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33
The Symbolic Integration of Exact PDEs
, 2000
"... An algorithm is described which decides if a given polynomial differential expression \Delta of multivariate functions is exact, i.e. whether there exists a first integral P such that D x P = \Delta for any one x of a set of n variables and to provide the integral P . A generalization is given to al ..."
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Cited by 8 (5 self)
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An algorithm is described which decides if a given polynomial differential expression \Delta of multivariate functions is exact, i.e. whether there exists a first integral P such that D x P = \Delta for any one x of a set of n variables and to provide the integral P . A generalization is given to allow integration in the case that the exactness is prevented by terms which contain only functions of less than n independent variables. 1 Motivation The common way to deal with problems that involve the solution of nonlinear differential equations is to try different ansatze which are either geometrically motivated or just chosen to simplify computations. Typical examples are the investigation of infinitesimal symmetries, the search for classes of integrating factors and related first integrals/conservation laws or the search for a variational principle equivalent to a given system of equations. In all these cases overdetermined systems of partial differential 1 equations (PDEs) have t...
Symmetry group analysis of the shallow water and semigeostrophic
"... The twodimensional shallow water equations and their semigeostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetr ..."
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The twodimensional shallow water equations and their semigeostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semigeostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semigeostrophic equations in such a way that a welldefined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an ‘infinite linear cascade’. 1.
Algorithmic Determination Of Structure Of Infinite Lie Pseudogroups Of Symmetries Of PDEs
 In Proceedings of the 1995 international symposium on Symbolic and algebraic computation
, 1995
"... We describe a method which uses a finite number of differentiations and linear operations to determine the Cartan structure coefficients of a structurally transitive Lie pseudogroup from its infinitesimal defining equations. If the defining system is of first order and the pseudogroup has no scalar ..."
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Cited by 6 (1 self)
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We describe a method which uses a finite number of differentiations and linear operations to determine the Cartan structure coefficients of a structurally transitive Lie pseudogroup from its infinitesimal defining equations. If the defining system is of first order and the pseudogroup has no scalar invariants, the structure coefficients can be simply extracted from the coefficients of the infinitesimal system. We give an algorithm which reduces the higher order case to the first order case. The reduction process uses only differentiation and linear eliminations, for which several wellknown algorithms are available. Our method makes feasible the calculation of the Cartan structure of infinite Lie pseudogroups of symmetries of differential equations. Examples including the KP equation and Liouville's equation are given. 1 INTRODUCTION This paper is one of a series in which we investigate the determination of structure of infinite Lie pseudogroups. The main results from the preprint [1...
Moving frames for pseudo–groups. I. The Maurer–Cartan forms
, 2002
"... Sur la théorie, si importante sans doute, mais pour nous si obscure, des ≪groupes ..."
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Cited by 6 (3 self)
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Sur la théorie, si importante sans doute, mais pour nous si obscure, des ≪groupes
Size Reduction and Partial Decoupling of Systems of Equations
 J. Symb. Comput
, 1999
"... A method is presented that reduces the number of terms of systems of linear equations (algebraic, ODEs or PDEs). As a byproduct these systems become partially decoupled. A variation of this method is applicable to nonlinear systems. Modifications to improve efficiency are given and examples are sho ..."
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Cited by 5 (5 self)
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A method is presented that reduces the number of terms of systems of linear equations (algebraic, ODEs or PDEs). As a byproduct these systems become partially decoupled. A variation of this method is applicable to nonlinear systems. Modifications to improve efficiency are given and examples are shown. This procedure can be used as a presimplification step of a computation of the radical of a differential ideal (pseudo differential Grobner basis). Algorithms for applying integrability conditions to a system of differential equations in a systematic way in order to generate simplified differential equations are implemented in a number of programs ([3, 4, 7, 8, 10, 11, 12] and more in [6]). Such calculations result in the radical or a (pseudo) differential Grobner Basis of the differential ideal generated by the original system. A common problem of these algorithms, and consequently their implementations, is an explosive expression swell. Optimizations like Buchbergers 2 nd criterion...
SymbolicNumeric Completion of Differential Systems by Homotopy Continuation
 Proc. ISSAC 2005. ACM
, 2005
"... Two ideas are combined to construct a hybrid symbolicnumeric differentialelimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagona ..."
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Cited by 4 (1 self)
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Two ideas are combined to construct a hybrid symbolicnumeric differentialelimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagonal homotopies to incrementally process new constraints, one at a time. The method is illustrated on several examples, combining symbolic differential elimination (using rifsimp) with numerical homotopy continuation (using phc).
Application of Numerical Algebraic Geometry and Numerical Linear Algebra to PDE
 ISSAC'06
, 2006
"... The computational difficulty of completing nonlinear pde to involutive form by differential elimination algorithms is a significant obstacle in applications. We apply numerical methods to this problem which, unlike existing symbolic methods for exact systems, can be applied to approximate systems ar ..."
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Cited by 4 (2 self)
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The computational difficulty of completing nonlinear pde to involutive form by differential elimination algorithms is a significant obstacle in applications. We apply numerical methods to this problem which, unlike existing symbolic methods for exact systems, can be applied to approximate systems arising in applications. We use Numerical Algebraic Geometry to process the lower order leading nonlinear parts of such pde systems. The irreducible components of such systems are represented by certain generic points lying on each component and are computed by numerically following paths from exactly given points on components of a related system. To check the conditions for involutivity Numerical Linear Algebra techniques are applied to constant matrices which are the leading linear parts of such systems evaluated at the generic points. Representations for the constraints result from applying a method based on Polynomial Matrix Theory. Examples to illustrate the new approach are given. The scope of the method, which applies to complexified problems, is discussed. Approximate ideal and differential ideal membership testing are also discussed.
Nonclassical reductions of a 3+1cubic nonlinear Schrödinger system
, 1998
"... An analytical study, strongly aided by computer algebra packages diffgrob2 by Mansfield and rif by Reid, is made of the 3+lcoupled nonlinear Schrödinger (CNLS) system i~,+V21y+(~ly~2 + ..."
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Cited by 4 (0 self)
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An analytical study, strongly aided by computer algebra packages diffgrob2 by Mansfield and rif by Reid, is made of the 3+lcoupled nonlinear Schrödinger (CNLS) system i~,+V21y+(~ly~2 +
Abel ODEs: Equivalence and Integrable Classes
"... A classification, according to invariant theory, of nonconstant invariant Abel ODEs known as solvable and found in the literature is presented. A set of new integrable classes depending on one or no parameters, derived from the analysis of the works by Abel, Liouville and Appell [2, 3, 4], is also ..."
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A classification, according to invariant theory, of nonconstant invariant Abel ODEs known as solvable and found in the literature is presented. A set of new integrable classes depending on one or no parameters, derived from the analysis of the works by Abel, Liouville and Appell [2, 3, 4], is also shown. Computer algebra routines were developed to solve ODEs members of these classes by solving their related equivalence problem. The resulting library permits a systematic solving of Abel type ODEs in the Maple symbolic computing environment. PROGRAM SUMMARY Title of the software package: Extension to the Maple ODEtools package Catalogue number: (supplied by Elsevier) Software obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland (see application form in this issue) Licensing provisions: none Operating systems under which the program has been tested: UNIX, Macintosh, Windows (95/98/NT). Programming language used: Maple V Release 5 Memory required to execut...
Determination of Maximal Symmetry Groups of Classes of Differential Equations
 in: Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation
, 2000
"... A symmetry of a dierential equation is a transformation which leaves invariant its family of solutions. As the functional form of a member of a class of dierential equations changes, its symmetry group can also change. We give an algorithm for determining the structure and dimension of the symmetry ..."
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Cited by 4 (1 self)
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A symmetry of a dierential equation is a transformation which leaves invariant its family of solutions. As the functional form of a member of a class of dierential equations changes, its symmetry group can also change. We give an algorithm for determining the structure and dimension of the symmetry group(s) of maximal dimension for classes of partial dierential equations. It is based on the application of dierential elimination algorithms to the linearized equations for the unknown symmetries. Existence and Uniqueness theorems are applied to the output of these algorithms to give the dimension of the maximal symmetry group. Classes of dierential equations considered include ode of form uxx = f(x; u; ux ), ReactionDiusion Systems of form u t uxx = f(u; v); v t vxx = g(u; v), and Nonlinear Telegraph Systems of form v t = ux ; vx = C(u; x)ux +B(u;x). 1. INTRODUCTION The symmetries, or transformations leaving invariant a system of partial dierential equations, are generally not...