equations", ISSAC'88.

• 

The Newton polygon is an important tool for the classification of singular points of linear differential equations, because the slopes give information on the asymptotic behavior of the solutions near the singularity. It is also used in the computation of formal solutions; see also DEtools[formal_sols] and DEtools[regular_parts].

• 

The output is a list of entries of the form k,Pku where k is a slope of the Newton polygon and Pk is the associated Newton polynomial. Note that P0u is also called an indicial equation. The roots of P0u are called the exponents.

• 

This function is part of the DEtools package, and so it can be used in the form newton_polygon(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[newton_polygon](..).

Examples

withDEtools:

odex7ⅆ4ⅆx4yxx+x7ⅆⅆxyxx9yx

odex7ⅆ4ⅆx4yxx7+xⅆⅆxyxx9yx

(1)

newton_polygonode,yx,u

0,u,1,u31

(2)

newton_polygonode,yx,u,x=1

0,u46u3+11u26u

(3)

newton_polygonode,yx,u,x=∞

32,u21

(4)

odex2+15xⅆ2ⅆx2yxx2+13ⅆⅆxyxx9yx

odex2+15xⅆ2ⅆx2yxx2+13ⅆⅆxyxx9yx

(5)

newton_polygonode,yx,u,x=RootOfxx2+1,xx

32,32u+RootOf_Z2+1

(6)

See Also

DEtools

DEtools/DEplot_polygon

DEtools/formal_sols

DEtools/regular_parts