The Homotopy Operator: From Integration by Parts to the Computation of Conservation Laws of Nonlinear PDEs in Multiple Dimensions

Douglas Poole and Willy Hereman

Department of Mathematical and Computer Sciences
Colorado School of Mines
Golden, Colorado 80401

Abstract

In many applications one needs to integrate exact expressions involving unspecified functions of several independent variables. Even in cases with one independent variable, Mathematica's Integrate fails to integrate elementary rational expressions. Neither can it integrate some polynomial expressions involving trigonometric functions. For example, unless c=0, Integrate fails to undo the x-derivative of

(((f'))^2)cos f - g cos f + cg' where f(x) and  g(x) are differentiable functions and c is constant.

The homotopy operator is a tool from differential geometry that reduces the integration by parts to a series of differentiations followed by a single integration over an auxiliary variable. By circumventing integration by parts, the homotopy operator succeeds where the Integrate function has failed. Applied to problems with more than one independent variable, the homotopy operator inverts the total divergence, i.e. automatically computes a vector for which the divergence was given. No functions in Mathematica can accomplish that important task.

For universal applicability and fast execution, the homotopy operator has been implemented in the language of standard calculus. Thus, the homotopy operator can be used as a stand alone tool for problems where integration by parts of expressions involving unspecified functions is needed.

One application is the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs) in multiple dimensions. Such conservation laws consist of a conserved density and an associated flux. Nonlinear PDEs that admit many, sometimes infinitely many, conservation laws arise in fluid dynamics, plasma physics, electrical networks, etc.

The Mathematica algorithm for the homotopy operator will be applied to the computation of fluxes which indeed requires reliable integration by parts of differential expressions in multiple independent variables.