The first order partial differential equations resolved with any derivatives

TL;DR

This paper presents a comprehensive method for solving first order partial differential equations involving any derivatives by transforming, converting, and applying Fourier transforms to obtain solutions.

Contribution

It introduces a novel approach that transforms PDEs with respect to any derivatives into linear systems and solves them using Fourier transforms, extending existing methods.

Findings

Transforming PDEs into linear systems simplifies their solution.

Using Fourier transforms converts PDEs into integral equations for easier solving.

The method applies to PDEs resolved with respect to any derivatives, broadening applicability.

Abstract

In this paper we discuss the first order partial differential equations resolved with any derivatives. At first, we transform the first order partial differential equation resolved with respect to a time derivative into a system of linear equations. Secondly, we convert it into a system of the first order linear partial differential equations with constant coefficients and nonlinear algebraic equations. Thirdly, we solve them by the Fourier transform and convert them into the equivalent integral equations. At last, we extend to discuss the first order partial differential equations resolved with respect to time derivatives and the general scenario resolved with any derivatives.