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# 数学代写|偏微分方程代考PARTIAL DIFFERENTIAL EQUATIONS代写|M-541z

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Multiplicity

Once we have asked the question of whether a solution to a given problem exists, it is natural to consider the question of how many solutions there are.
Uniqueness for initial-value problems in ODEs
The prototype for uniqueness results is for initial-value problems in ODEs.
Theorem 1.4 (ODE uniqueness). Let the function $\mathbf{F}$ satisfy the hypotheses of Theorem 1.1. Then the initial-value problem (1.2), (1.3) has at most one solution.
A proof of this based on Gronwall’s inequality is given below.
It should be noted that although this result covers a very wide range of initial-value problems, there are some standard, simple examples for which uniqueness fails. For instance, the problem
\begin{aligned} \frac{d y}{d t} & =y^{1 / 3}, \ y(0) & =0 \end{aligned}
has an entire family of solutions parameterized by $\gamma \in[0,1]$ :
$$y_\gamma(t):=\left{\begin{array}{cc} 0, & 0 \leq t \leq \gamma \ {\left[\frac{2}{3}(t-\gamma)\right]^{3 / 2},} & \gamma<t \leq 1 . \end{array}\right.$$
Nonuniqueness for linear and nonlinear boundary-value problems
While uniqueness is often a desirable property for a solution of a problem (often for physical reasons), there are situations in which multiple solutions are desirable. A common mathematical problem involving multiple solutions is an eigenvalue problem. The reader should, of course, be familiar with the various existence and multiplicity results from finite-dimensional linear algebra, but let us consider a few problems from ordinary differential equations. We consider the following second-order ODE depending on the parameter $\lambda$ :
$$u^{\prime \prime}+\lambda u=0$$
Of course, if we imposed two initial conditions (at one point in space) Theorem 1.4 would imply that we would have a unique solution. (To apply the theorem directly we need to convert the problem from a second-order equation to a first-order system.) However, if we impose the two-point boundary conditions
\begin{aligned} u(0) & =0, \ u^{\prime}(1) & =0, \end{aligned}
the uniqueness theorem does not apply. Instead we get the following result.

## 数学代写|偏微分方程代考Partial Differential Equations代写|Elementary Solutions of the Diffusion Equation

The term stability is one that has a variety of different meanings within mathematics. One often says that a problem is stable if it is “continuous with respect to the data”; i.e., a problem is stable if when we change the problem “slightly,” the solution changes only slightly. We make this precise below in the context of initial-value problems for ODEs. Another notion of stability is that of “asymptotic stability.” Here we say a problem is stable if all of its solutions get close to some “nice” solution as time goes to infinity. We make this notion precise with a result on linear systems of ODEs with constant coefficients.
Stability with respect to initial conditions
In this section we assume that $\mathbf{F}$ satisfies the hypotheses of Theorem 1.1, and we define $\hat{\mathbf{y}}\left(t, t_0, \mathbf{y}_0\right)$ to be the unique solution of (1.2), (1.3). We then have the following standard result.
Theorem 1.6 (Continuity with respect to initial conditions). The function $\hat{\mathbf{y}}$ is well defined on an open set
$$U \subset \mathbb{R} \times D .$$
Furthermore, at every $\left(t, t_0, \mathbf{y}_0\right) \in U$ the function
$$\left(t_0, \mathbf{y}_0\right) \mapsto \hat{\mathbf{y}}\left(t, t_0, \mathbf{y}_0\right)$$
is continuous; i.e., for any $\epsilon>0$ there exists $\delta$ (depending on $\left(t, t_0, \mathbf{y}_0\right)$ and $\epsilon)$ such that if
$$\left|\left(t_0, \mathbf{y}_0\right)-\left(\tilde{t}_0, \tilde{\mathbf{y}}_0\right)\right|<\delta,$$
then $\hat{\mathbf{y}}\left(t, \tilde{t}_0, \tilde{\mathbf{y}}_0\right)$ is well defined and
$$\left|\hat{\mathbf{y}}\left(t, t_0, \mathbf{y}_0\right)-\hat{\mathbf{y}}\left(t, \tilde{t}_0, \tilde{\mathbf{y}}_0\right)\right|<\epsilon .$$
Thus, we see that small changes in the initial conditions result in small changes in the solutions of the initial-value problem.

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Multiplicity

ode中初值问题的唯一性

\begin{aligned} \frac{d y}{d t} & =y^{1 / 3}, \ y(0) & =0 \end{aligned}

$$y_\gamma(t):=\left{\begin{array}{cc} 0, & 0 \leq t \leq \gamma \ {\left[\frac{2}{3}(t-\gamma)\right]^{3 / 2},} & \gamma<t \leq 1 . \end{array}\right.$$

$$u^{\prime \prime}+\lambda u=0$$

\begin{aligned} u(0) & =0, \ u^{\prime}(1) & =0, \end{aligned}

## 数学代写|偏微分方程代考Partial Differential Equations代写|Elementary Solutions of the Diffusion Equation

$$U \subset \mathbb{R} \times D .$$

$$\left(t_0, \mathbf{y}_0\right) \mapsto \hat{\mathbf{y}}\left(t, t_0, \mathbf{y}_0\right)$$

$$\left|\left(t_0, \mathbf{y}_0\right)-\left(\tilde{t}_0, \tilde{\mathbf{y}}_0\right)\right|<\delta,$$

$$\left|\hat{\mathbf{y}}\left(t, t_0, \mathbf{y}_0\right)-\hat{\mathbf{y}}\left(t, \tilde{t}_0, \tilde{\mathbf{y}}_0\right)\right|<\epsilon .$$

## 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。