Limit residual function method and applications to PDE models

The article focuses on finding analytical series solutions for linear and nonlinear partial differential equations. It introduces a new technique named the limit residual function method, which is used to determine the coefficients of the power series solution of the equations. This method does not require transforming the target equation into another space and rely on the concept of the limit and the residual function. The article discusses various applications to demonstrate the proposed method’s effectiveness, reliability, and ease. These applications cover five types of partial differential equations: Navier–Stokes, reaction–diffusion, Fisher, Klein-Gordon, Poisson, and Hunter–Saxton equations.