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# 数学代写|应用数学代考APPLIED MATHEMATICS代写|MATH311 Natural Boundary Condition

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## 数学代写|应用数学代考APPLIED MATHEMATICS代写|Natural Boundary Condition

Finally, let us address natural boundary conditions instead of fixed boundary conditions, which we hav considered so far. These can be motivated, for instance, by the following problem.
Example 7.5.3: Motivation for natural boundary conditions
A river with parallel straight banks $b$ units apart has a stream velocity given by
$$\boldsymbol{v}(x, y)=\left(\begin{array}{c} 0 \ v(x) \end{array}\right) .$$
Assuming that one of the banks is the $y$-axis and that the point $(0,0)$ is the point of departure, what route should a boat take to reach the opposite bank in the shortest possible time? Assume that the speed of the boat in still water is $c \in \mathbb{R}^{+}$with $c>v(x)$ for all $x$. This problem differs from those in earlier sections in that right-hand endpoint, the point of arrival on the line $x=b$, is not specified. Instead, it must be determined as part of the solution. It can be shown that the time required for the boat to cross the river along a given path $y=y(x)$ is
$$F(y)=\int_a^b \frac{\sqrt{c^2\left[1+y^{\prime}(x)^2\right]-v(x)^2}-v(x) y^{\prime}(x)}{c^2-v(x)^2} \mathrm{~d} x .$$
Thus, the variational problem is to minimize $F$ subject to the conditions
$$y(0)=0, y(b) \text { free. }$$

## 数学代写|应用数学代考APPLIED MATHEMATICS代写|Best Approximations in Inner Product Spaces

The potentially excellent approximation properties of the Fourier series might be best understood by noting that the (truncated) Fourier series can be characterized as the best approximation in certain inner product spaces. Getting to the heart of the underlying theory – presented in this and the next section – will also allow us to formulate generalized Fourier series, therefore extending the concept of classical Fourier series to a much broader class of function spaces.

Definition 8.1.1: Best approximations
Let $(X,|\cdot|)$ be a normed linear space, $f \in X$, and $V \subset X$ be a linear subspace. An element $v^* \in V$ is called best approximation of $f$ from $V$ with respect to $|\cdot|$ if
$$\left|f-v^\right| \leq|f-v|$$ holds for all $v \in V$. This means that there is no element $v \in V$ which is closer to $f$ that $v^$.
Example 8.1.2: $\left(\mathbb{R}^2,|\cdot|_{\infty}\right)$
Given is the linear space $\left(\mathbb{R}^2,|\cdot|_{\infty}\right)$ with $\left|(x, y)^{\top}\right|_{\infty}=\max {|x|,|y|}$, the element $f=(0,1)^{\top}$, and the linear subspace $V=\mathbb{R} \times{0}$ which corresponds to the $x$-axis of the $x y$-plane. Then, every element
$$v_r^=(r, 0)^{\boldsymbol{\top}} \in V, \quad r \in[-1,1],$$ is a best approximation of $f=(0,1)^{\top}$ from $V$ with respect to $|\cdot|_{\infty}$. This can be noted by observing that $$\left|f-v_r^\right|_{\infty}=\left|(-r, 1)^{\top}\right|_{\infty}=\max {|r|, 1}=1$$
for $r \in[-1,1]$ and
$$|f-v|_{\infty}>1$$
for all other elements from $V$.

# 应用数学代考

## 数学代写|应用数学代考APPLIED MATHEMATICS代写|Natural Boundary Condition

$$\boldsymbol{v}(x, y)=(0 v(x))$$

$$F(y)=\int_a^b \frac{\sqrt{c^2\left[1+y^{\prime}(x)^2\right]-v(x)^2}-v(x) y^{\prime}(x)}{c^2-v(x)^2} \mathrm{~d} x .$$

$$y(0)=0, y(b) \text { free. }$$

## 数学代写|应用数学代考APPLIED MATHEMATICS代写|Best Approximations in Inner Product Spaces

$$|f-v|_{\infty}>1$$

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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。