符号分部积分和发散求逆的同伦算子方法及其应用
The homotopy operator method for symbolic integration by parts and inversion of divergences with applications
摘要：
使用标准的微积分，明确的公式，一维，二维和三维同伦算子。一维同伦算子的推导。可以使用类似的方法来导出多维版本。同伦算子的基于微积分的公式很容易在Mathematica、Maple和REDUCE等计算机代数系统中实现。几个例子说明了同伦算子的使用，范围和限制。同伦算子可用于非线性偏微分方程守恒律的符号计算。守恒定律提供了对偏微分方程的物理和数学性质的深入了解。例如，无穷多个守恒律的存在建立了非线性偏微分方程的完全可积性。
英文摘要：
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.
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