### 数学代写|偏微分方程代写partial difference equations代考|MATH4310

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|偏微分方程代写partial difference equations代考|The Laplace transform

We have seen that in order to apply the method of Fourier transforms to solve partial differential equations it is necessary to assume suitable decay of the solutions, that is, asymptotic boundary conditions. If, however, other boundary conditions are given, then we need a different integral transform. For the sake of completeness we will give a brief introduction to the Laplace transform here, without however going into much detail. This transform is also highly significant in many parts of the theory of partial differential equations; a systematic treatment can be found in [6], for example. We denote by
$$L_{1, \text { loc }}\left(\mathbb{R}{+}, \mathbb{C}\right):=\left{f:[0, \infty) \rightarrow \mathbb{C} \text { measurable }: \int_0^c|f(t)| d t<\infty \text { for all } c>0\right}$$ the space of locally integrable functions on $\mathbb{R}{+}:=[0, \infty)$.
Definition 3.61 For a function $f \in L_{1, \text { loc }}\left(\mathbb{R}{+}, \mathbb{C}\right)$ we set $$\mathcal{L} f(s)=F(s):=\int_0^{\infty} f(t) e^{-s t} d t=\lim {c \rightarrow \infty} \int_0^c f(t) e^{-s t} d t, s \in \mathbb{C},$$
if the indefinite integral exists, and call the resulting function the Laplace transform of $f . \Delta$
Theorem 3.62 (Existence of the Laplace transform) Let $f \in L_1\left(\mathbb{R}{+}, \mathbb{C}\right)$ be exponentially bounded, that is, we have the bound $|f(t)| \leq M e^{\gamma t}, t \geq 0$, for some constants $M \geq 0$ and $\gamma \in \mathbb{R}$. Then $\mathcal{L} f(s)$ exists for all $s \in \mathbb{C}$ for which $\operatorname{Re} s>\gamma$. Proof See Exercise 3.11. We call the number $\gamma$ in Theorem $3.62$ an exponential bound for the function $f$. Remark $3.63$ The pair $f(t), F(s)=\mathcal{L} f(s)$ is sometimes known as a Laplace correspondence, especially in the engineering literature, written $f(t) \leadsto F(s)$. $\Delta$ For functions $f, g \in L{1, \text { loc }}\left(\mathbb{R}_{+}, C\right)$, we define their convolution via
$$(f * g)(t):=\int_0^t f(t-s) g(s) d s$$

## 数学代写|偏微分方程代写partial difference equations代考|Inner product spaces

Since all Hilbert spaces are inner product spaces, sometimes also called pre-Hilbert spaces, we will naturally start with these. Let $E$ be a vector space over the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$. A mapping
$$(\cdot, \cdot): E \times E \rightarrow \mathbb{K} \quad f, g \mapsto(f, g)$$
is called an inner product or scalar product if the following conditions are satisfied:
(a) $(f+g, h)=(f, h)+(g, h), \quad f, g, h \in E$;
(b) $(\lambda f, g)=\lambda(f, g), \quad f, g \in E, \lambda \in \mathbb{K}$;
(c) $(f, g)=\overline{(g, f)}, \quad f, g \in E$;
(d) $(f, f)>0 \quad(f \neq 0), \quad f \in E$.
Notice that (c) implies that $(f, f)=\overline{(f, f)} \in \mathbb{R}$, for all $f \in E$. Thus (d) does in fact make sense when $\mathbb{K}=\mathbb{C}$. We call (c) symmetry and (d) positive definiteness. The symmetry property also implies
(a’) $(f, g+h)=(f, g)+(f, h), \quad f, g, h \in E$;
(b’) $(f, \lambda g)=\bar{\lambda}(f, g), \quad f, g \in E$.
Here and in what follows, $\bar{\lambda}$ denotes the complex conjugate of the number $\lambda \in \mathbb{C}$. Inner products are thus linear in the first variable (that is, (a) and (b) hold), while they are antilinear in the second (that is, (a’) and (b’) hold). We shall now consider a few examples.

Example $4.1$ (a) Let $E=\mathbb{R}^d$, then $(x, y):=\sum_{j=1}^d x_j y_j=x^T y$ defines the natural inner product on $\mathbb{R}^d$.
(b) Let $E=\mathbb{C}^d$, then $(x, y):=\sum_{j=1}^d x_j \overline{y_j}$ is the natural inner product on $\mathbb{C}^d$.
(c) Let $a<b$ and set $C([a, b]):={f:[a, b] \rightarrow \mathbb{K}: f$ continuous $}$ to be the space of continuous functions on $[a, b]$. Then
$$(f, g):=\int_a^b f(t) \overline{g(t)} d t$$
defines an inner product on $C([a, b])$. Observe that $C([a, b])$ is infinite dimensional, while $\mathbb{R}^d$ and $\mathbb{C}^d$ are finite dimensional. $\quad \Delta$
We call a vector space $E$ equipped with an inner product, or more precisely the pair $(E,(\cdot, \cdot)$ ), an inner product space (or sometimes pre-Hilbert space). We now wish to establish a number of geometric properties of inner products.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|The Laplace transform

L_{1, Itext ${$ loc $}} \backslash l e f t(\backslash m a t h b b{R}{+}, \backslash m a t h b b{C} \backslash$ right): $=\backslash$ eft ${f:[0$, \infty) \rightarrow $\backslash m a t h b b{C} \backslash$ Itext ${$ 可测量 $}$ :

$$\mathcal{L} f(s)=F(s):=\int_0^{\infty} f(t) e^{-s t} d t=\lim c \rightarrow \infty \int_0^c f(t) e^{-s t} d t, s \in \mathbb{C},$$

$\operatorname{Re} s>\gamma$. 证明见练习 3.11。我们拨打号码 $\gamma$ 在定理中 $3.62$ 函数的指数界限 $f$. 评论 $3.63$ 这对 $f(t), F(s)=\mathcal{L} f(s)$ 有时被称为拉普拉斯对应，特别是在工程文献中，写成 $f(t) \rightsquigarrow F(s)$. $\Delta$ 对于函数 $f, g \in L 1, \operatorname{loc}\left(\mathbb{R}_{+}, C\right)$ ，我们通过定义它们的卷积
$$(f * g)(t):=\int_0^t f(t-s) g(s) d s$$

## 数学代写|偏微分方程代写partial difference equations代考|Inner product spaces

$$(\cdot, \cdot): E \times E \rightarrow \mathbb{K} \quad f, g \mapsto(f, g)$$

(a) $(f+g, h)=(f, h)+(g, h), \quad f, g, h \in E$ ；
(乙) $(\lambda f, g)=\lambda(f, g), \quad f, g \in E, \lambda \in \mathbb{K}$;
(C) $(f, g)=\overline{(g, f)}, \quad f, g \in E$;
(四) $(f, f)>0 \quad(f \neq 0), \quad f \in E$.

(a’) $(f, g+h)=(f, g)+(f, h), \quad f, g, h \in E$;
(b’) $(f, \lambda g)=\bar{\lambda}(f, g), \quad f, g \in E$.

(b) 让 $E=\mathbb{C}^d$ ， 然后 $(x, y):=\sum_{j=1}^d x_j \overline{y_j}$ 是上的自然内积 $\mathbb{C}^d$.
(c) 让 $a<b$ 并设置 $C([a, b]):=f:[a, b] \rightarrow \mathbb{K}: f$ Scontinuous $\$$是连续函数的空间 [a, b]. 然后$$ (f, g):=\int_a^b f(t) \overline{g(t)} d t$$定义一个内积$C([a, b])$. 观察那个$C([a, b])$是无限维的，而$\mathbb{R}^d$和$\mathbb{C}^d$是有限维的。$\Delta$我们称向量空间$E$配备了一个内积，或者更准确地说是一对$(E,(\cdot, \cdot))\$ ，一个内积空间（有时是前莃尔伯 特空间）。我们现在希望建立内积的一些几何特性。

## 有限元方法代写

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## MATLAB代写

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