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# 数学代写|数值分析代写Numerical analysis代考|MATH2200 Boundary-Value Problems

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## 数学代写|数值分析代写Numerical analysis代考|Boundary-Value Problems

More general than initial-value problems are boundary-value problems. In these one seeks a solution $y(x)$ of a system of $n$ ordinary differential equations,
(7.3.0.1a) $y^{\prime}=f(x, y), \quad y=\left[\begin{array}{c}y_1 \ \vdots \ y_n\end{array}\right], \quad f(x, y)=\left[\begin{array}{c}f_1\left(x, y_1, \ldots, y_n\right) \ \vdots \ f_n\left(x, y_1, \ldots, y_n\right)\end{array}\right]$, satisfying a boundary condition of the form
$$A y(a)+B y(b)=c .$$
Here, $a \neq b$ are given numbers, $A, B$ square matrices of order $n$, and $c$ a vector in $\mathbb{R}^n$. In practice, the boundary conditions are usually separated:

$\left(7.3 .0 .1 \mathrm{~b}^{\prime}\right) \quad A_1 y(a)=c_1, \quad B_2 y(b)=c_2$,
i.e., in (7.3.0.1b) the rows of the matrix $[A, B, c]$ can be permuted such that for the rearranged matrix $[\bar{A}, \bar{B}, \bar{c}]$,
$$[\bar{A}, \bar{B}, \bar{c}]=\left[\begin{array}{c|c|c} A_1 & 0 & c_1 \ \hline 0 & B_2 & c_2 \end{array}\right] .$$
The boundary conditions (7.3.0.1b) are linear (more precisely, affine) in $y(a), y(b)$

Occasionally, in practice, one encounters also nonlinear boundary conditions of the type
$\left(7.3 .0 .1 \mathrm{~b}^{\prime \prime}\right) \quad r(y(a), y(b))=0$,
which are formed by means of a vector $r$ of $n$ functions $r_i, i=1, \ldots, n$, of $2 n$ variables:
$$r(u, v) \equiv\left[\begin{array}{c} r_1\left(u_1, \ldots, u_n, v_1, \ldots, v_n\right) \ \vdots \ r_n\left(u_1, \ldots, u_n, v_1, \ldots, v_n\right) \end{array}\right]$$

## 数学代写|数值分析代写Numerical analysis代考|The Simple Shooting Method

We want to explain the simple shooting method first by means of an example. Suppose we are given the boundary-value problem
$$\begin{gathered} w^{\prime \prime}=f\left(x, w, w^{\prime}\right), \ w(a)=\alpha, \quad w(b)=\beta, \end{gathered}$$

with separated boundary conditions. The initial-value problem
(7.3.1.2) $\quad w^{\prime \prime}=f\left(x, w, w^{\prime}\right), \quad w(a)=\alpha, \quad w^{\prime}(a)=s$
in general has a uniquely determined solution $w(x) \equiv w(x ; s)$ which of course depends on the choice of the initial value $s$ for $w^{\prime}(a)$. To solve the boundary-value problem (7.3.1.1), we must determine $s=: \bar{s}$ so as to satisfy the second boundary condition, $w(b)=w(b ; \bar{s})=\beta$. In other words: one has to find a zero $\bar{s}$ of the function $F(s): \equiv w(b ; s)-\beta$. For every argument $s$ the function $F(s)$ can be computed. For this, one has to determine (e.g., with the methods of Section 7.2) the value $w(b)=w(b ; s)$ of the solution $w(x ; s)$ of the initial-value problem $(7.3 .1 .2)$ at the point $x=b$. The computation of $F(s)$ thus amounts to the solution of an initial-value problem.

## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|BOUNDARY-VALUE PROBLEMS

7.3.0.1a $y^{\prime}=f(x, y), \quad y=\left[y_1 \vdots y_n\right], \quad f(x, y)=\left[f_1\left(x, y_1, \ldots, y_n\right) \vdots f_n\left(x, y_1, \ldots, y_n\right)\right]$, 满足形式的边界条件
$$A y(a)+B y(b)=c$$

$\left(7.3 .0 .1 \mathrm{~b}^{\prime}\right) \quad A_1 y(a)=c_1, \quad B_2 y(b)=c_2 ，$

$\left(7.3 .0 .1 \mathrm{~b}^{\prime \prime}\right) \quad r(y(a), y(b))=0$ ，

$$r(u, v) \equiv\left[r_1\left(u_1, \ldots, u_n, v_1, \ldots, v_n\right) \vdots r_n\left(u_1, \ldots, u_n, v_1, \ldots, v_n\right)\right]$$

## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|THE SIMPLE SHOOTING METHOD

$$w^{\prime \prime}=f\left(x, w, w^{\prime}\right), w(a)=\alpha, \quad w(b)=\beta,$$

7.3.1.2 $w^{\prime \prime}=f\left(x, w, w^{\prime}\right), \quad w(a)=\alpha, \quad w^{\prime}(a)=s$

e.g., withthemethodsofSection $7.2$ 价值 $w(b)=w(b ; s)$ 的解决方案 $w(x ; s)$ 初值问题 $(7.3 .1 .2)$ 在这一点上 $x=b$. 的计算 $F(s)$ 因此相当于解决了一个初值问题。

## 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。