### 数学代写|数值方法作业代写numerical methods代考|Ordinary Differential Equations

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值方法作业代写numerical methods代考|An Example

We take a simple autonomous non-linear scalar ODE to show how to calculate Picard iterates:
$$y^{\prime}=f(y)=y^{2}, \quad y\left(t_{0}\right)=a$$
whose solution is given by:
$$y(t)=\frac{a}{1-a\left(t-t_{0}\right)}$$
We now compute the Picard iterates (3.4) for this ODE in order to determine the values of $t$ for which the ODE has a solution. For convenience, let us take $a=1, t_{0}=0$. Some simple integration shows that:
\begin{aligned} &\phi_{1}(t)=1 \ &\phi_{1}(t)=1+\int_{0}^{t} f\left(\phi_{0}\right) d t=1+t \ &\phi_{2}(t)=1+\int_{0}^{t} f\left(\phi_{1}\right) d t=1+t+t^{2}+t^{3} / 3 \ &\phi_{3}(t)=1+t+t^{2}+t^{3}+\frac{2 t^{4}}{3}+\frac{t^{5}}{3}+\frac{t^{6}}{9}+\frac{t^{7}}{63} \end{aligned}
We can see that the series is beginning to look like $\frac{1}{1-t}=\sum_{j=0}^{\infty} t^{\jmath}$. We know that this series is convergent for $|t|<1$. A nice exercise is to compute the Picard iterates in the most general case (that is, $a \neq 1, t_{0} \neq 0$ ) and to determine under which circumstances the ODE (3.6) has a solution. In this case we have represented the solution of an ODE as a series, and we then analysed this series for which there are many convergence results, such as the root test and the ratio test.

## 数学代写|数值方法作业代写numerical methods代考|Riccati ODE

The Riccati ODE is a non-linear ODE of the form:
$$y^{\prime}=P(x)+Q(x) y+R(x) y^{2}+N(x, y)$$
This ODE has many applications, for example to interest-rate models (Duffie and Kan (1996)). In some cases a closed-form solution to Equation (3.10) is possible, but in this book our focus is on approximating it using the finite difference method.

We now discuss the relationship between the Riccati equation and the pricing of a zero-coupon bond $P(t, T)$, which is a contract that offers one dollar at maturity $T$. By definition, an affine term structure model assumes that $P(t, T)$ has the form:
$$P(t, T)=\exp [A(t, T)-B(t, T) r(t)]$$
Let us assume that the short-term interest rate is described by the following stochastic differential equation (SDE):
$$d r=\mu(t, r) d t+\sigma(t, r) d W_{t}$$
where $W_{t}$ is a standard Brownian motion under the risk-neutral equivalent measure and $\mu$ and $\sigma$ are given functions.

Duffie and Kan proved that $P(t, T)$ is exponential-affine if and only if the drift $\mu$ and volatility $\sigma$ have the form:
$$\mu(t, r)=\alpha(t) r+\beta(t), \quad \sigma(t, r)=\sqrt{\gamma(t) r+\delta(t)}$$
where $\alpha(t), \beta(t), \gamma(t)$ and $\delta(t)$ are given functions of $t$.
The coefficients $A(t, T)$ and $B(t, T)$ in this case are determined by the following ordinary differential equations:
$$\frac{d B}{d t}=\frac{\gamma(t)}{2} B(t, T)^{2}-\alpha(t) B(t, T)-1, B(T, T)=0$$
and:
$$\frac{d A}{d t}=\beta(t) B(t, T)-\frac{\delta(t)}{2} B(t, T)^{2}, A(T, T)=0$$
The first Equation (3.11) for $B(t, T)$ is the Riccati equation and the second one (3.12) is solved easily from the first one by integration.

## 数学代写|数值方法作业代写numerical methods代考|Predator-Prey Models

ODEs can be used as simple models of population growth, for example, by assuming that the rate of reproduction of a population of size $P$ is proportional to the existing population and to the amount of available resources. The ODE is:
$$\frac{d P}{d t}=r P\left(1-\frac{P}{K}\right), P(0)=P_{0}$$
where $r$ is the growth rate and $K$ is the carrying capacity. The initial population is $P_{0}$. It is easy to check the following identities:
$$P(t)=\frac{K P_{0} e^{r t}}{K+P_{0}\left(e^{r t}-1\right)} \text { and } \lim _{t \rightarrow \infty} P(t)=K .$$
Transformation of this equation leads to the logistic ODE:
$$\frac{d n}{d \tau}=n(1-n)$$
where $n$ is the population in units of carrying capacity $(n=P / K)$ and $\tau$ measures time in units of $1 / r$.

For systems, we can consider the predator-prey model in an environment consisting of foxes and rabbits:
\begin{aligned} &\frac{d r(t)}{d t}=-a r(t) f(t)+b r(t) \ &\frac{d f(t)}{d t}=-p f(t)+q f(t) r(t) \end{aligned}
where:
\begin{aligned} r(t) &=\text { number of rabbits at time } t \ f(t) &=\text { number of foxes at time } t \ b r(t) &=\text { birth rate of rabbits } \ -a r(t) f(t) &=\text { death rate of rabbits } \ b &=\text { unit birth rate of rabbits } \ -p f(t) &=\text { death rate of foxes } \ q f(t) r(t) &=\text { birth rate of foxes } \ q &=\text { unit birth rate of foxes. } \end{aligned}
The ODE system (3.14) is a model of a closed ecological environment in which foxes and rabbits are the only kinds of animals. Rabbits eat grass (of which there is a constant supply), procreate and are eaten by foxes. All foxes eat rabbits, procreate and die of geriatric diseases.

System (3.14) is sometimes called the Lotka-Volterra equations, which are an example of a more general Kolmogorov model to model the dynamics of ecological systems with predator-prey interactions, competition, disease and mutualism (Lotka (1956)).

## 数学代写|数值方法作业代写numerical methods代考|An Example

φ1(吨)=1 φ1(吨)=1+∫0吨F(φ0)d吨=1+吨 φ2(吨)=1+∫0吨F(φ1)d吨=1+吨+吨2+吨3/3 φ3(吨)=1+吨+吨2+吨3+2吨43+吨53+吨69+吨763

## 数学代写|数值方法作业代写numerical methods代考|Riccati ODE

Riccati ODE 是以下形式的非线性 ODE：

dr=μ(吨,r)d吨+σ(吨,r)d在吨

Duffie 和 Kan 证明了磷(吨,吨)是指数仿射的当且仅当漂移μ和波动性σ有以下形式：
μ(吨,r)=一种(吨)r+b(吨),σ(吨,r)=C(吨)r+d(吨)

d乙d吨=C(吨)2乙(吨,吨)2−一种(吨)乙(吨,吨)−1,乙(吨,吨)=0

d一种d吨=b(吨)乙(吨,吨)−d(吨)2乙(吨,吨)2,一种(吨,吨)=0

## 数学代写|数值方法作业代写numerical methods代考|Predator-Prey Models

ODE 可以用作人口增长的简单模型，例如，通过假设磷与现有人口和可用资源量成正比。ODE 是：
d磷d吨=r磷(1−磷ķ),磷(0)=磷0

dndτ=n(1−n)

dr(吨)d吨=−一种r(吨)F(吨)+br(吨) dF(吨)d吨=−pF(吨)+qF(吨)r(吨)

r(吨)= 一次兔子的数量 吨 F(吨)= 一次狐狸的数量 吨 br(吨)= 兔子的出生率  −一种r(吨)F(吨)= 兔子的死亡率  b= 兔单位出生率  −pF(吨)= 狐狸的死亡率  qF(吨)r(吨)= 狐狸出生率  q= 狐狸的单位出生率。
ODE系统（3.14）是一个封闭的生态环境模型，其中狐狸和兔子是唯一的动物。兔子吃草（其中有源源不断的供应），繁殖并被狐狸吃掉。所有的狐狸都吃兔子，生育并死于老年病。

## 有限元方法代写

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## MATLAB代写

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