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

TL;DR

This paper presents explicit calculus-based formulas for homotopy operators in multiple dimensions, facilitating symbolic integration by parts and divergence inversion, with applications to computing conservation laws of nonlinear PDEs.

Contribution

It derives explicit formulas for multi-dimensional homotopy operators using standard calculus, enabling easier implementation in computer algebra systems for PDE analysis.

Findings

Formulas for 1D, 2D, and 3D homotopy operators are provided.

Homotopy operators can be used to compute conservation laws of nonlinear PDEs.

Examples demonstrate the scope and limitations of the method.

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.