A MuPAD Library for Differential Equations

We consider linear overdetermined systems of partial differential equations. We show that the introduction of weights classically used for the definition of ellipticity is not necessary, as any system that is elliptic with respect to some weights becomes elliptic without weights during its completion to involution. Furthermore, it turns out that there are systems which are not elliptic for any choice of weights but whose involutive form is nevertheless elliptic. We also show that reducing the given system to lower order or to an equivalent system with only one unknown function preserves ellipticity.

show abstract

“…Computing with MuPAD 8 [5] an involutive completion of Ay = 0, we get One easily verifies that it is equivalent to A (3) . Remark 6.16.…”

Section: Theorem 612 If An Operator a Is Dn-elliptic Then Its Compmentioning

confidence: 90%

“…By the same reasoning as used in the above proof, one sees immediately that if such a system is elliptic, it must be either in CauchyKovalevskaya form or overdetermined. 5 …”

Section: Proposition 42mentioning

confidence: 97%

Overdetermined Elliptic Systems

2005

Self Cite

View full text Add to dashboard Cite

show abstract

“…There is the following important result. In practice to complete a system to the involutive form one may use DETOOLS package [4] in computer algebra system MUPAD [7].…”

Section: Definition 24 a Differential Operator A (Of Order Q) Is Camentioning

confidence: 99%

Generalised elliptic boundary problems

Krupchyk

Тарханов

Tuomela

2007

Journal of Differential Equations

View full text Add to dashboard Cite

For elliptic systems of differential equations on a manifold with boundary, we prove the Fredholm property of a class of boundary problems which do not satisfy the Shapiro-Lopatinskii property. We name these boundary problems generalised elliptic, for they preserve the main properties of elliptic boundary problems. Moreover, they reduce to systems of pseudodifferential operators on the boundary which are generalised elliptic in the sense of Saks [R.S. Saks, On properties of generalised elliptic pseudodifferential operators on closed manifolds, Zap. Nauchn. Sem. LOMI 243 (1997) 215-269].

show abstract

“…For a given sufficiently regular system R q , there are numbers r and s such that R (s) q+r is involutive. In practice, to complete a system to the involutive form we have used the DETools package [6] in MuPAD [11] (see also Appendix A.6 for appropriate MuPad commands).…”

Section: The Shapiro-lopatinskij Conditionmentioning

confidence: 99%

The Sharpiro–Lopatinskij Condition for Elliptic Boundary Value Problems

Krupchyk¹,

Tuomela²

2006

LMS J. Comput. Math.

View full text Add to dashboard Cite

Elliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. We propose here a criterion (which also covers over-determined elliptic systems) for checking this condition. We present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables we give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Our approach is based on formal theory of PDEs, and we use constructive module theory and polynomial factorisation in our test. Actual computations were carried out with computer algebra systems Singular and MuPad.

show abstract

A MuPAD Library for Differential Equations

Cited by 6 publications

References 12 publications

Overdetermined Elliptic Systems

Overdetermined Elliptic Systems

Generalised elliptic boundary problems

The Sharpiro–Lopatinskij Condition for Elliptic Boundary Value Problems

Contact Info

Product

Resources

About