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# 数学代写|偏微分方程代考Partial Differential Equations代写|MTH3023 Laplace Transform

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Laplace Transform

Laplace transform has been introduced in an ODE course, and is used especially to solve ODEs having pulse sources. In this chapter we review the Laplace transform and its properties and show how it is used in analyzing PDEs. It should be noted that most problems that can be analyzed by Laplace transform, can also be analyzed by one of the other techniques in this book.
Definition 22: The Laplace transform of a function $f(t)$, denoted by $\mathcal{L}[f(t)]$ is defined by
$$\mathcal{L}[f]=\int_0^{\infty} f(t) e^{-s t} d t,$$
assuming the integral converges (real part of $s>0$ ).
We will denote the Laplace transform of $f$ by $F(s)$, exactly as with Fourier transform. The Laplace transform of some elementary functions can be obtained by definition. See Table at the end of this Chapter.
The inverse transform is given by
$$f(t)=\mathcal{L}^{-1}[F(s)]=\frac{1}{2 \pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} F(s) e^{s t} d s,$$
where $\gamma$ is chosen so that $f(t) e^{-\gamma t}$ decays sufficiently rapidly as $t \rightarrow \infty$, i.e. we have to compute a line integral in the complex $s$-plane.

From the theory of complex variables, it can be shown that the line integral is to the right of all singularities of $F(s)$. To evaluate the integral we need Cauchy’s theorem from the theory of functions of a complex variable which states that if $f(s)$ is analytic (no singularities) at all points inside and on a closed contour $C$, then the closed line integral is zero,
$$\oint_C f(s) d s=0 .$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Solution of Wave Equation

In this section, we show how to use Laplace transform to solve the one dimensional wave equation in the semi-infinite and finite domain cases.

Consider the vibrations of a semi-infinite string caused by the boundary condition, i.e.
$$\begin{gathered} u_{t t}-c^2 u_{x x}=0, \quad 00, \ u(x, 0)=0, \ u_t(x, 0)=0, \ u(0, t)=f(t) . \end{gathered}$$
A boundary condition at infinity would be
$$\lim {x \rightarrow \infty} u(x, t)=0 .$$ Using Laplace transform for the time variable, we get upon using the zero initial conditions, $$s^2 U(x, s)-c^2 U{x x}=0 .$$
This is an ordinary differential equation in $x$ (assuming $s$ is fixed). Transforming the boundary conditions,
\begin{aligned} &U(0, s)=F(s), \ &\lim _{x \rightarrow \infty} U(x, s)=0 . \end{aligned}

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Laplace Transform

$$\mathcal{L}[f]=\int_0^{\infty} f(t) e^{-s t} d t,$$

$$f(t)=\mathcal{L}^{-1}[F(s)]=\frac{1}{2 \pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} F(s) e^{s t} d s,$$
$$\oint_C f(s) d s=0 .$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Solution of Wave Equation

$$u_{t t}-c^2 u_{x x}=0, \quad 00, u(x, 0)=0, u_t(x, 0)=0, u(0, t)=f(t) .$$

$$\lim x \rightarrow \infty u(x, t)=0 .$$
$$s^2 U(x, s)-c^2 U x x=0 .$$

$$U(0, s)=F(s), \quad \lim _{x \rightarrow \infty} U(x, s)=0$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。