Improved equivalent transformation method for reduction NLPDEs with time-dependent variables

Abstract

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.