  The method of separation of variables can be used to solve many separable
linear partial differential equations (LPDEs). Moreover, variable separation
solutions usually are some trigonometric series. In the paper, base on some
ideas of this method, we introduce a new technique to solve the Cauchy problem
for some LPDEs with the initial conditions consisting of some trigonometric
series, power series and exponential series. Then many LPDEs which are not
separable are solved, such as some second order elliptic equations, Stokes
equations and so on. In addition, the solutions of them can be expressed by
trigonometric series, power series or exponential series. Moreover, by using
power amd exponential series and an iterative method, we can solve many LPDEs
and nonlinear PDEs for the first time.
