一個解特定二階駐定方程的新方法

摘 要：本文對文獻[1]中提出的一類二階駐定方程[G/G=j=0mpjG/Gj]的求法進行探討，提出了全新的求解方法——特征伴隨方程法，通過該求法得到這類方程當[m]取不同非負整數（本文僅討論[m=2]）時的通解，并相應給出該方程作為求解非線性偏微分方程的輔助方程時所需要的[G/G]的解析表達式；同時給出一個作為常微分方程中駐定方程的應用實例，以及該方程作為非線性偏微分方程的輔助方程時利用擴展[G/G]展開法的求解例子，通過該實例給出方程的精確行波解。

關鍵詞：二階駐定方程；特征方程；指數變換；特征伴隨方程（法）；降階法；異型通解

中圖分類號：O175 DOI：10.16375/j.cnki.cn45-1395/t.2024.01.018

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A new method for solving specific second order stationary

equations：characteristic adjoint equation method

HAN Song， ZHANG Mingjun， HE Xiaoying

（College of Science， Guangxi University of Science and Technology， Liuzhou 545006， China）

Abstract：The characteristic adjoint equation method is presented for solving a type of second-order

stationary equation， which is denoted as [G/G=j=0mpjG/Gj]，[m=0，1，2，…]，and was initially proposed in paper [1]. With the new method， the general solution of the equation can be obtained when [m] is equal to different non-negative integers （we only discuss [m=2] in this text）. The analytical expression [G/G] of the equation can also be obtained when the second-order stationary equation is used as an auxiliary equation for solving nonlinear partial differential equations. Two application examples are shown. In the first case， the second-order stationary equation is used as a stationary equation in the ordinary differential equation; in the second example， it is regarded as an auxiliary equation for solving the nonlinear partial differential equation by extending [G/G] using the expansion method， which obtains the corresponding exact traveling wave solution.

Keywords：second-order stationary equation; characteristic equation; exponential transformation; characteristic adjoint equation（method）; reduction method; heterotypic general solution

（責任編輯：羅小芬