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# 数学代写|偏微分方程代考PARTIAL DIFFERENTIAL EQUATIONS代写|Math112A

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Basic Mathematical Questions

Questions of existence occur naturally throughout mathematics. The question of whether a solution exists should pop into a mathematician’s head any time he or she writes an equation down. Appropriately, the problem of existence of solutions of partial differential equations occupies a large portion of this text. In this section we consider precursors of the PDE theorems to come.
Initial-value problems in ODEs
The prototype existence result in differential equations is for initial-value problems in ODEs.

Theorem 1.1 (ODE existence, Picard-Lindelöf). Let $D \subseteq \mathbb{R} \times \mathbb{R}^n$ be an open set, and let $\mathbf{F}: D \rightarrow \mathbb{R}^n$ be continuous in its first variable and uniformly Lipschitz in its second; i.e., for $(t, \mathbf{y}) \in D, \mathbf{F}(t, \mathbf{y})$ is continuous as a function of $t$, and there exists a constant $\gamma$ such that for any $\left(t, \mathbf{y}_1\right)$ and $\left(t, \mathbf{y}_2\right)$ in $D$ we have
$$\left|\mathbf{F}\left(t, \mathbf{y}_1\right)-\mathbf{F}\left(t, \mathbf{y}_2\right)\right| \leq \gamma\left|\mathbf{y}_1-\mathbf{y}_2\right|$$
Then, for any $\left(t_0, \mathbf{y}_0\right) \in D$, there exists an interval $I:=\left(t^{-}, t^{+}\right)$containing $t_0$, and at least one solution $\mathbf{y} \in C^1(I)$ of the initial-value problem
$$\begin{gathered} \frac{d \mathbf{y}}{d t}(t)=\mathbf{F}(t, \mathbf{y}(t)), \ \mathbf{y}\left(t_0\right)=\mathbf{y}_0 . \end{gathered}$$

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

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Basic Mathematical Questions

ode中的初值问题

$$\left|\mathbf{F}\left(t, \mathbf{y}_1\right)-\mathbf{F}\left(t, \mathbf{y}_2\right)\right| \leq \gamma\left|\mathbf{y}_1-\mathbf{y}_2\right|$$

$$\begin{gathered} \frac{d \mathbf{y}}{d t}(t)=\mathbf{F}(t, \mathbf{y}(t)), \ \mathbf{y}\left(t_0\right)=\mathbf{y}_0 . \end{gathered}$$

## 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}

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

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

## Matlab代写

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