广义泰勒级数方法与同伦分析方法并不等价
The generalized Taylor series approach is not equivalent to the homotopy analysis method
摘要：
在最近关于非线性微分方程解的近似方法领域的工作中，有人提出所谓的广义泰勒级数方法等价于同伦分析方法。在本文中，我们证明了这种观点只在非常特殊的情况下是有效的，一般同伦分析方法是更强大的。特别是，只有当解表示为自变量的幂级数时，等价性才有效。正如已经多次证明的那样，交替基函数可以大大改善同伦解的误差性质，并且当基函数不是多项式或幂函数时，不再有广义泰勒级数方法与同伦分析方法等价的说法。我们证明了这一点考虑的一个例子，其中广义泰勒级数必须始终有一个有限的收敛半径（因此有限的适用性），而同伦解决方案是有效的整个无限域。然后，我们给出第二个例子，其中的精确解不是解析的，因此它将不同意在域上的广义泰勒级数。我们的结论是广义泰勒级数方法是不等价的同伦分析方法。特别地，广义泰勒级数只能恢复解的局部信息，而同伦分析方法只要选择适当的基函数，就能全局地恢复解.这样的结果有重要的影响，如何计算系列时，近似解非线性微分方程。
英文摘要：
In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the present paper, we demonstrate that such a view is only valid in very special cases, and in general the homotopy analysis method is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the homotopy analysis method. We demonstrate this by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence it will not agree with the generalized Taylor series over the domain. We conclude that the generalized Taylor series approach is not equivalent to the homotopy analysis method. In particular, the generalized Taylor series can at best recover local information about solutions, whereas the homotopy analysis method can recover the solutions globally if appropriate base functions are selected. Such results have important implications for how series are calculated when approximating solutions to nonlinear differential equations.
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