## Computer Algebra and Differential Equations - An Overview

Citations: | 4 - 0 self |

### BibTeX

@MISC{Seiler_computeralgebra,

author = {Werner M. Seiler},

title = {Computer Algebra and Differential Equations - An Overview},

year = {}

}

### OpenURL

### Abstract

We present an informal overview of a number of approaches to differential equations which are popular in computer algebra. This includes symmetry and completion theory, local analysis, differential ideal and Galois theory, dynamical systems and numerical analysis. A large bibliography is provided. 1 Introduction Differential equations represent one of the largest fields within mathematics. Besides being an interesting subject of their own right one can hardly overestimate their importance for applications. They appear in natural and engineering sciences and increasingly often in economics and social sciences. Whenever a continuous process is modeled mathematically, chances are high that differential equations are used. Thus it is not surprising that differential equations also play an important role in computer algebra and most general purpose computer algebra systems provide some kind of solve command. Many casual users believe that designing and improving such procedures is a centra...

### Citations

1029 |
Application of Lie groups to differential equations
- Olver
- 1986
(Show Context)
Citation Context ...pansions are useful for constructing Backlund transformations, Lax pairs and much more [121]. There also exist relations to non-classical symmetry reductions [30]. Symmetry Analysis Symmetry analysis =-=[9, 77, 111]-=- has made the strongest impact on computer algebra applications to differential equations. The most general definition of a symmetry is that of a transformation that maps solutions into solutions. Dep... |

537 |
Ordinary Differential Equations
- Ince
- 1950
(Show Context)
Citation Context ...initial or boundary data. One important direction is the Painleve 3 Especially in the linear case local solutions can also be very useful for the construction of closed-form solutions. mathPAD theory =-=[54]-=-. It was introduced by Painleve while searching for new special functions (there still exists a strong connection between the local analysis of ordinary differential equations and special function the... |

275 |
Advanced Mathematical Methods for Scientists and Engineers
- Bender, Orszag
- 1978
(Show Context)
Citation Context ...o sense to construct an approximation; instead one tries to capture its asymptotic behavior which requires the addition of an exponential part. An elementary introduction can be found in the textbook =-=[10]-=-. There exist various algorithms for the construction of approximate or asymptotic solutions, partly dating back at least to Frobenius. Some are discussed together with implementations in [94, 126]. A... |

246 |
Differential algebra and algebraic groups
- Kolchin
- 1973
(Show Context)
Citation Context ...addition closed under the derivation of the differential ring. Many of the basic ideas in differential ideal theory can be traced back to Ritt [93]; the most advanced book is still the one by Kolchin =-=[60]-=-. Like in the purely algebraic theory one would like to introduce something like a Grobner basis. As the ring of differential polynomials is not Noetherian, algorithms along the lines of the Buchberge... |

174 |
Exterior Differential Systems
- Chern, Goldschmidt, et al.
- 1991
(Show Context)
Citation Context ... criterion for the termination of the completion. As an intrinsic concept involution requires no coordinate dependent ingredients like a ranking. Involution analysis based on the Cartan-Kahler theory =-=[17]-=- for exterior systems is discussed from an algorithmic point of view in [55, 56]. A completion algorithm for the jet bundle formalism based on the formal theory of Pommaret [96] was presented in [108]... |

169 |
Symmetries and differential equations
- Bluman, Kumei
- 1989
(Show Context)
Citation Context ... solutions. Other goals are classifications, a proof of (complete) integrability, separation ansatze, conservation laws and much more. Several excellent textbooks on this subject are available, e. g. =-=[12, 86, 122]-=-. Symmetry analysis goes back to the seminal work of Lie. He developed the concept of Lie groups in his quest for a Galois theory for differential equations. As we will see later in Sect. 7, not much ... |

145 |
Differentialgleichungen: Lösungsmethoden und Lösungen, Band 1: Gewöhnliche Differentialgleichungen (3
- KAMKE
- 1944
(Show Context)
Citation Context ...s can solve some differential equations. 1 They mainly apply some standard techniques like those in Zwillinger's handbook [127] or try some "pattern matching" in a list of solved equations l=-=ike Kamke [58]-=-. Heuristics often extend the applicability of these techniques, for example by finding a transformation such that a given equation can be handled by the implemented methods. Although this approach so... |

143 |
Application of Centre Manifold Theory
- CARR
- 1981
(Show Context)
Citation Context ...y bifurcation phenomena. For equivariant systems (see below) they automatically classify them using Grobner bases. Computer algebra is also much used to determine (approximations of) center manifolds =-=[20]-=-, a special form of invariant manifolds. If a dynamical system possesses a center manifold, it often suffices to study its behavior on this manifold. If the zero solution of the reduced system is stab... |

122 |
Differential equations. Their solution using symmetries
- Stephani
- 1989
(Show Context)
Citation Context ...pansions are useful for constructing Backlund transformations, Lax pairs and much more [121]. There also exist relations to non-classical symmetry reductions [30]. Symmetry Analysis Symmetry analysis =-=[9, 77, 111]-=- has made the strongest impact on computer algebra applications to differential equations. The most general definition of a symmetry is that of a transformation that maps solutions into solutions. Dep... |

118 |
Differential Algebra
- Ritt
- 1950
(Show Context)
Citation Context ... not apply. A differential ideal is an ideal which is in addition closed under the derivation of the differential ring. Many of the basic ideas in differential ideal theory can be traced back to Ritt =-=[93]-=-; the most advanced book is still the one by Kolchin [60]. Like in the purely algebraic theory one would like to introduce something like a Grobner basis. As the ring of differential polynomials is no... |

107 | Representation for the Radical of a Finitely Generated Differential Ideal
- Boulier, Lazard, et al.
- 1995
(Show Context)
Citation Context ...ansfield [79]. They are computed with pseudo-reductions and have thus weaker properties than their algebraic counterpart. Especially, it may happen that one leaves the ideal. Recently, Boulier et al. =-=[13]-=- presented a Rosenfeld-Grobner algorithm which computes a representation for perfect differential ideals in the following form. The ideal is written as a finite intersection of saturations ideals; the... |

97 | An Algorithm for Solving Second Order Linear Homogeneous Differential Equations
- Kovacic
- 1986
(Show Context)
Citation Context ...the solution algorithms. The original work of Singer covered only equations with rational coefficients. Later, he extended it to Liouvillian coefficients [12, 106]. For second order equations Kovacic =-=[57, 28]-=- developed independently a solution algorithm. Only much later it could be shown that the classification underlying this algorithm can also be derived within the Singer theory [108]. The Kovacic algor... |

74 |
The general similarity solution of the heat equation
- Bluman, Cole
- 1969
(Show Context)
Citation Context ... theory but on the theory of over-determined systems of partial differential equations and thus on questions of completion (cf. [111]). The first non-classical method was developed by Bluman and Cole =-=[11]-=- and uses the invariant surface condition as constraint. Although this leads for many differential equations to new reductions, the drawback is that the determining system becomes nonlinear. The direc... |

67 |
Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach Science
- Pommaret
- 1978
(Show Context)
Citation Context ...e CartanK ahler theory [14] for exterior systems is discussed from an algorithmic point of view in [48, 49]. A completion algorithm for the jet bundle formalism based on the formal theory of Pommaret =-=[84]-=- was presented in [97]. Completion algorithms are very useful in the symmetry analysis of differential equations. Once a system is either passive or involutive, one can make statements about the size ... |

62 |
Elementary first integrals of differential equations
- Prelle, Singer
- 1983
(Show Context)
Citation Context ...15]. Factorization (although only of polynomials) is also an issue in differential ideal theory. Differential Galois theory also gives algorithms for the construction of (Liouvillian) first integrals =-=[69, 88, 120]-=-. 4 A MuPAD implementation of this approach (and some related code) can be down-loaded from Fakler's WWW page with the URL http://iaks-www.ira.uka.de/home/fakler/index.html. 8 mathPAD Vol n No m Date ... |

51 | Elementary and Liouvillian solutions of linear differential equations
- DAVENPORT, SINGER
- 1986
(Show Context)
Citation Context ...ass) and gives the basic case distinctions in the solution algorithms. The original work of Singer covered only equations with rational coefficients. Later, he extended it to Liouvillian coefficients =-=[12, 106]-=-. For second order equations Kovacic [57, 28] developed independently a solution algorithm. Only much later it could be shown that the classification underlying this algorithm can also be derived with... |

50 |
The symmetry approach to the classification of nonlinear equations. Complete lists of integrable systems
- Mikhailov, Shabat, et al.
- 1987
(Show Context)
Citation Context ... with respect to generalized symmetries is an important tool for the construction of soliton solutions. It is also possible to classify nonlinear partial differential equations using these symmetries =-=[73]-=-. Some MuPAD packages for symmetries of integrable systems are described in [33]. Non-classical reductions can be understood within the general scheme of augmenting a given differential equation with ... |

47 |
Local methods in nonlinear differential equations
- Bruno
- 1989
(Show Context)
Citation Context ...s a classical topic in computer algebra. For dynamical systems normal forms have already been introduced by Poincar'e, Birkhoff, Gustavson and many others, often in the context of celestial mechanics =-=[16, 30]. They for-=-m the basis for the solution of many problems in dynamical systems theory like for example stability or bifurcation analysis. One should however note that the word "normal form" is used here... |

47 | Liouvillian and algebraic solutions of second and third order linear differential equations
- Singer, Ulmer
- 1993
(Show Context)
Citation Context ...uations Kovacic [57, 28] developed independently a solution algorithm. Only much later it could be shown that the classification underlying this algorithm can also be derived within the Singer theory =-=[108]-=-. The Kovacic algorithm has been implemented in several computer algebra systems. An alternative approach based on the invariant ring of the differential Galois group was presented by Fakler [31] foll... |

45 | Symbolic software for Lie symmetry analysis. In: Lie Group Analysis of Differential Equations, vol 3, ed by N.H
- Hereman
- 1996
(Show Context)
Citation Context ... mention the survey [104] by Singer. It gives much more details, especially on the more algebraic approaches, and contains a large bibliography. The same holds for the more focused surveys by Hereman =-=[50, 51]-=- covering symmetry theory and related fields and the one by MacCallum [65] on the integration of ordinary differential equations. In addition there have been three conferences devoted exclusively to d... |

43 |
Differential Gröbner Bases
- Mansfield
- 1991
(Show Context)
Citation Context ...l Grobner basis. The completion algorithm of the Janet-Riquier theory can be considered as a simple example for the first strategy. An example for the second one are the bases introduced by Mansfield =-=[70]-=-. They are computed with pseudo-reductions and have thus weaker properties than their algebraic counterpart. Especially, it may happen that one leaves the ideal. Recently, Boulier et al. [10] presente... |

41 | On solutions of linear ordinary differential equations in their coefficient field
- BRONSTEIN
- 1992
(Show Context)
Citation Context ...ss) and gives the basic case distinctions in the solution algorithms. The original work of Singer covered only equations with rational coefficients. Later, it was extended to Liouvillian coefficients =-=[15, 117]-=-. For second order equations Kovacic [63, 34] developed independently a solution algorithm. Only much later one could show that the classification behind this algorithm can also be derived within the ... |

41 |
Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution, Euro
- Reid
- 1991
(Show Context)
Citation Context ...if all arising integrability conditions are also quasi-linear), as it must be possible to solve for the leading derivative. In this case the resulting passive system is sometimes call a standard form =-=[91]-=-. In geometric theories the notion of a passive system is replaced by involution. It combines a geometric definition of formal integrability with an algebraic criterion for the termination of the comp... |

40 | Review of symbolic software for the computation of Lie symmetries of differential equations
- Hereman
- 1994
(Show Context)
Citation Context ... mention the survey [104] by Singer. It gives much more details, especially on the more algebraic approaches, and contains a large bibliography. The same holds for the more focused surveys by Hereman =-=[50, 51]-=- covering symmetry theory and related fields and the one by MacCallum [65] on the integration of ordinary differential equations. In addition there have been three conferences devoted exclusively to d... |

40 |
An example of a smooth linear partial differential equation without solution
- Lewy
(Show Context)
Citation Context ... like the Cartan-Kahler theorem (the well-known Cauchy-Kowalevsky theorem is a special case of it). For non-analytic equations solvability is a much more complicated question due to Lewy type effects =-=[63]-=-. The first systematic approach to the problem of completion was probably provided by the Janet-Riquier theory [55] with the introduction of passive systems. Their definition is based on a ranking of ... |

37 |
Some algorithmic questions on ideals of differential operators
- Galligo
- 1985
(Show Context)
Citation Context ...on's law of gravity from the three Kepler laws [125]. Besides ideals of differential polynomials there has also been some work on ideals of linear differential operators or ideals of the Weil algebra =-=[34]-=-. However, here one is dealing with non-commutative rings. One could also consider the Cartan-Kahler theory as a kind of differential ideal theory, as it represents differential equations by closed id... |

37 |
Liouvillian solutions of n-th order homogeneous linear differential equations
- Singer
- 1981
(Show Context)
Citation Context ...number of extensions, thus it is algorithmically constructible. Most expressions one would call "closed-form" are in fact Liouvillian. Most solution algorithms are based on the seminal work =-=of Singer [102]-=-. He showed that the logarithmic derivative of any Liouvillian solution is algebraic and determined an a priori bound for the degree of the minimal polynomial, namely the Jordan bound for the index of... |

35 |
Nonlinear evolution equations and ordinary differential equations of Painlevé type
- Ablowitz, Ramani, et al.
- 1978
(Show Context)
Citation Context ...'e conjecture states that every ordinary differential equation obtained by symmetry reduction (see Sect. 4) of an integrable system is of Painlev'e type; only weakened versions of it have been proven =-=[2, 80]-=-. Truncated series expansions are useful for constructing Backlund transformations, Lax pairs and much more [132]. There also exist relations to non-classical symmetry reductions [36]. Comparing with ... |

35 |
The index of general nonlinear DAE’s
- Campbell, Gear
- 1995
(Show Context)
Citation Context ...ysis are differential algebraic equations. The index of such a system comprising differential and algebraic equations measures in a certain sense, how far it is away from a pure differential equation =-=[18]-=-. This gives an indication of the difficulties one must expect in a numerical integration. The determination of the index is essentially equivalent to the completion procedures described in Sect. 5 [7... |

31 |
A computable ordinary differential equation which possesses no computable solution
- Richards
- 1979
(Show Context)
Citation Context ...g situation. Computability theory yields principal limits to what can be solved. For example if one restricts to computable functions some classical existence theorems for differential equations fail =-=[1, 87]-=-. More precisely, one has constructed examples of differential equations where one can show that solutions exists but that it is not possible to compute them. Some further (positive and negative) resu... |

30 |
A and Kruskal M D, New similarity reductions of the Boussinesq equation
- Clarkson
- 1989
(Show Context)
Citation Context ...ondition as constraint. Although this leads for many differential equations to new reductions, the drawback is that the determining system becomes nonlinear. The direct method of Clarkson and Kruskal =-=[25]-=- tries to reduce a given partial differential equation to a system of ordinary differential equations by constructing a good ansatz; it corresponds to a special case of the method of Bluman and Cole. ... |

27 | Direct reduction and differential constraints
- Olver
- 1994
(Show Context)
Citation Context ... symmetries of integrable systems are described in [33]. Non-classical reductions can be understood within the general scheme of augmenting a given differential equation with differential constraints =-=[78]-=-. This corresponds to requiring that only some solutions are mapped into solutions, therefore one hopes to find more symmetries (these are sometimes called weak symmetries). In this approach the empha... |

27 |
Differential Galois Theory
- Put, Singer
- 2003
(Show Context)
Citation Context ...ifferential Galois Theory Already Lie was looking for a differential analog of the (algebraic) Galois theory, when he introduced Lie groups. What is nowadays usually called differential Galois theory =-=[66, 105]-=- has however no connection to Lie symmetry theory. The latter one uses continuous transformation groups and can be applied to any differential equations. But as discussed above it is not completely al... |

25 |
A perturbative Painlevé approach to nonlinear differential equations
- Conte, Fordy, et al.
- 1993
(Show Context)
Citation Context ...ciently many resonances or Fuchsian indices (free coefficients) to represent the general solution and if these occur at non-negative powers. In the case of negative resonances a perturbation approach =-=[27]-=- yields further information. Some references concerning implementations can be found in [106]. Weiss et al. [133] generalized the Painlev'e theory to partial differential equations where a whole singu... |

21 | The Painlevé Approach to Nonlinear Ordinary Differential Equations
- Conte
- 1999
(Show Context)
Citation Context ...ons the situation becomes much more complicated as spontaneous or movable singularities may occur, i. e. their location depends on initial or boundary data. One usually speaks of the Painlev'e theory =-=[26, 61, 69]-=-. It was introduced by Painlev'e while searching for new special functions and there still exists a strong connection to special function theory. If all singularities are poles, no branch points appea... |

21 | Algorithmic methods for Lie pseudogroups
- Schü, Seiler, et al.
- 1993
(Show Context)
Citation Context ... [14] for exterior systems is discussed from an algorithmic point of view in [48, 49]. A completion algorithm for the jet bundle formalism based on the formal theory of Pommaret [84] was presented in =-=[97]-=-. Completion algorithms are very useful in the symmetry analysis of differential equations. Once a system is either passive or involutive, one can make statements about the size of the solution space ... |

19 | The Theory of Involutive Divisions and an Application to Hilbert Function
- Apel
(Show Context)
Citation Context ...er basis. In some cases the new algorithms are considerably faster than the classical Buchberger algorithm. Involutive bases also allow for a straightforward determination of the Hilbert polynomial 4 =-=[6]. 6 D-=-ifferential Ideal Theory Differential ideal theory belongs to the field of differential algebra. It can be informally described as an attempt "to write differential in front of everything in alge... |

19 |
Lecons sur les Systemes d’ Equations aux Derivees Partielles
- Janet
- 1929
(Show Context)
Citation Context ...c equations solvability is a much more complicated question due to Lewy type effects [63]. The first systematic approach to the problem of completion was probably provided by the Janet-Riquier theory =-=[55]-=- with the introduction of passive systems. Their definition is based on a ranking of the derivatives which decides in what order the integrability conditions are constructed. The completion can be don... |

19 | On the Arbitrariness of the General Solution of an Involutive Partial Differential Equation
- Seiler
- 1994
(Show Context)
Citation Context .... Completion algorithms are very useful in the symmetry analysis of differential equations. Once a system is either passive or involutive, one can make statements about the size of the solution space =-=[91, 98]-=-. Thus it is possible to compute the size of the symmetry group without explicitly solving the determining system or to determine the loss of generality in a symmetry reduction [99]. One can even comp... |

18 |
Geometric approach to invariance groups and solution of partial differential systems
- Harrison, Estabrook
- 1971
(Show Context)
Citation Context ...ather surprising how often this suffices to obtain the complete symmetry algebra. The symmetry package of MAPLE is somewhat unusual, as it uses the exterior systems approach of Harrison and Estabrook =-=[47]-=-. Although Lie point symmetries proved to be very useful in many applications, many differential equations of practical interest have no such symmetries. There are two basic approaches to generalize t... |

17 |
A rational approach to the Prelle-Singer algorithm
- Man, MacCallum
- 1997
(Show Context)
Citation Context ...15]. Factorization (although only of polynomials) is also an issue in differential ideal theory. Differential Galois theory also gives algorithms for the construction of (Liouvillian) first integrals =-=[69, 88, 120]-=-. 4 A MuPAD implementation of this approach (and some related code) can be down-loaded from Fakler's WWW page with the URL http://iaks-www.ira.uka.de/home/fakler/index.html. 8 mathPAD Vol n No m Date ... |

16 | A constructive implementation of the Cartan–Kähler theory of exterior differential systems
- Hartley, Tucker
- 1991
(Show Context)
Citation Context ...volution requires no coordinate dependent ingredients like a ranking. Involution analysis based on the CartanK ahler theory [14] for exterior systems is discussed from an algorithmic point of view in =-=[48, 49]-=-. A completion algorithm for the jet bundle formalism based on the formal theory of Pommaret [84] was presented in [97]. Completion algorithms are very useful in the symmetry analysis of differential ... |

16 |
Computing Closed Form Solutions of First Order ODEs Using the Prelle–Singer Procedure
- Man
- 1983
(Show Context)
Citation Context ...with the URL http://iaks-www.ira.uka.de/home/fakler/index.html. 8 mathPAD Vol n No m Date Computer Algebra and Differential Equations --- An Overview These can be used to construct explicit solutions =-=[68]-=-. Other applications appear in the theory of completely integrable systems. Ziglin has given an algebraic characterization of such systems based on their monodromy group. His criterion for integrabili... |

16 | Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs
- Reid, Lisle, et al.
- 1992
(Show Context)
Citation Context ... the determining system or to determine the loss of generality in a symmetry reduction [99]. One can even compute the abstract structure of the symmetry algebra without solving the determining system =-=[64, 92]-=-. These concepts are closely related to Grobner bases in commutative algebra. This holds especially for the JanetRiquier theory where rankings play a similar role as in the definition of a Grobner bas... |

15 | An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system
- Barkatou
- 1997
(Show Context)
Citation Context ...a careful analysis of the algorithms can often significantly reduce the necessary amount of computations with algebraic numbers. Recent work concerns an extension of the theory to first order systems =-=[8, 93]-=-. In principle, one can transform any system into a single equation of higher order, e. g. using cyclic vectors. But this approach is rather inefficient, especially in higher dimensions. Hence one is ... |

14 |
Power series solutions of algebraic differential equations
- Denef, Lipshitz
- 1980
(Show Context)
Citation Context ...examples of differential equations where one can show that solutions exists but that it is not possible to compute them. Some further (positive and negative) results in this direction can be found in =-=[29]-=-. Ideally, a solution algorithm should return the general solution. But for nonlinear equations it is surprisingly difficult even just to define this term. A rigorous resolution of this problem (for o... |

14 |
Algorithmic derivation of centre conditions
- Pearson, Lloyd, et al.
- 1996
(Show Context)
Citation Context ...stinguish between a focus and a center. The derivation of sufficient and especially of necessary conditions for a center can be very involved and is sometimes hardly feasible without computer algebra =-=[80]-=-. In a recent study of cubic systems [29] a CRAY-J90 had to be used. Numerical Analysis It was already mentioned above that the capabilities of computer algebra systems to explicitly solve differentia... |

13 |
Gröbner bases and differential algebra
- Ferro
- 1987
(Show Context)
Citation Context ... theory one would like to introduce Grobner bases. But as the Ritt-Raudenbush theorem is weaker than full Noetherianity, algorithms along the lines of the Buchberger algorithm do not always terminate =-=[21]-=-. More generally, one can prove that the ideal membership problem is undecidable for arbitrary differential ideals [43]. However, this result is more of theoretical interest, as for finitely generated... |

13 | The General Solution of an Ordinary Differential Equation
- Hubert
- 1996
(Show Context)
Citation Context ... solution. But for nonlinear equations it is surprisingly difficult even just to define this term. A rigorous resolution of this problem based on differential ideal theory was only recently presented =-=[53]-=-. Intuitively one would expect that the general solution depends on some arbitrary parameters (constants or functions) and that every solution of the differential equation can be obtained by a suitabl... |

13 |
Formal solutions of differential equations
- Singer
- 1990
(Show Context)
Citation Context ...es or books and not the historically first or the most "ground breaking" work. The bibliography is of course far from being exhaustive. As a further source of references one should mention t=-=he survey [104]-=- by Singer. It gives much more details, especially on the more algebraic approaches, and contains a large bibliography. The same holds for the more focused surveys by Hereman [50, 51] covering symmetr... |