通过渐近近似量求和发散、截断和欠指定的幂级数
On the summation of divergent, truncated, and underspecified power series via asymptotic approximants
AI总结：
本文提出一种紧凑且精确的方法，用于求解发散或仅知有限项幂级数的问题。通过简单的递推关系构建渐近近似量，可预测未知系数及其他重要性质，为边界层和单极问题提供新基准值。
摘要：
本文提供了一种紧凑且精确的求解方法，用于处理无限幂级数解发散或其系数仅知有限项的问题。该方法仅需幂级数解或其截断解可用，并且已知解在展开点以外的渐近行为。我们正式化了渐近近似量方法，该方法在热力学virial级数中的应用取得近期成功。通过简单的递推关系，利用少量级数操作规则构建渐近近似量，从而在两个渐近区域之间架起桥梁，保持中间区域的准确性。提供了一种通用算法来构建此类近似量。为展示方法的通用性，为三个数学物理相关的非线性问题构建了近似量：Sakiadis边界层、Blasius边界层和Flierl-Petviashvili单极。每个问题的幂级数解欠指定，因为在没有数值模拟的情况下，一个低阶系数未知，导致更高阶系数也未知。构建的近似量能够预测未知系数及其他重要性质。近似量为Sakiadis边界层提供了新基准值，并与Blasius边界层和Flierl-Petviashvili单极的近期数值值一致。
英文摘要：
A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.
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