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# 数学代写|Matlab代考|ME1311 INITIAL CONDITIONS: CAUCHY PROBLEM

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Matlab 是由数学家和计算机程序员Cleve Moler发明的。MATLAB的想法是基于他1960年代的博士论文。Moler成为新墨西哥大学的一名数学教授，并开始为他的学生开发MATLAB作为一种爱好。他在1967年与他曾经的论文导师George Forsythe开发了MATLAB的最初线性代数编程。随后在1971年开发了线性方程的Fortran 代码。

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## 数学代写|Matlab代考|INITIAL CONDITIONS: CAUCHY PROBLEM

Any mathematical model of a physical process must include not only the governing differential equation but also any conditions that are imposed on the solution. For example, in time-dependent problems the solution must conform to the initial condition of the modeled process. Finding those solutions that satisfy the initial conditions (initial data) is called the Cauchy problem.

In the case of partial differential equations with second-order derivatives in time, such as the wave equation, we correctly pose the Cauchy boundary condition if we specify the value of the solution $u\left(x, t_{0}\right)=f(t)$ and its time derivative $u_{t}\left(x, t_{0}\right)=g(t)$ at some initial time $t_{0}$, usually taken to be $t_{0}=0$. The functions $f(t)$ and $g(t)$ are called the Cauchy data. We require two conditions involving time because the differential equation has two time derivatives.

In addition to the initial conditions, we must specify boundary conditions in the spatial direction. For example, we may require that the end of the string be fixed. In the next chapter, we discuss boundary conditions in greater depth. However, one boundary condition that is uniquely associated with the wave equation on an open domain is the radiation condition. It requires that the waves radiate off to infinity and remain finite as they propagate there.

In summary, Cauchy boundary conditions, along with the appropriate spatial boundary conditions, uniquely determine the solution to the wave equation; any additional information is extraneous. Having developed the differential equation and initial conditions necessary to solve the wave equation, let us now turn to the actual methods used to solve this equation.

## 数学代写|Matlab代考|SEPARATION OF VARIABLES

Separation of variables is the most popular method for solving the wave equation. Despite its current widespread use, its initial application to the vibrating string problem was controversial because of the use of a half-range Fourier sine series to represent the initial conditions. On one side, Daniel Bernoulli claimed (in 1775) that he could represent any general initial condition with this technique. To d’Alembert and Euler, however, the halfrange Fourier sine series, with its period of $2 L$, could not possibly represent any arbitrary function. ${ }^{2}$ However, by 1807 Bernoulli was proven correct by the use of separation of variables in the heat conduction problem and it rapidly grew in acceptance. ${ }^{3}$ In the following examples we show how to apply this method.

Separation of variables consists of four distinct steps that convert a second-order partial differential equation into two ordinary differential equations. First, we assume that the solution equals the product $X(x) T(t)$. Direct substitution into the partial differential equation and boundary conditions yields two ordinary differential equations and the corresponding boundary conditions. Step two involves solving a boundary-value problem. In step three we find the corresponding time dependence. Finally we construct the complete solution as a sum of all product solutions. Upon applying the initial conditions, we have a Fourier half-range expansion and must compute the Fourier coefficients. The substitution of these coefficients into the summation yields the complete solution.

# Matlab代写

## 数学代写|Matlab代考|SEPARATION OF VARIABLES

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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。