摘要数学发展的历史告诉我们，300年来数学分析是数学的首要分支，而微分方程又是数学分析的重要研究分支之一，特别是Laplace方程的研究有着特别重要的意义。半个世纪以来，偏微分方程的理论得到重大的发展，计算机的出现为Laplace方程的研究提供了新线索和新方法。Laplace方程不仅仅是一个数学的微分方程，近几年来，分数阶偏微分方程在流体力学、材料力学、生物学、金融学、化学等许多领域中被提出，并有着丰富的研究，包括分数阶Fokker-Planck方程，分数阶Navier-Stokes方程，分数阶 Landau-lifshitz方程等。

本文将先介绍分数阶Laplace方程的产生背景以及发展，论述其定义与性质以及相关概念的证明。其次，研究分数阶Laplace方程的求解，并对解做出解释。然后参阅Caffarelli和Silvestre的文献对分数阶Laplace方程给出另一种解释。65252

毕业论文关键词  Laplace方程  偏微分方程  分数阶Laplace方程   

毕业设计说明书（论文）外文摘要

Title    An explanation of the fractional Laplace equation                     

Abstract

The history of the development of mathematics tells us that, mathematical analysis of 300 years is the primary branch of mathematics, mathematical analysis and differential equations is the heart, especially the research on Laplace equation has special significance. For half of a century, the theory of partial differential equations has obtained a significant development, the advent of computers provides the research on the Laplace equation with new clues and new methods. Laplace equation is not simply a field of mathematical concepts，in recent years, fractional partial differential equations have been proposed in fluid mechanics, mechanics of materials, biology, finance, chemistry and many other fields, and has a lot of researches, including fractional Fokker-Planck equation, fractional Navier-Stokes equations, fractional Landau-lifshitz equation. 

the background and development of fractional Laplace equation are introduced in this article, discusses the definition and properties and the related proof. Secondly, do research on fractional Laplace equations, and explain solutions. Then see Caffarelli and Silvestre’s article on fractional Laplace equation and give another explanation.

Keywords  the Laplace equation   Partial Differential Equations  Fractional Laplace equation

目   次

1  绪论 5

1.1 偏微分方程的产生背景以及发展 5

1.2 本文的主要内容和结构安排 9

1.3  一个分数阶Laplace算子的引入 9

2  偏微分方程(PDEs)的性质 11

2.1  n+1+a维的调和方程 11

2.2  初始条件下的基本解 12

2.3  共轭方程 12

2.4  泊松公式 13

3  和分数阶Laplace算子的联系 14