数学代写|傅里叶分析代写Fourier analysis代考|MAT3105

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数学代写|傅里叶分析代写Fourier analysis代考|Fourier’s Solution of Laplace Equation

In 1804 , the French mathematician and egyptologist Jean Baptiste Joseph Fourier (1768-1830) began his studies on the heat propagation in solid bodies. In 1807, he finished a first paper about heat propagation. He discovered the fundamental partial differential equation of heat propagation and developed a new method to solve this equation. The mathematical core of Fourier’s idea was that each periodic function can be well approximated by a linear combination of sine and cosine terms. This theory contradicted the previous views on functions and was met with resistance by some members of the French Academy of Sciences, so that a publication was initially prevented. Later, Fourier presented these results in the famous book “The Analytical Theory of Heat” published firstly 1822 in French, cf. [119]. For an image of Fourier, see Fig. 1.1 (Image source: https://commons.wikimedia.org/wiki/File: Joseph_Fourier.jpg).

In the following, we describe Fourier’s idea by a simple example. We consider the open unit disk $\Omega=\left{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}<1\right}$ with the boundary $\Gamma=$ $\left{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right}$. Let $v(x, y, t)$ denote the temperature at the point $(x, y) \in \Omega$ and the time $t \geq 0$. For physical reasons, the temperature fulfills the heat equation $$\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}=c \frac{\partial v}{\partial t}, \quad(x, y) \in \Omega, t>0$$
with some constant $c>0$. At steady state, the temperature is independent of the time such that $v(x, y, t)=v(x, y)$ satisfies the Laplace equation
$$\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}=0, \quad(x, y) \in \Omega .$$
What is the temperature $v(x, y)$ at any point $(x, y) \in \Omega$, if the temperature at each point of the boundary $\Gamma$ is known?
Using polar coordinates
$$x=r \cos \varphi, \quad y=r \sin \varphi, \quad 0<r<1,0 \leq \varphi<2 \pi,$$

数学代写|傅里叶分析代写Fourier analysis代考|Fourier Coefficients and Fourier Series

A complex-valued function $f: \mathbb{R} \rightarrow \mathbb{C}$ is $2 \pi$-periodic or periodic with period $2 \pi$, if $f(x+2 \pi)=f(x)$ for all $x \in \mathbb{R}$. In the following, we identify any $2 \pi$-periodic function $f: \mathbb{R} \rightarrow \mathbb{C}$ with the corresponding function $f: \mathbb{T} \rightarrow \mathbb{C}$ defined on the torus $\mathbb{T}$ of length $2 \pi$. The torus $\mathbb{T}$ can be considered as quotient space $\mathbb{R} /(2 \pi \mathbb{Z})$ or its representatives, e.g. the interval $[0,2 \pi]$ with identified endpoints 0 and $2 \pi$. For short, one can also geometrically think of the unit circle with circumference $2 \pi$. Typical examples of $2 \pi$-periodic functions are $1, \cos (n \cdot), \sin (n \cdot)$ for each angular frequency $n \in \mathbb{N}$ and the complex exponentials $\mathrm{e}^{\mathrm{i} k \cdot}$ for each $k \in \mathbb{Z}$.

By $C(\mathbb{T})$ we denote the Banach space of all continuous functions $f: \mathbb{T} \rightarrow \mathbb{C}$ with the norm
$$|f|_{C(\mathbb{T})}:=\max {x \in \mathbb{T}}|f(x)|$$ and by $C^{r}(\mathbb{T}), r \in \mathbb{N}$ the Banach space of $r$-times continuously differentiable functions $f: \mathbb{T} \rightarrow \mathbb{C}$ with the norm $$|f|{C^{r}(\mathbb{T})}:=|f|_{C(\mathbb{T})}+\left|f^{(r)}\right|_{C(\mathbb{T})} .$$
Clearly, we have $C^{r}(\mathbb{T}) \subset C^{s}(\mathbb{T})$ for $r>s$.
Let $L_{p}(\mathbb{T}), 1 \leq p \leq \infty$ be the Banach space of measurable functions $f: \mathbb{T} \rightarrow$ $\mathbb{C}$ with finite norm
\begin{aligned} &|f|_{L_{p}(\mathbb{T})}:=\left(\frac{1}{2 \pi} \int_{-\pi}^{\pi}|f(x)|^{p} \mathrm{~d} x\right)^{1 / p}, \quad 1 \leq p<\infty \ &|f|_{L_{\infty}(\mathbb{T})}:=\operatorname{ess} \sup {|f(x)|: x \in \mathbb{T}} \end{aligned}
where we identify almost equal functions. If a $2 \pi$-periodic function $f$ is integrable on $[-\pi, \pi]$, then we have
$$\int_{-\pi}^{\pi} f(x) \mathrm{d} x=\int_{-\pi+a}^{\pi+a} f(x) \mathrm{d} x$$ for all $a \in \mathbb{R}$ so that we can integrate over any interval of length $2 \pi$.

数学代写|傅里叶分析代写Fourier analysis代考|Fourier’s Solution of Laplace Equation

1804 年，法国数学家和埃及学家让·巴蒂斯特.约瑟夫·傅里叶 (Jean Baptiste Joseph Fourier， 1768-1830) 开始研 究固体中的热传播。1807 年，他完成了第一篇关于热传播的论文。他发现了热传播的基本偏微分方程，并开发了 一种求解该方程的新方法。傅里叶思想的数学核心是每个周期函数都可以很好地近似为正弦和余弦项的线性组 合。这一理论与之前关于函数的观点相矛盾，并遭到法国科学院一些成员的抵制，因此最初阻止发表。后来，傅 立叶在 1822 年首次以法语出版的著名著作《热的分析理论》中介绍了这些结果，参见。[119]。傅里叶图像见图 $1.1$ (图片来源:

IOmega=\left } { ( x , y ) \backslash \text { in } \backslash \text { mathbb } { R } \wedge { 2 } : x ^ { \wedge } { 2 } + y ^ { \wedge } { 2 } < 1 \backslash r _ { i } \text { ght } } \text { 与边界 } \Gamma = 理原因，温度满足热方程 $$\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}=c \frac{\partial v}{\partial t}, \quad(x, y) \in \Omega, t>0$$

$$\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}=0, \quad(x, y) \in \Omega$$

$$x=r \cos \varphi, \quad y=r \sin \varphi, \quad 0<r<1,0 \leq \varphi<2 \pi$$

数学代写|傅里叶分析代写Fourier analysis代考|Fourier Coefficients and Fourier Series

$$|f|{C(\mathrm{~T})}:=\max x \in \mathbb{T}|f(x)|$$ 并通过 $C^{r}(\mathbb{T}), r \in \mathbb{N}$ 拿赫空间 $r$ – 次连续可微函数 $f: \mathbb{T} \rightarrow \mathbb{C}$ 与规范 $$|f| C^{r}(\mathbb{T}):=|f|{C(\mathbb{T})}+\left|f^{(r)}\right|{C(\mathbb{T})} .$$ 显然，我们有 $C^{r}(\mathbb{T}) \subset C^{s}(\mathbb{T})$ 为了 $r>s$. 让 $L{p}(\mathbb{T}), 1 \leq p \leq \infty$ 是可测函数的巴拿赫空间 $f: \mathbb{T} \rightarrow \mathbb{C}$ 具有有限范数
$$|f|{L{p}(\mathbb{T})}:=\left(\frac{1}{2 \pi} \int_{-\pi}^{\pi}|f(x)|^{p} \mathrm{~d} x\right)^{1 / p}, \quad 1 \leq p<\infty \quad|f|{L{\infty}(\mathbb{T})}:=\operatorname{ess} \sup |f(x)|: x \in \mathbb{T}$$

$$\int_{-\pi}^{\pi} f(x) \mathrm{d} x=\int_{-\pi+a}^{\pi+a} f(x) \mathrm{d} x$$

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