For dealing with exact solutions and other properties, the variable-coefficient equations are far more complicated than its constant-coefficient counterparts. So far, there are few systematic and effective methods to deal with this problem, in this paper, we develop an effective approach to doing this. The present paper is concerned with the nonlinear partial differential equations (NLPDEs) with time-dependent. By the improved equivalent transformation (ET) method, the variable-coefficient NLPDEs (vc-NLPDEs) are transformed into different constant-coefficient partial differential equations (cc-PDEs) under different conditions, one of the cc-PDEs is simpler, so it is easy to solve. Moreover, we give the vc-PDE which can be transformed into the corresponding cc-PDE based on the condition. Thus, the exact solutions to vc-PDEs are obtained by exact solutions to the cc-PDEs. In general, this method can be applied to reduction all time-dependent coefficients NLPDEs.

为了处理精确解和其他属性，可变系数方程比其常数系数方程要复杂得多。到目前为止，很少有系统有效的方法来解决此问题，在本文中，我们开发了一种有效的方法来解决此问题。本文涉及具有时间相关性的非线性偏微分方程（NLPDE）。通过改进的等效变换（ET）方法，将变系数NLPDE（vc-NLPDE）在不同条件下转换为不同的常系数偏微分方程（cc-PDE），其中一个cc-PDE更简单，因此它很容易解决。此外，我们根据条件给出了可以转换为相应的cc-PDE的vc-PDE。因此，通过cc-PDE的精确解可以获得vc-PDE的精确解。通常，此方法可用于减少所有与时间相关的系数NLPDE。