The homotopy operator method for symbolic integration by parts and inversion of divergences with applications

Abstract

Using standard calculus, explicit formulas for one-, two- and three-dimensional homotopy operators are presented. A derivation of the one-dimensional homotopy operator is given. A similar methodology can be used to derive the multi-dimensional versions. The calculus-based formulas for the homotopy operators are easy to implement in computer algebra systems such as Mathematica, Maple and REDUCE. Several examples illustrate the use, scope and limitations of the homotopy operators. The homotopy operator can be applied to the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs). Conservation laws provide insight into the physical and mathematical properties of the PDE. For instance, the existence of infinitely many conservation laws establishes the complete integrability of a nonlinear PDE.

Keywords:

• homotopy operator
• exact differential function
• Euler operator
• total divergence
• conservation law
• complete integrability

AMS Subject Classifications::

• Primary: 37J35
• 37K40
• 35Q51
• Secondary: 68W30
• 47J35
• 70H06

Acknowledgements

This material is based in part upon research supported by the National Science Foundation (NSF) under Grant No. CCF-0830783. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF. Willy Hereman is grateful for the hospitality and support of the Department of Mathematics and Statistics of the University of Canterbury (Christchurch, New Zealand) during his sabbatical visit in Fall 2007. M. Hickman (University of Canterbury) and B. Deconinck (University of Washington) are thanked for sharing their insights about the scope and limitations of homotopy operators.