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21
Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations
 Euromath Bull
, 1999
"... A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briey outlined. Examples illustrate the use of the software. Directions for further research and development are ..."
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Cited by 41 (3 self)
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A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briey outlined. Examples illustrate the use of the software. Directions for further research and development are indicated.
Symmetry reductions and exact solutions of a class of nonlinear heat equations
 Physica D
, 1993
"... Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f(u), (1) are considered. The method of differential Gröbner bases is used both to find the conditions on f(u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to sol ..."
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Cited by 33 (2 self)
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Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f(u), (1) are considered. The method of differential Gröbner bases is used both to find the conditions on f(u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f(u) in terms of the roots of f(u) =0. 0 Symmetry Reductions of a Nonlinear Heat Equation 1
A Precise Definition Of Reduction Of Partial Differential Equations
, 1999
"... We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and nonclassical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the noncla ..."
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Cited by 15 (12 self)
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We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and nonclassical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the nonclassical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of nonclassical reduction of the nonlinear wave equation in 1+3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of nonLie reductions which are farreaching generalization of the wellknown symmetry reductions of the nonlinear wave equations.
Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems
 Adv. Math
, 2000
"... Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinar ..."
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Cited by 8 (3 self)
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Abstract. We completely characterize all nonlinear partial differential equations leaving a given finitedimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasiexactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented.
Involution and Symmetry Reductions
 Math. Comp. Model
, 1995
"... After reviewing some notions of the formal theory of differential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the effects of local solvability and of gauge symmetries, respectively. We consider nonclassical symmetry reducti ..."
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Cited by 6 (5 self)
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After reviewing some notions of the formal theory of differential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the effects of local solvability and of gauge symmetries, respectively. We consider nonclassical symmetry reductions and more general reductions using differential constraints.
Nonclassical And Conditional Symmetries
, 1996
"... features. Through the process of prolongation, which requires the group transformations preserve the intrinsic contact structure on the jet space, they define local groups of contact transformations on the kth order jet spaces J k . Such a transformation group will be a symmetry group of the sys ..."
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Cited by 5 (0 self)
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features. Through the process of prolongation, which requires the group transformations preserve the intrinsic contact structure on the jet space, they define local groups of contact transformations on the kth order jet spaces J k . Such a transformation group will be a symmetry group of the system of differential equations E ae J k if the transformations of the symmetry group leave E invariant. This implies that the group transformations map solutions of E onto solutions of E. The classical Lie symmetries are sometimes called external symmetries. To date, several extensions of the classical Lie approach have been proposed. Each of them relaxes one or more of the basic properties obeyed by classical symmetry groups. If we relax the restriction that the infinitesimal generators determine geometrical transformations on a finite order
Higher Conditional Symmetry and Reduction of Initial Value Problems
"... We give the exposition of a generalized symmetry approach to reduction of initial value problems for nonlinear evolution equations in one spatial variable. Using this approach we classify the initial value problems for thirdorder evolution equations that admit reduction to Cauchy problems for syste ..."
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Cited by 4 (2 self)
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We give the exposition of a generalized symmetry approach to reduction of initial value problems for nonlinear evolution equations in one spatial variable. Using this approach we classify the initial value problems for thirdorder evolution equations that admit reduction to Cauchy problems for systems of two ordinary di#erential equations. These reductions are shown to correspond to higher conditional symmetries admitted by the corresponding nonlinear evolution equations.
Differential constraints and exact solutions of nonlinear diffusion equations
, 2002
"... The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries. ..."
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Cited by 4 (1 self)
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The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries.
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 +
Computer Algebra and Differential Equations  An Overview
"... 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 ..."
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Cited by 4 (0 self)
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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...