## 数学代写|常微分方程代写ordinary differential equation代考|MTH225

2023年2月2日

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## 数学代写|常微分方程代写ordinary differential equation代考|On the Existence and Uniqueness of Solutions

Unfortunately, not all problems are solvable, and those that are solvable sometimes have several solutions. This is true in mathematics just as it is true in real life.

Before attempting to solve a problem involving some given differential equation and auxiliary condition (such as an initial value), it would certainly be nice to know that the given differential equation actually has a solution satisfying the given auxiliary condition. This would be especially true if the given differential equation looks difficult and we expect that considerable effort will be required in solving it (effort which would be wasted if that solution did not exist). And even if we can find a solution, we normally would like some assurance that it is the only solution.

The following theorem is the standard theorem quoted in most elementary differential equation texts addressing these issues for fairly general first-order initial-value problems.
Theorem $3.1$ (on existence and uniqueness)
Consider a first-order initial-value problem
$$\frac{d y}{d x}=F(x, y) \quad \text { with } \quad y\left(x_0\right)=y_0$$
in which both $F$ and ${ }^{\partial F} / \partial y$ are continuous functions on some open region of the $X Y$-plane containing the point $\left(x_0, y_0\right) .^2$ The initial-value problem then has exactly one solution over some open interval $(\alpha, \beta)$ containing $x_0$. Moreover, this solution and its derivative are continuous over that interval.

This theorem assures us that, if we can write a first-order differential equation in the derivative formula form,
$$\frac{d y}{d x}=F(x, y),$$
and that $F(x, y)$ is a ‘reasonably well-behaved’ formula on some region of interest, then our differential equation has solutions – with luck and skill, we will be able to find them. Moreover, if we can find a solution to this equation that also satisfies some initial value $y\left(x_0\right)=y_0$ corresponding to a point at which $F$ is ‘reasonably well-behaved’, then that solution is unique (i.e., it is the only solution) – there is no need to worry about alternative solutions – at least over some interval $(\alpha, \beta)$. Just what that interval $(\alpha, \beta)$ is, however, is not explicitly described in this theorem. It turns out to depend in subtle ways on just how well behaved $F(x, y)$ is. More will be said about this in a few paragraphs.

## 数学代写|常微分方程代写ordinary differential equation代考|Confirming the Existence of Solutions

So let us consider the first-order initial-value problem
$$\frac{d y}{d x}=F(x, y) \quad \text { with } \quad y\left(x_0\right)=y_0 \quad,$$
assuming that both $F$ and ${ }^{\partial F} / \partial y$ are continuous on some open region in the $X Y$-plane containing the point $\left(x_0, y_0\right)$. Our goal is to verify that a solution $y$ exists over some interval. (This is the existence claim of Theorem 3.1. The uniqueness claim of that theorem will be left as an exercise using material developed in the next section – see Exercise $3.2$ on page 56.)
The gist of our proof consists of three steps:

1. Observe that the initial-value problem is equivalent to a corresponding integral equation.
2. Derive a sequence of functions $-\psi_0, \psi_1, \psi_2, \psi_3, \ldots-$ using a formula inspired by that integral equation.
3. Show that this sequence of functions converges on some interval to a solution $y$ of the original initial-value problem.

The “hard” part of the proof is in the details of the last step. We can skip over these details initially, returning to them in the next section.

1. The $\psi_k$ ‘s end up being approximations to the solution $y$, and, in theory at least, the method we are about to describe can be used to find approximate solutions to an initial-value problem. Other methods, however, are often more practical.
2. This method was developed by the French mathematician Emile Picard and is often referred to as the (Picard’s) method of successive approximations or as Picard’s iterative method (because of the way the $\psi_k$ ‘s are generated).
To simplify discussion let us assume $x_0=0$, so that our initial-value problem is
$$\frac{d y}{d x}=F(x, y) \quad \text { with } \quad y(0)=y_0 .$$
There is no loss of generality here. After all, if $x_0 \neq 0$, we can apply the change of variable $s=x-x_0$ and convert our original problem into problem (3.3) (with $x$ replaced by $s$ ).

# 常微分方程代写

## 数学代写|常微分方程代写ordinary differential equation代考|On the Existence and Uniqueness of Solutions

$$\frac{d y}{d x}=F(x, y) \quad \text { with } \quad y\left(x_0\right)=y_0$$

$$\frac{d y}{d x}=F(x, y),$$

## 数学代写|常微分方程代写ordinary differential equation代考|Confirming the Existence of Solutions

$$\frac{d y}{d x}=F(x, y) \quad \text { with } \quad y\left(x_0\right)=y_0 \quad,$$

1. 观察到初始值问题等价于相应的积分方程。
2. 导出函数序列 $-\psi_0, \psi_1, \psi_2, \psi_3, \ldots$-使用受该 积分方程启发的公式。
3. 证明这个函数序列在某个区间收敛到一个解 $y$ 的原 始初值问题。
证明的”困难”部分在于最后一步的细节。我们最初可以 跳过这些细节，在下一节中返回到它们。
这里应该提出两点意见：1. 这 $\psi_k$ 最终成为解决方案的近似值 $y$ ，并且至少在 理论上，我们将要描述的方法可用于找到初始值 问题的近似解。然而，其他方法通常更实用。
4. 这种方法由法国数学家 Emile Picard 开发，通常 被称为 (Picard 的) 逐次逼近法或 Picard 的迭代 法 (因为 $\psi_k$ 的生成)。
为了简化讨论让我们假设 $x_0=0$, 所以我们的初 始值问题是
$$\frac{d y}{d x}=F(x, y) \quad \text { with } \quad y(0)=y_0 .$$
这里不失一般性。毕竟，如果 $x_0 \neq 0$ ，我们可以 应用变量的变化 $s=x-x_0$ 并将我们的原始问题 转换为问题 (3.3) (与 $x$ 取而代之 $s$ ).

## 有限元方法代写

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## MATLAB代写

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