Series solutions to the cauchy problem for partial differential equations

TL;DR

This paper introduces a new technique based on separation of variables to solve a broader class of linear and nonlinear partial differential equations, including non-separable cases, using series solutions.

Contribution

It presents a novel approach that extends series solution methods to non-separable LPDEs and nonlinear PDEs, which were previously difficult to solve.

Findings

Successfully solved non-separable LPDEs like elliptic and Stokes equations.

Expressed solutions using trigonometric, power, and exponential series.

Developed an iterative method for solving nonlinear PDEs.

Abstract

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.