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15
Notes on triangular sets and triangulationdecomposition algorithms II: Differential Systems
 SYMBOLIC AND NUMERICAL SCIENTIFIC COMPUTING
, 2003
"... This is the second in a series of two tutorial articles devoted to triangulationdecomposition algorithms. The value of these notes resides in the uniform presentation of triangulationdecomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We em ..."
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Cited by 69 (8 self)
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This is the second in a series of two tutorial articles devoted to triangulationdecomposition algorithms. The value of these notes resides in the uniform presentation of triangulationdecomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.
Fast Differential Elimination in C: The CDiffElim Environment
, 2000
"... We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in different ..."
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Cited by 14 (10 self)
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We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in differential elimination algorithms, such as exhaustion of computer memory due to intermediate expression swell, and failure to complete due to the massive number of calculations involved. These strategies include lowlevel memory management strategies and data representations that are tailored for efficient differential elimination algorithms. These strategies, which are coded in a lowlevel C implementation, seem much more difficult to implement in highlevel general purpose computer algebra systems. A differential elimination algorithm written in this environment is applied to the determination of symmetry properties of classes of n+1dimensional coupled nonlinear partial differential equations of form iut+r2u+ i a(t)jxj2 + b(t) \Delta x + c(t) + djuj 4n j u = 0; where u is an mcomponent vectorvalued function. The resulting systems of differential equations for the symmetries have been made available on the web, to be used as benchmark systems for other researchers. The new differential elimination algorithm in C, runs on the test suite an average of 400 times faster than our RifSimp algorithm in Maple.
Geometric Completion of Differential Systems using NumericSymbolic Continuation
 SIGSAM Bulletin
, 2002
"... Symbolic algorithms using a finite number of exact differentiations and eliminations are able to reduce over and underdetermined systems of polynomially nonlinear differential equations to involutive form. The output involutive form enables the identification of consistent initial values, and eases ..."
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Cited by 11 (9 self)
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Symbolic algorithms using a finite number of exact differentiations and eliminations are able to reduce over and underdetermined systems of polynomially nonlinear differential equations to involutive form. The output involutive form enables the identification of consistent initial values, and eases the application of exact or numerical integration methods. Motivated to avoid expression swell of pure symbolic approaches and with the desire to handle systems with approximate coefficients, we propose the use of homotopy continuation methods to perform the differentialelimination process on such nonsquare systems. Examples such as the classic index 3 Pendulum illustrate the new procedure. Our approach uses slicing by random linear subspaces to intersect its jet components in finitely many points. Generation of enough generic points enables irreducible jet components of the differential system to be interpolated. 1
Computing power series solutions of a nonlinear PDE system
 ISSAC ’03: PROCEEDINGS OF THE 2003 INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION
, 2003
"... This paper presents a new algorithm to compute the power series solutions of a significant class of nonlinear systems of partial differential equations. The algorithm is very different from previous algorithms to perform this task. Those relie on differentiating iteratively the differential equation ..."
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Cited by 5 (0 self)
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This paper presents a new algorithm to compute the power series solutions of a significant class of nonlinear systems of partial differential equations. The algorithm is very different from previous algorithms to perform this task. Those relie on differentiating iteratively the differential equations to get coefficients of the power series, one at a time. The algorithm presented here relies on using the linearisation of the system and the associated recurrences. At each step the order up to which the power series solution is known is doubled. The algorithm can be seen as belonging to the family of Newton iteration methods.
Symbolicnumeric Computation of Implicit Riquier Bases for PDE
 Proc. ISSAC ’07, ACM
, 2007
"... Riquier Bases for systems of analytic pde are, loosely speaking, a differential analogue of Gröbner Bases for polynomial equations. They are determined in the exact case by applying a sequence of prolongations (differentiations) and eliminations to an input system of pde. We present a symbolicnumer ..."
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Cited by 5 (3 self)
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Riquier Bases for systems of analytic pde are, loosely speaking, a differential analogue of Gröbner Bases for polynomial equations. They are determined in the exact case by applying a sequence of prolongations (differentiations) and eliminations to an input system of pde. We present a symbolicnumeric method to determine Riquier Bases in implicit form for systems which are dominated by pure derivatives in one of the independent variables and have the same number of pde and unknowns. The method is successful provided the prolongations with respect to the dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are nonsingular when evaluated at points on the zero sets defined by the functions of the pde. For polynomially nonlinear pde, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. We give a differential algebraic interpretation of Pryce’s method for ode, which generalizes to the pde case. A major aspect of the method’s efficiency is that only prolongations with respect to a single (dominant) independent variable are made, possibly after a random change of coordinates. Potentially expensive and numerically unstable eliminations are not made. Examples are given to illustrate theoretical features of the method, including a curtain of Pendula and the control of a crane.
Symmetry classification using noncommutative invariant differential operators
 Found. Comput. Math
"... Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G f, or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associate ..."
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Cited by 4 (0 self)
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Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G f, or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated overdetermined ‘defining system ’ of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations. We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F. The method is applied to a class of nonlinear diffusion convection equations vx = u, vt = B(u)ux − K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of Ginvariant differential operators.
Implicit Riquier Bases for PDAE and their SemiDiscretizations
, 2008
"... Complicated nonlinear systems of pde with constraints (called pdae) arise frequently in applications. Missing constraints arising by prolongation (differentiation) of the pdae need to be determined to consistently initialize and stabilize their numerical solution. In this article we review a fast pr ..."
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Complicated nonlinear systems of pde with constraints (called pdae) arise frequently in applications. Missing constraints arising by prolongation (differentiation) of the pdae need to be determined to consistently initialize and stabilize their numerical solution. In this article we review a fast prolongation method, a development of (explicit) symbolic Riquier Bases, suitable for such numerical applications. Our symbolicnumeric method to determine Riquier Bases in implicit form, without the unstable eliminations of the exact approaches, applies to square systems which are dominated by pure derivatives in one of the independent variables. The method is successful provided the prolongations with respect to a single dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are nonsingular when evaluated at points on the zero sets defined by the functions of the pdae. For polynomially nonlinear pdae, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. Our method generalizes Pryce’s method for dae to pdae. Given a dominant independent time variable, for an initial value problem for a system of pdae we show that its semidiscretization
A Normal Form Algorithm For Regular Differential Chains
, 2013
"... Abstract. This paper presents a new algorithm for computing the normal form of a differential rational fraction modulo differential ideals presented by regular differential chains. An application to the computation of power series solutions is presented and illustrated with the new DifferentialAlgeb ..."
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Abstract. This paper presents a new algorithm for computing the normal form of a differential rational fraction modulo differential ideals presented by regular differential chains. An application to the computation of power series solutions is presented and illustrated with the new DifferentialAlgebra MAPLE package.
Noncommutative Riquier Theory in Moving Frames of Differential Operators
, 2003
"... Moving frames chosen to be invariant under a known Lie group G provide a powerful generalization of the idea of choosing Ginvariant coordinates to cases where Ginvariant coordinates do not exist. Such Ginvariant formulations are of great current interest in areas such as Geometric Integration whe ..."
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Cited by 1 (1 self)
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Moving frames chosen to be invariant under a known Lie group G provide a powerful generalization of the idea of choosing Ginvariant coordinates to cases where Ginvariant coordinates do not exist. Such Ginvariant formulations are of great current interest in areas such as Geometric Integration where Ginvariant integrators (e.g. symplectic integrators), can often substantially outperform noninvariant integrators. They are also of substantial interest in applications where one would like to factor out a known group. One form of classical existence and uniqueness theory for analytic PDE referred to (standard) commuting partial derivatives is that of Riquier, which was formulated and generalized by Rust using a Gröbner style development. We extend the RustRiquier existence and uniqueness theory to analytic PDE written in terms of moving frames of noncommuting Partial Dierential Operators (PDO). The main idea for the theoretical development is to use the commutation relations between the PDO to place them in a standard order. This normalization is exploited to generalize the corresponding steps of the commuting RustRiquier Theory to the noncommuting case. Given an equivalence group G Lisle has given a Ginvariant method for determining the structure of Lie symmetry groups of classes of PDE. Lisle's method for such group classication problems was illustrated on a number of challenging examples, which lead to unmanageable expression explosion for computer algebra programs using the standard (commuting) frame. He obtained new results, which for want of an existence and uniqueness theorem for PDE in noncommuting frames, had to be individually checked. We provide an existence and uniqueness theorem making rigorous the output from Lisle's method. For the finite parameter group case, the output is reformulated in terms of the integration of a system of Frobenius type, which can be numerically integrated by integrating an ODE system along a curve.