积分变换在偏微分方程中的应用

摘要人们在处理与分析工程实际中的一些问题时，常常采取某种方法将问题进行转换，从另一个角度进行处理与分析，这就是所谓的变换。傅里叶变换与拉普拉斯变换是两种常见的积分变换，它们在数学、物理、工程技术等领域中有着广泛的应用。傅里叶变换与拉普拉斯变换都可以用来求解偏微分方程，由于在求解偏微分方程时两者都可以将方程化为某个变量的代数方程，使得问题得以简化，因此用积分变换法求解偏微分方程是一种非常有效的方法。本文从两种积分变换的基本概念性质等入手，对比分析了两种积分变换的区别与联系，并介绍了这两种积分变换在偏微分方程中的应用，如傅里叶变换在热传导方程和波动方程中的应用，拉普拉斯变换在解波动信号问题及非稳态热传导问题中的应用，并以非线性薛定谔方程为例，介绍了傅里叶变换在非线性偏微分方程中的应用。65254

毕业论文关键词积分变换傅里叶变换拉普拉斯变换偏微分方程

毕业设计（论文）外文摘要

Title The application of integral transformations in partial differential equations

Abstract

During the process of analyzing some problems in engineering, one should convert the problems into another ones and then processing and analyzing the new problems ,this process is called the transformation from mathematical point. In mathematics, physics, engineering and other fields, the most widely used transformation are Fourier transform and Laplace transform. Fourier transform and Laplace transform can be used to solve partial differential equations by converting the PDEs into simple algebraic equations, therefore the two kinds of integration transforms do have great significance in solving partial differential equations. In this paper, we start from the two basic concepts of integral transforms, comparing and analyzing the difference and contact between two types of integral transforms. We introduce the application of the two integral transforms in partial differential equations, such as the application of Fourier transform in heat conduction equation and volatility equation and Laplace transform in wave signal problems and non-steady-state heat conduction problems respectively. We also give the application of Fourier transform in a nonlinear partial differential equation named Schrodinger equation.

Keywords Integral Transformation Fourier transform Laplace transform Partial Differential Equations

第1章绪论 1

1.1 研究背景及意义 1

1.2 主要内容 2

第2章傅里叶变换 4

2.1 傅里叶变换的由来 4

2.3 傅里叶变换的定义 5

2.3 傅里叶变换与逆变换的性质 6

2.4 常见傅里叶变换 8

2.5 单位脉冲函数 10

第3章拉普拉斯变换 13

3.1 拉普拉斯变换的历史由来 13

3.2 拉普拉斯变换的定义 13

3.3 拉普拉斯变换与逆变换的性质 14

3.4 常见拉普拉斯变换 15

第4章傅里叶变换在偏微分方程中的应用积分变换在偏微分方程中的应用:http://www.751com.cn/shuxue/lunwen_72800.html