Partial Differential Equations in Applied Mathematics (Dec 2023)
The Z5 transformation for solving quasi-linear partial differential equations
Abstract
The Z1-Z4 transformations proposed in our previous papers have acquired remarkable success in the field of linear partial differential equations, general solutions of a large number of typical equations were obtained for the first time, and exact solutions of many definite solution problems of general significance were also successfully obtained. In this paper we will present the Z5 transformation and use it to get the general solutions of a variety of first-order quasi-linear partial differential equations and, for the first time, to obtain analytical solutions of a class of second-order quasi-linear partial differential equations containing arbitrary functions. Using the general solutions, we obtain exact solutions of two definite solution problems. By comparison we find that the general solutions of some first-order quasi-linear partial differential equations that can be obtained using the characteristic equation method are incomplete.
