Allan D. Wittkopf

Differential Elimination–Completion Algorithms for DAE and PDAE

Differential–algebraic equations (DAE) and partial differential–algebraic equations (PDAE) are systems of ordinary equations and PDAEs with constraints. They occur frequently in such applications as constrained multibody mechanics, spacecraft control, and incompressible fluid dynamics. A DAE has differential index r if a minimum of r+1 differentiations of it are required before no new constraints are obtained. Although DAE of low differential index (0 or 1) are generally easier to solve numerically, higher index DAE present severe difficulties. Reich et al. have presented a geometric theory and an algorithm for reducing DAE of high differential index to DAE of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for DAE of low differential index. We show that for analytic autonomous first-order DAE, this algorithm is equivalent to the Cartan–Kuranishi algorithm for completing a system of differential equations to involutive form. The Cartan–Kuranishi algorithm has the advantage that it also applies to PDAE and delivers an existence and uniqueness theorem for systems in involutive form. We present an effective algorithm for computing the differential index of polynomially nonlinear DAE. A framework for the algorithmic analysis of perturbed systems of PDAE is introduced and related to the perturbation index of DAE. Examples including singular solutions, the Pendulum, and the Navier–Stokes equations are given. Discussion of computer algebra implementations is also provided.

Existence and uniqueness theorems for formal power series solutions of analytic differential systems

We present Existence ant1 Uniqueness Theorems for formal pO\Wr series solutions Of ilnd~t.i(’ s\lSteIlls Of PDF. in il cmtain form. ‘This form can be obtained by it finil.c number of differediations and cliulinat.ic)us of the original systen~~ and allows its formal power series solut,ious t.0 lx coniput~ed in a11 alg0rithmic fashion. The result.ing reduced involutiw form (rif’ form) produced by our rif’ algorit,liui is a generalizitt.ion of the ClassiCal fornl of Riquier and .Janet; and that of CauchKOV~l.lC?\?h~iL I;(: waken the assumpt.ion of linearity iu the highest dermdves iu t~hosc approaches t.O allow for systcrns which are n0nlineiK in their highest deriva.t.ives. A new fornml developn~cx~t. of Riqnicr’s theory is given: with proofs. n~otleled after t,how in Griilmcr Basis Theory. For the uonlincar tllcoryz the concept of rclatiw Riquiel Bases is introduced. This allows for t.he easy esteusion of ideas from the linear t0 tlw nonlinear t,hrory. Tile essent.ial idea is that an arbitrary noulincar system can Ix writ.teu (aft.cr tliffcrcutiatiou if necessary), as il syst.cmi which is liw ear in its highw, dcrivat.ivcs, and a constraint syst,em. which is n0nlinear in its highclst, derivatives. Our t,heorems iwe applied t,o S6T~rill eximplcs.

Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs

We present an algorithm COMMUTATION.RELATIONS, which can calculate the commutation relations for the Lie symmetry algebra of symmetry operators for any system of PDEs. Unlike existing methods, COMMUTATION_RELATIONS does not depend on the heuristic process of integrating the associated differential equations for the symmetry operators (i.e. integrating the ‘determining equations’), An algorithm INITIAL-DATA, developed in previous work, is used to calculate lists of initial data which are in l-to-l correspondence with solutions of determining equations. COMMUTATION.RELATIONS exploits this correspondence by calculating commutators in terms of initial data. The method has been implemented in the symbolic language MAPLE and can be applied to both finiteand infinite-dimensional Lie symmetry algebras. We show how knowledge of the Lie symmetry algebra calculated by CoMMUTATION.RELATIONS can simplify the task of explicitly integrating determining equations.

Fast differential elimination in C: The CDiffElim environment

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 low-level memory management strategies and data representations that are tailored for efficient differential elimination algorithms. These strategies, which are coded in a low-level C implementation, seem much more difficult to implement in high-level 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+1)-dimensional coupled nonlinear partial differential equations of form iut+∇2u+(a(t)|x|2+b(t)·x+c(t)+d|u|4/n)u=0, where u is an m-component vector-valued 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. New algorithms, including an enhanced GCD algorithm, and a hybrid symbolic-numeric differential elimination algorithm, are also described.

Algorithms for the non-monic case of the sparse modular GCD algorithm

Let <i>G</i> = (4<i>y</i>²+2<i>z</i>)<i>x</i>² + (10<i>y</i>²+6<i>z</i>) be the greatest common divisor (<sc>Gcd</sc>) of two polynomials <i>A, B</i> ∈ ℤ[<i>x,y,z</i>]. Because <i>G</i> is not monic in the main variable <i>x</i>, the sparse modular <sc>Gcd</sc> algorithm of Richard Zippel cannot be applied directly as one is unable to scale univariate images of <i>G</i> in <i>x</i> consistently. We call this the <i>normalization problem</i>.We present two new sparse modular <sc>Gcd</sc> algorithms which solve this problem without requiring any factorizations. The first, a modification of Zippels algorithm, treats the scaling factors as unknowns to be solved for. This leads to a structured coupled linear system for which an efficient solution is still possible. The second algorithm reconstructs the monic <sc>Gcd</sc> <i>x</i>² + (5<i>y</i>²+3<i>z</i>)/(2<i>y</i>²+<i>z</i>) from monic univariate images using a sparse, variable at a time, rational function interpolation algorithm.

Determination of maximal symmetry groups of classes of differential equations

A symmetry of a differential equation is a transformation which leaves invariant its family of solutions. As the functional form of a member of a class of differential 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 differential equations. It is based on the application of differential 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 differential equations considered include ODE of form <italic>u<subscrpt>xx</subscrpt></italic> = ƒ(<italic>x, u, u<subscrpt>x</subscrpt></italic>), Reaction-Diffusion Systems of form <italic>u<subscrpt>t</subscrpt></italic> - <italic>u<subscrpt>xx</subscrpt></italic> = ƒ(<italic>u, v</italic>), <italic>v<subscrpt>t</subscrpt></italic> - <italic>v<subscrpt>xx</subscrpt></italic> = <italic>g</italic>(<italic>u, v</italic>), and Nonlinear Telegraph Systems of form <italic>v<subscrpt>t</subscrpt></italic> = <italic>u<subscrpt>x</subscrpt></italic>, <italic>v<subscrpt>x</subscrpt></italic> = <italic>C</italic>(<italic>u, x</italic>)<italic>u<subscrpt>x</subscrpt></italic> + <italic>B</italic>(<italic>u, x</italic>).

Symbolic-numeric completion of differential systems by homotopy continuation

Two ideas are combined to construct a hybrid symbolic-numeric differential-elimination 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).

On the design and implementation of Brown's algorithm over the integers and number fields

We study the design and implementation of the dense modular GCD algorithm of Brown applied to bivariate polynomial GCDs over the integers and number fields. We present an improved design of Browns algorithm and compare it asymptotically with Browns original algorithm, with GCD-HEU, the heuristic GCD algorithm, and with the EEZGCD algorithm. We also make an empirical comparison based on Maple implementations of the algorithms. Our findings show that a careful implementation of our improved version of Browns algorithm is much better than the other algorithms in theory and in practice.