# 数学代写|偏微分方程代写partial difference equations代考|Math462

## 数学代写|偏微分方程代写partial difference equations代考|Existence and Uniqueness Theorem

Let $U \subseteq \mathbb{R}^n$ be an open connected set, $I \subseteq \mathbb{R}$ be an interval, and $f: I \times U \rightarrow \mathbb{R}^n$ be at least continuous. We may write $\boldsymbol{f}=\left(f_1, \ldots, f_n\right)$. Consider an initial value problem for a first order system of differential equations given by
$$\frac{\mathrm{d} x}{\mathrm{~d} t}=\boldsymbol{f}(t ; \boldsymbol{x}(t) ; \boldsymbol{b}), \quad \text { with } x\left(t_0\right)=\boldsymbol{a}, \quad \text { for }\left(t_0, x_0\right) \in I \times U,$$
where $\boldsymbol{a}=\left(a_1, \ldots, a_n\right) \in \mathbb{R}^n$ is the vector of initial values assigned at $t=t_0 \in I$. As in the case when $n=1$, in geometrical terms, the graph $\Gamma_x$ of the function $x=\boldsymbol{x}(t)$ is a curve in $\mathbb{R}^{n+1}$, and $f$ defines a direction field in the domain $I \times U$ such that, if $\boldsymbol{c}=\boldsymbol{f}(s, \boldsymbol{y})$, then the vector $(1, \boldsymbol{c}) \in \mathbb{R}^{n+1}$ or, equivalently, the line
$$\boldsymbol{x}=\boldsymbol{y}+(t-s) \boldsymbol{c}$$
gives a direction at the point $(s, y) \in I \times U$. The graphs of solutions of the system $\boldsymbol{x}^{\prime}(t)=\boldsymbol{f}(t, \boldsymbol{x})$ fits on the direction field. The existence and uniqueness theorem for the initial value problem (3.5.6), as given below, is very useful to solve many practical problems concerning different types of dynamical systems.

Theorem $3.18$ (Existence Theorem) Let $U \subset \mathbb{R}^n$ and $V \subset \mathbb{R}^k$ be open sets, $c>$ 0 , and $f_i \in C^1[(-c, c) \times V \times U]$, for $i=1, \ldots, n$. Consider a first order system as in (3.5.4), with time-dependent parameters $\boldsymbol{b}=\left(b_1, \ldots, b_k\right) \in V$. For any $\boldsymbol{a}=$ $\left(a_1, \ldots, a_n\right) \in U$, there exists $n$ smooth functions $x_i=x_i(t ; b):(-\delta, \varepsilon) \times V \rightarrow \mathbb{R}$ satisfying the system, and also the initial values given by
$$x_i(0, b)=a_i, \text { for } i=1, \ldots, n .$$
Proof The statement of the theorem is a straightforward generalisation of Theorem 3.1, and so the proof requires minor modifications. The details are left for the reader as an exercise.

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

Notice that, by introducing new dependent variables $x_1$ and $x_2$, the second order Eq. (3.3.26) can be written as a first order system given by

\begin{aligned} & x_1^{\prime}(t)=x_2(t) ; \ & x_2^{\prime}(t)=-\frac{c}{m} x_2(t)-\frac{k}{m} x_1(t)+\frac{a_0}{m} \sin \omega t . \end{aligned}
We can also write (3.5.9) in vector form as
\begin{aligned} & A=\left(\begin{array}{cc} 0 & 1 \ -k / c-c / m \end{array}\right) \text { and } \boldsymbol{b}(t)=\left(\begin{array}{c} 0 \ \left(a_0 / m\right) \sin \omega t \end{array}\right) \text {. } \ & \end{aligned}
For many applications related to dynamical system, it is desirable to have a differential equation model given by a linear system.

Definition 3.27 For a given $n \times n$ matrix function $A: I \rightarrow \mathbb{R}^{n^2}$, with $n^2$ entries $a_{i j} \in C(I)$, and a function $g: I \rightarrow \mathbb{R}^n$, a linear system of order $n$ for a differentiable function $x: I \rightarrow \mathbb{R}^n$ is a first order system of the form
$$\frac{\mathrm{d} x}{\mathrm{~d} t}=A \cdot \boldsymbol{x}+g,$$
where both $\boldsymbol{x}=\left(x_1, \ldots, x_n\right)$ and $g=\left(g_1, \ldots, g_n\right)$ are taken as column vectors. The matrix $A=A(t)$ is called the coefficients matrix of the linear system. We say (3.5.11) is a homogeneous linear system if $g \equiv 0$. Otherwise, it is called a nonhomogeneous linear system.
For example, the initial value problem
$$x^{\prime}(t)=a x(t), \quad \text { with } x(0)=x_0 .$$
defines a dynamical system given by the smooth function $\varphi_t\left(x_0\right)=x_0 \mathrm{e}^{a t}$. In general, by taking a square matrix $A$ of order $n$ in place of the scalar $a$, and a vector function $\boldsymbol{x}=\boldsymbol{x}(t): \mathbb{R} \rightarrow \mathbb{R}^n$ for $x=x(t)$, we obtain a dynamical system given by an initial value problems for a first order linear system of the form (3.5.11) (with $g \equiv 0$ ). Clearly, by using
$$\boldsymbol{f}(t, \boldsymbol{x})=A \cdot \boldsymbol{x}+\boldsymbol{g}, \text { for } t \in I,$$
every first order linear system is a system of the form (3.5.4).

# 偏微分方程代考

## 数学代写|偏微分方程代写partial difference equations代考|Existence and Uniqueness Theorem

$$\frac{\mathrm{d} x}{\mathrm{~d} t}=\boldsymbol{f}(t ; \boldsymbol{x}(t) ; \boldsymbol{b}), \quad \text { with } x\left(t_0\right)=\boldsymbol{a}, \quad \text { for }\left(t_0, x_0\right) \in I \times U,$$

$$\boldsymbol{x}=\boldsymbol{y}+(t-s) \boldsymbol{c}$$

$$x_i(0, b)=a_i, \text { for } i=1, \ldots, n .$$

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

$$x_1^{\prime}(t)=x_2(t) ; \quad x_2^{\prime}(t)=-\frac{c}{m} x_2(t)-\frac{k}{m} x_1(t)+\frac{a_0}{m} \sin \omega t .$$

$$\frac{\mathrm{d} x}{\mathrm{~d} t}=A \cdot \boldsymbol{x}+g,$$

$$x^{\prime}(t)=a x(t), \quad \text { with } x(0)=x_0 .$$

$$\boldsymbol{f}(t, \boldsymbol{x})=A \cdot \boldsymbol{x}+\boldsymbol{g}, \text { for } t \in I,$$

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