摘要步入21世纪，物理、医学等学科有了进一步发展和完善，随之微分方程相关理论被更加广泛的应用在各个专业领域中。 微分方程理论在物理、机械和其他学科研究中占有很重要的作用，微分方程不仅应用在传统的应用数学内容之中，并且也是现代数学的重要组成部分之一。微分方程是实践应用与数学理论之间的重要的桥梁，微分方程特别是非线性微分方程和非线性偏微分方程一直非常活跃在拓展应用研究领域。从这个角度说，偏微分方程变成了数学的中心。78417

本文从微分方程的产生背景出发，阐述了微分方程的定义和微分方程模型，并深入研究了一阶齐次线性偏微分方程、一阶线性偏微分方程、一阶拟线性非齐次偏微分方程、双曲型偏微分方程、抛物型偏微分方程、椭圆型偏微分方程、两个自变量的二阶线性偏微分方程、n个自变量的二阶线性偏微分方程，并且探讨了偏微分方程在人口问题中和传染病动力学中的应用，最后给出了本文的结论。

毕业论文关键词：微分方程；线性偏微分方程；椭圆型方程

Abstract With physics, medicine and other disciplines in the study of the phenomenon of expansion of both the breadth and depth of application of partial differential equations more broadly。 Differential equation in physical, mechanical and other background studies, or problems of other disciplines, not only is one of the main traditions of applied mathematics, it is also an important part of modern mathematics。 Differential is an important bridge between mathematical theory and practical application, particularly non-linear differential equations and nonlinear differential equations partial differential equations has been very active in expanding research area。 From this perspective, partial differential equations became the center of mathematics。

From the background differential equations starting to explain the definition of differential equation model and differential equations, and in-depth study of the first-order homogeneous linear partial differential equations, first order linear partial differential equations, first order quasilinear inhomogeneous partial differential equations , hyperbolic partial differential equations, parabolic partial differential equations, elliptic partial differential equations, second order linear partial differential equations of the two arguments, the second order linear n argument of partial differential equations, partial differential equations and discusses population issues and dynamics of infectious disease applications。 Finally, the conclusion of this article。

Keywords: Differential equations; Linear partial differential equations; Elliptic equations

目  录

第一章 绪 论 1

1。1 研究背景 1

1。2 研究意义和目的 1

1。3 研究方法 1

第二章  相关理论概述 2

2。1 微分方程的定义 2

2。2 微分方程模型 3

第三章  微分方程的分类与化简 5

3。1 一阶齐次线性偏微分方程 5

3。2 一阶线性偏微分方程 7

3。3 一阶拟线性非齐次偏微分方程 11

3。4 两个自变量的二阶线性偏微分方程 14

3。4。1  双曲型偏微分方程 15

3。4。2  抛物型偏微分方程