formal solutions of a homogeneous linear ODE

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DEtools[formal_sol] - formal solutions of a homogeneous linear ODE

	Calling Sequence
	formal_sol(L, domain, T, opt) formal_sol(eqn, dvar, T, opt)

Parameters

eqn	-	homogeneous linear differential equation
dvar	-	dependent variable, for instance
L	-	differential operator representing an homogeneous linear differential equation (see DEtools[de2diffop])
domain	-	list containing two names representing the differential algebra, for instance
T	-	(optional) name, used in the output to represent , where is the expansion point, is the ramification index and some constant
opt	-	(optional), any of x = x0 to expand around x0, order = n to compute the first n terms or groundfield = [RootOf(...), ...] .

Description

•	The input is a differential operator L or a linear ODE (ordinary differential equation) eqn having rational function coefficients.

•	The output is a list of lists. Each of these lists contains one type of formal solution; those solutions that can be written with the same (see below for a description) are grouped together in the same list.

•	Let be the independent variable. If a differential operator is specified, then is the second element of the list domain. If an ODE is specified, then is implicitly given in dvar, which is of the form .

•

To set the notations, we first discuss formal solutions at the point . Denote as the set of all formal power series in . Formal power series are power series that are not necessarily convergent. If is a positive integer, then denote as the set of all formal power series in . These series can be written as Sum(a[i]*x^(i/r), i = 0 .. infinity) where the are in . If is such a series, then the ramification index of is the smallest positive integer for which is in .

•	Elements of are also called Puiseux series. However, Puiseux series are more general in the sense that there one also allows negative exponents (so then the valuation (defined below) could be negative). One denotes as the set of all Puiseux series whose ramification index divides .

•

Denote as the set of expressions that can be written in the form Sum(Sum(a[i, j]*x^(i/r)*log(x)^j, i = 0 .. infinity), j = 0 .. M) for some in and some non-negative integer . If is such an expression, and if is not zero, then the valuation of is the smallest rational number for which the -term in is not zero. So the valuation is the smallest for which is not zero for some .

•	Denote as the set of all expressions of the form where in and is a non-negative integer.

•

A formal solution is a solution that can be presented in the form for some in and in for some integer (if one chooses this integer minimal, then it is called the ramification index). A homogeneous linear differential equation of order has precisely linearly independent formal solutions, which are computed by formal_sol.

•

If the ODE has a formal solution of the form and if the valuation of is zero, then the expression is called the generalized exponent of . This generalized exponent is an element of . Note that if the valuation of is not zero, then one can rewrite to the above form with valuation 0 for . For example, if where is for example then one can rewrite as so the generalized exponent of this would be . As another example, if then this can be rewritten as so this would have generalized exponent .

•	The generalized exponents of the ODE are the generalized exponents of the formal solutions of the differential equation. The number of generalized exponents equals the order if we count with multiplicity.

•	If a generalized exponent is a constant (if is in ) then it is called an exponent. The exponents are the solutions of the indicial equation.

•	If the ODE has a generalized exponent that is not a constant, then the differential equation is called irregular singular. Such has negative powers of and so the expression will have an essential singularity at .

•	The name T, when specified, is used to denote times a constant, where r is the ramification index. formal_sol computes the formal solutions and expresses them in terms of T. The relation between T and is given in the output as well, in each list containing one type of formal solution.

•

If the optional argument where in $(P)^(1) (=C union {infinity})$ is given, then this procedure first applies a transformation DEtools[translate] to move the point to the point , then computes the formal solutions, and then substitutes in the result (or if ). Note that this substitution only affects the part of the output that gives the relation between T and . If and the ramification index is , the relation between and T given in the output will be .

•	The formal solutions are computed only up to conjugation over the field , where is the minimal field of constants over which the input is defined. A larger field can be specified by the option groundfield = a list of RootOfs.

•	By default the number of terms to be computed is given by the variable Order. To obtain a different number of terms, use the optional argument where is the desired number of terms.

•	The argument domain describes the differential algebra. If this argument is the list , then the differential operators are notated with the symbols and . They are viewed as elements of the differential algebra where is the field of constants, and denotes the differentiation operator.

•	If the argument domain is omitted then the differential algebra specified by the environment variable _Envdiffopdomain is used. If this environment variable is not set then the argument domain may not be omitted.

•	Instead of a differential operator, the input can also be a linear homogeneous ODE having rational function coefficients. In this case, the second argument dvar must be the dependent variable.

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

Examples

(1)

[series(1+O((x-1)^3),x = 1,3), (x-1)^3*exp(-RootOf(1+_Z^2, index = 1)/(x-1))*(series(1+(3*RootOf(1+_Z^2, index = 1))*(x-1)-12*(x-1)^2+O((x-1)^3),x = 1,3)), (x-1)^3*exp(-RootOf(1+_Z^2, index = 2)/(x-1))*(series(1+(3*RootOf(1+_Z^2, index = 2))*(x-1)-12*(x-1)^2+O((x-1)^3),x = 1,3))]

(2)

Giving the optional argument T, a 'name', the output is expansion in T, making more evident the structure of the series, and the equation relating T to the independent variable x indicates what the ramification index is (in this example: one)

(3)

Note also that the output is given only up to conjugation over the field of definition of the input. So, to obtain a basis of the formal solutions, the conjugate (replace by ) solution needs to be added as well. One can increase the field of definition with the option groundfield:

[[series(1+O(T^3),T,3), T = x-1], [T^3*exp(RootOf(1+_Z^2)/T)*(series(1-(3*RootOf(1+_Z^2))*T-12*T^2+O(T^3),T,3)), T = x-1], [T^3*exp(-RootOf(1+_Z^2)/T)*(series(1+(3*RootOf(1+_Z^2))*T-12*T^2+O(T^3),T,3)), T = x-1]]

(4)

An example with solutions of ramification index 2, so x = some constant times T^2, as well as solutions of ramification index 3 (so x = some constant times T^3):

ode := diff(y(x), x, x, x, x, x)+(diff(y(x), x, x, x))/x^3+y(x)/x^7

ode := diff(y(x), `$`(x, 5))+(diff(y(x), `$`(x, 3)))/x^3+y(x)/x^7

(5)

[[T^5*exp(-3/T)*(series(1-(43/9)*T+O(T^2),T,2)), -T^3 = x], [T^(19/2)*exp(-2/T)*(series(1+(3/16)*T+O(T^2),T,2)), -T^2 = x]]

(6)

Compare the series structures above with the structure of the output obtained without giving the optional name T:

(7)

The next ode has (generalized) exponents and at . This means that the dominant term in the formal solutions at will be resp. . The variable is here, so the ramification index is 1.

ode := x*(diff(y(x), x, x))+(nu-x)*(diff(y(x), x))-mu*y(x)

ode := x*(diff(y(x), `$`(x, 2)))+(nu-x)*(diff(y(x), x))-mu*y(x)

(8)

(9)

The generalized exponents at are and where now (since is , the ramification index is again 1). Each generalized exponent describes the dominant term in a formal solution. So the dominant term in the formal solutions at will be resp. which is resp. . Since is not a constant, the ode is irregular singular at .

[[T^mu*(series(1+(-mu+nu*mu-mu^2)*T+O(T^2),T,2)), T = 1/x], [T^(nu-mu)*exp(1/T)*(series(1+(-nu*mu+nu+mu^2-mu)*T+O(T^2),T,2)), T = 1/x]]

(10)

By solving the indicial equation one finds the exponents (exponents = those generalized exponents that are constants). At both generalized exponents are constant, so both are exponents. At one finds only one exponent, because only one of the two generalized exponents is a constant.

(11)

(12)

Note that formal_sol also allows ode's with coefficients that are not rational functions, but only if they are meromorphic at the specified point. For example, sin(x) is meromorphic at x=0 but not at , so the following example will fail at .

ode := sin(x)*(diff(y(x), x, x))-(diff(y(x), x))/x-y(x) = 0

ode := sin(x)*(diff(y(x), `$`(x, 2)))-(diff(y(x), x))/x-y(x) = 0

(13)

Error, (in `series/trig`) unable to compute series

[[series(1-(1/2)*T^2-(1/3)*T^3-(3/8)*T^4-(4/5)*T^5+O(T^6),T,6), T = x], [T^2*exp(-1/T)*(series(1-(11/6)*T+(433/72)*T^2-(159613/6480)*T^3+(19390253/155520)*T^4-(24592869013/32659200)*T^5+O(T^6),T,6)), T = x]]

(14)

	See Also
	DEtools/gen_exp, DEtools[indicialeq], DEtools[translate], diffop, Order, Slode/hypergeom_formal_sol

References

Ince, E.L. Ordinary Differential Equations, Chap. XVI-XVII. New York: Dover Publications, 1956.

van Hoeij, M. "Formal Solutions and Factorization of Differential Operators with Power Series Coefficients." Journal of Symbolic Computation. Vol. 24, (1997): 1-30.

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