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# 数学代写|数值分析代写Numerical analysis代考|MATH408 Systems of equations

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## 数学代写|数值分析代写Numerical analysis代考|Systems of equations

In most applications we do not have a simple scalar equation, but rather a system of equations describing the coupled dynamics of several variables. Such situations give rise to vector-valued functions $\mathbf{x}(t) \in \mathbb{R}^n$. In particular, the initial value problem becomes
Given: $\quad \mathbf{x}^{\prime}(t)=\mathbf{f}(t, \mathbf{x}(t))$, with $\mathbf{x}\left(t_0\right)=\mathbf{x}_0$
Determine: $\mathbf{x}(t)$ for all $t \geq t_0$.
All the techniques for solving scalar initial value problems described in this course can be applied to systems of this type.

Higher-order ODEs
Newton’s Second Law, $\mathbf{F}(t)=m \mathbf{a}(t)$, leads to many important second-order differential equations. Noting the acceleration $\mathbf{a}(t)$ is the second derivative of the position $\mathbf{x}(t)$, we arrive at
$$\mathbf{x}^{\prime \prime}(t)=m^{-1} \mathbf{F}(t)$$
Thus, we are often interested in systems of higher-order ODEs.
To keep the notation simple, consider the scalar second-order problem
Given: $\quad x^{\prime \prime}(t)=f\left(t, x(t), x^{\prime}(t)\right)$, with $x\left(t_0\right)=x_0, x^{\prime}\left(t_0\right)=y_0$
Determine: $\quad x(t)$ for all $t \geq t_0$.

## 数学代写|数值分析代写Numerical analysis代考|Picard’s Theorem: Existence and Uniqueness of Solutions

Before constructing numerical solutions to these differential equations, it is important to understand when solutions exist at all. Picard’s theorem establishes existence and uniqueness.
Theorem 5.1 (Picard’s Theorem).
Let $f(t, x)$ be a continuous function on the rectangle
$$D=\left{(t, x): t \in\left[t_0, t_{\text {final }}\right], x \in\left[x_0-c, x_0+c\right]\right}$$
for some fixed $c>0$. Furthermore, suppose $\left|f\left(t, x_0\right)\right| \leq K$ for all $t \in\left[t_0, t_{\text {final }}\right]$, and suppose there exists some Lipschitz constant $L>0$ such that
$$|f(t, u)-f(t, v)| \leq L|u-v|$$

for all $u, v \in\left[x_0-c, x_0+c\right]$ and all $t \in\left[t_0, t_{\text {final }}\right]$. Finally, suppose that
$$c \geq \frac{K}{L}\left(\mathrm{e}^{L\left(t_{\text {final }}-t_0\right)}-1\right) .$$
(That is, the box $D$ must be sufficiently large to compensate for large values of $K$ and $L$.) Then there exists a unique $x \in C^1\left[t_0, t_{\text {final }}\right]$ such that $x\left(t_0\right)=x_0, x^{\prime}(t)=f(t, x)$ for all $t \in\left[t_0, t_{\text {final }}\right]$, and $\left|x(t)-x_0\right| \leq c$ for all $t \in\left[t_0, t_{\text {final }}\right]$.

In simpler words, these hypotheses ensure the existence of a unique $C^1$ solution to the initial value problem, and this solution stays within the rectangle $D$ for all $t \in\left[t_0, t_{\text {final }}\right]$, i.e., as $t$ increases the solution will ‘exit’ from the right-side of the rectangle, not the top or bottom.

## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|PICARD’S THEOREM: EXISTENCE AND UNIQUENESS OF SOLUTIONS

$$|f(t, u)-f(t, v)| \leq L|u-v|$$

$$c \geq \frac{K}{L}\left(\mathrm{e}^{L\left(t_{\text {fral }}-t_0\right)}-1\right)$$
Thatis, thebox $\$ D \$$mustbesufficientlylargetocompensateforlargevaluesof \ K \ a n d \ L \$$.那么存在唯一性 $x \in C^1\left[t_0, t_{\text {final }}\right]$ 这样 $x\left(t_0\right)=x_0, x^{\prime}(t)=f(t, x)$ 对所有人 $t \in\left[t_0, t_{\text {final }}\right]$ ， 和 $\left|x(t)-x_0\right| \leq c$ 对所有人 $t \in\left[t_0, t_{\text {final }}\right]$.

## Matlab代写

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