Loewy decomposition

In the study of differential equations, the Loewy decomposition breaks every linear ordinary differential equation (ODE) into what are called largest completely reducible components. It was introduced by Alfred Loewy.[1]

Solving differential equations is one of the most important subfields in mathematics. Of particular interest are solutions in closed form. Breaking ODEs into largest irreducible components, reduces the process of solving the original equation to solving irreducible equations of lowest possible order. This procedure is algorithmic, so that the best possible answer for solving a reducible equation is guaranteed. A detailed discussion may be found in.[2]

Loewy's results have been extended to linear partial differential equations (PDEs) in two independent variables. In this way, algorithmic methods for solving large classes of linear pde's have become available.

Decomposing linear ordinary differential equationsEdit

Let   denote the derivative w.r.t. the variable  . A differential operator of order   is a polynomial of the form

 

where the coefficients  ,   are from some function field, the base field of  . Usually it is the field of rational functions in the variable  , i.e.  . If   is an indeterminate with  ,   becomes a differential polynomial, and   is the differential equation corresponding to  .

An operator   of order   is called reducible if it may be represented as the product of two operators   and  , both of order lower than  . Then one writes  , i.e. juxtaposition means the operator product, it is defined by the rule  ;   is called a left factor of  ,   a right factor. By default, the coefficient domain of the factors is assumed to be the base field of  , possibly extended by some algebraic numbers, i.e.   is allowed. If an operator does not allow any right factor it is called irreducible.

For any two operators   and   the least common left multiple   is the operator of lowest order such that both   and   divide it from the right. The greatest common right divisior   is the operator of highest order that divides both   and   from the right. If an operator may be represented as   of irreducible operators it is called completely reducible. By definition, an irreducible operator is called completely reducible.

If an operator is not completely reducible, the   of its irreducible right factors is divided out and the same procedure is repeated with the quotient. Due to the lowering of order in each step, this proceeding terminates after a finite number of iterations and the desired decomposition is obtained. Based on these considerations, Loewy [1] obtained the following fundamental result.

Theorem 1 (Loewy 1906) Let   be a derivative and  . A differential operator

 

of order   may be written uniquely as the product of completely reducible factors   of maximal order   over   in the form

 

with  . The factors   are unique. Any factor  ,   may be written as

 

with  ;   for  , denotes an irreducible operator of order   over  .

The decomposition determined in this theorem is called the Loewy decomposition of  . It provides a detailed description of the function space containing the solution of a reducible linear differential equation  .

For operators of fixed order the possible Loewy decompositions, differing by the number and the order of factors, may be listed explicitly; some of the factors may contain parameters. Each alternative is called a type of Loewy decomposition. The complete answer for   is detailed in the following corollary to the above theorem.[3]

Corollary 1 Let   be a second-order operator. Its possible Loewy decompositions are denoted by  , they may be described as follows;   and   are irreducible operators of order  ;   is a constant.

     
    
 

The decomposition type of an operator is the decomposition   with the highest value of  . An irreducible second-order operator is defined to have decomposition type  .

The decompositions  ,   and   are completely reducible.

If a decomposition of type  ,   or   has been obtained for a second-order equation  , a fundamental system may be given explicitly.

Corollary 2 Let   be a second-order differential operator,  ,   a differential indeterminate, and  . Define   for   and  ,   is a parameter; the barred quantities   and   are arbitrary numbers,  . For the three nontrivial decompositions of Corollary 1 the following elements   and   of a fundamental system are obtained.

 :  ;   

 

 
 :  
 

  is not equivalent to  .

 :  
 
 

Here two rational functions   are called equivalent if there exists another rational function   such that

 .

There remains the question how to obtain a factorization for a given equation or operator. It turns out that for linear ode's finding the factors comes down to determining rational solutions of Riccati equations or linear ode's; both may be determined algorithmically. The two examples below show how the above corollary is applied.

Example 1 Equation 2.201 from Kamke's collection.[4] has the   decomposition

 

The coefficients   and   are rational solutions of the Riccati equation  , they yield the fundamental system

 
 

Example 2 An equation with a type   decomposition is

 

The coefficient of the first-order factor is the rational solution of  . Upon integration the fundamental system   and   for   and   respectively is obtained.

These results show that factorization provides an algorithmic scheme for solving reducible linear ode's. Whenever an equation of order 2 factorizes according to one of the types defined above the elements of a fundamental system are explicitly known, i.e. factorization is equivalent to solving it.

A similar scheme may be set up for linear ode's of any order, although the number of alternatives grows considerably with the order; for order   the answer is given in full detail in.[2]

If an equation is irreducible it may occur that its Galois group is nontrivial, then algebraic solutions may exist.[5] If the Galois group is trivial it may be possible to express the solutions in terms of special function like e.g. Bessel or Legendre functions, see [6] or.[7]

Basic facts from differential algebraEdit

In order to generalize Loewy's result to linear pde's it is necessary to apply the more general setting of differential algebra. Therefore, a few basic concepts that are required for this purpose are given next.

A field   is called a differential field if it is equipped with a derivation operator. An operator   on a field   is called a derivation operator if   and   for all elements  . A field with a single derivation operator is called an ordinary differential field; if there is a finite set containing several commuting derivation operators the field is called a partial differential field.

Here differential operators with derivatives   and   with coefficients from some differential field are considered. Its elements have the form  ; almost all coefficients   are zero. The coefficient field is called the base field. If constructive and algorithmic methods are the main issue it is  . The respective ring of differential operators is denoted by   or  . The ring   is non-commutative,   and similarly for the other variables;   is from the base field.

For an operator   of order   the symbol of L is the homogeneous algebraic polynomial   where   and   algebraic indeterminates.

Let   be a left ideal which is generated by  ,  . Then one writes  . Because right ideals are not considered here, sometimes   is simply called an ideal.

The relation between left ideals in   and systems of linear pde's is established as follows. The elements   are applied to a single differential indeterminate  . In this way the ideal   corresponds to the system of pde's  ,   for the single function  .

The generators of an ideal are highly non-unique; its members may be transformed in infinitely many ways by taking linear combinations of them or its derivatives without changing the ideal. Therefore, M. Janet[8] introduced a normal form for systems of linear pde's (see Janet basis).[9] They are the differential analog to Gröbner bases of commutative algebra (which were originally introduced by Bruno Buchberger);[10] therefore they are also sometimes called differential Gröbner basis.

In order to generate a Janet basis, a ranking of derivatives must be defined. It is a total ordering such that for any derivatives  ,   and  , and any derivation operator   the relations  , and   are valid. Here graded lexicographic term orderings   are applied. For partial derivatives of a single function their definition is analogous to the monomial orderings in commutative algebra. The S-pairs in commutative algebra correspond to the integrability conditions.

If it is assured that the generators   of an ideal   form a Janet basis the notation   is applied.

Example 3 Consider the ideal

  

    

in   term order with  . Its generators are autoreduced. If the integrability condition

 

is reduced w.r.t. to  , the new generator   is obtained. Adding it to the generators and performing all possible reductions, the given ideal is represented as  . Its generators are autoreduced and the single integrability condition is satisfied, i.e. they form a Janet basis.

Given any ideal   it may occur that it is properly contained in some larger ideal   with coefficients in the base field of  ; then   is called a divisor of  . In general, a divisor in a ring of partial differential operators need not be principal.

The greatest common right divisor (Gcrd) or sum of two ideals   and   is the smallest ideal with the property that both   and   are contained in it. If they have the representation   and    ,   for all   and  , the sum is generated by the union of the generators of   and  . The solution space of the equations corresponding to   is the intersection of the solution spaces of its arguments.

The least common left multiple (Lclm) or left intersection of two ideals   and   is the largest ideal with the property that it is contained both in   and  . The solution space of   is the smallest space containing the solution spaces of its arguments.

A special kind of divisor is the so-called Laplace divisor of a given operator  ,[2] page 34. It is defined as follows.

Definition Let   be a partial differential operator in the plane; define

  and
 

be ordinary differential operators w.r.t.   or  ;   for all i;   and   are natural numbers not less than 2. Assume the coefficients  ,   are such that   and   form a Janet basis. If   is the smallest integer with this property then   is called a Laplace divisor of  . Similarly, if  ,   are such that   and   form a Janet basis and   is minimal, then   is also called a Laplace divisor of  .

In order for a Laplace divisor to exist the coeffients of an operator   must obey certain constraints.[3] An algorithm for determining an upper bound for a Laplace divisor is not known at present, therefore in general the existence of a Laplace divisor may be undecidable

Decomposing second-order linear partial differential equations in the planeEdit

Applying the above concepts Loewy's theory may be generalized to linear pde's. Here it is applied to individual linear pde's of second order in the plane with coordinates   and  , and the principal ideals generated by the corresponding operators.

Second-order equations have been considered extensively in the literature of the 19th century,.[11][12] Usually equations with leading derivatives   or   are distinguished. Their general solutions contain not only constants but undetermined functions of varying numbers of arguments; determining them is part of the solution procedure. For equations with leading derivative   Loewy's results may be generalized as follows.

Theorem 2 Let the differential operator   be defined by

   where   for all  .

Let   for   and  , and   be first-order operators with  ;   is an undetermined function of a single argument. Then   has a Loewy decomposition according to one of the following types.

         

The decomposition type of an operator   is the decomposition   with the highest value of  . If   does not have any first-order factor in the base field, its decomposition type is defined to be  . Decompositions  ,   and   are completely reducible.

In order to apply this result for solving any given differential equation involving the operator   the question arises whether its first-order factors may be determined algorithmically. The subsequent corollary provides the answer for factors with coefficients either in the base field or a universal field extension.

Corollary 3 In general, first-order right factors of a linear pde in the base field cannot be determined algorithmically. If the symbol polynomial is separable any factor may be determined. If it has a double root in general it is not possible to determine the right factors in the base field. The existence of factors in a universal field, i.e. absolute irreducibility, may always be decided.

The above theorem may be applied for solving reducible equations in closed form. Because there are only principal divisors involved the answer is similar as for ordinary second-order equations.

Proposition 1 Let a reducible second-order equation

  where  .

Define  ,   for  ;   is a rational first integral of  ;   and the inverse  ; both   and   are assumed to exist. Furthermore, define

  for  .

A differential fundamental system has the following structure for the various decompositions into first-order components.

 ,  

 

 

The   are undetermined functions of a single argument;  ,   and   are rational in all arguments;   is assumed to exist. In general  , they are determined by the coefficients  ,   and   of the given equation.

A typical example of a linear pde where factorization applies is an equation that has been discussed by Forsyth,[13] vol. VI, page 16,

Example 5 (Forsyth 1906)} Consider the differential equation  . Upon factorization the representation

  is obtained. There follows

 ,

 

Consequently, a differential fundamental system is

   

  and   are undetermined functions.

If the only second-order derivative of an operator is  , its possible decompositions involving only principal divisors may be described as follows.

Theorem 3 Let the differential operator   be defined by

  where   for all  .

Let   and   are first-order operators.   has Loewy decompositions involving first-order principal divisors of the following form.

         

The decomposition type of an operator   is the decomposition   with highest value of  . The decomposition of type   is completely reducible

In addition there are five more possible decomposition types involving non-principal Laplace divisors as shown next.

Theorem 4 Let the differential operator   be defined by

  where   for all  .

  and   as well as   and   are defined above; furthermore  ,  ,  .   has Loewy decompositions involving Laplace divisors according to one of the following types;   and   obey  .

 

 

 

   

If   does not have a first order right factor and it may be shown that a Laplace divisor does not exist its decomposition type is defined to be  . The decompositions  ,  ,   and   are completely reducible.

An equation that does not allow a decomposition involving principal divisors but is completely reducible w.r.t. non-principal Laplace divisors of type   has been considered by Forsyth.

Example 6 (Forsyth 1906) Define

 

generating the principal ideal  . A first-order factor does not exist. However, there are Laplace divisors

  and  

The ideal generated by   has the representation  , i.e. it is completely reducible; its decomposition type is  . Therefore, the equation   has the differential fundamental system

  and  .

Decomposing linear pde's of order higher than 2Edit

It turns out that operators of higher order have more complicated decompositions and there are more alternatives, many of them in terms of non-principal divisors. The solutions of the corresponding equations get more complex. For equations of order three in the plane a fairly complete answer may be found in.[2] A typical example of a third-order equation that is also of historical interest is due to Blumberg .[14]

Example 7 (Blumberg 1912) In his dissertation Blumberg considered the third order operator

 .

It allows the two first-order factors   and  . Their intersection is not principal; defining

 

 

it may be written as  . Consequently, the Loewy decomposition of Blumbergs's operator is

 

It yields the following differential fundamental system for the differential equation  .

  ,    

  and   are an undetermined functions.

Factorizations and Loewy decompositions turned out to be an extremely useful method for determining solutions of linear differential equations in closed form, both for ordinary and partial equations. It should be possible to generalize these methods to equations of higher order, equations in more variables and system of differential equations.

ReferencesEdit

  1. ^ a b Loewy, A. (1906). "Über vollständig reduzible lineare homogene Differentialgleichungen" (PDF). Mathematische Annalen. 62: 89–117. doi:10.1007/bf01448417.
  2. ^ a b c d , F.Schwarz, Loewy Decomposition of Linear Differential Equations, Springer, 2012
  3. ^ a b Schwarz, F. (2013). "Loewy Decomposition of linear Differential Equations". Bulletin of Mathematical Sciences. 3: 19–71. doi:10.1007/s13373-012-0026-7.
  4. ^ E. Kamke, Differentialgleichungen I. Gewoehnliche Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig, 1964
  5. ^ M. van der Put, M.Singer, Galois theory of linear differential equations, Grundlehren der Math. Wiss. 328, Springer, 2003
  6. ^ M.Bronstein, S.Lafaille, Solutions of linear ordinary differential equations in terms of special functions, Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation; T.Mora, ed., ACM, New York, 2002, pp. 23–28
  7. ^ F. Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations, CRC Press, 2007, page 39
  8. ^ Janet, M. (1920). "Les systemes d'equations aux derivees partielles". Journal de Mathematiques. 83: 65–123.
  9. ^ Janet Bases for Symmetry Groups, in: Gröbner Bases and Applications Lecture Notes Series 251, London Mathematical Society, 1998, pages 221–234, B. Buchberger and F. Winkler, Edts.
  10. ^ Buchberger, B. (1970). "Ein algorithmisches Kriterium fuer die Loesbarkeit eines algebraischen Gleichungssystems". Aequ. Math. 4 (3): 374–383. doi:10.1007/bf01844169.
  11. ^ E. Darboux, Leçons sur la théorie générale des surfaces, vol. II, Chelsea Publishing Company, New York, 1972
  12. ^ Édouard Goursat, Leçon sur l'intégration des équations aux dérivées partielles, vol. I and II, A. Hermann, Paris, 1898
  13. ^ A.R.Forsyth, Theory of Differential Equations, vol. I,...,VI, Cambridge, At the University Press, 1906
  14. ^ H.Blumberg, Ueber algebraische Eigenschaften von linearen homogenen Differentialausdruecken, Inaugural-Dissertation, Goettingen, 1912