A note for derivation of the formulas of solution to some evolution equations

Abstract

In this paper I studied the partial differential equations of Walter A Strauss which introduced many common partial differential equations This paper will try to derive the solutions of these non-homogeneous partial differential equations hoping to find their general solution or the expression of the solution In order to derive the higher order differential equations we will start from the solution of the one-dimensional linear partial differential equation to the higher order which includes the second and third orders We hope to recognize the commonality between more partial differential equations in the derivation process We will describe and introduce in detail in the derivation process In this article we refer to the Fourier analysis and application of Professor C K Lin and use Fourier transform to try to find correlations in various partial differential equations and to understand about the application of high-dimensional Fourier transformation and spherical coordinate transformation Fourier transform belongs to vector calculus in the function of multivariables We adopt the skill of spherical coordinate conversion in some different partial differential equations For these common partial differential equations we can more easily derive Before the article starts we need to introduce some related formulas and derivations including: Fourier transform spherical coordinate transform Gamma function Beta function and Gaussian function In the article we introduced in detail the homogeneous and non-homogeneous solutions of the wave equation Dirac equation Klein-Gordon equation Schrodinger equation diffusion equation

Date of Award	2020
Original language	English
Supervisor	Yung-Fu Fang (Supervisor)