数学代写|微分方程代写differential equation代考|MATH4403

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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|微分方程代写differential equation代考|An Agent-Based Approach

A popular way to simulate a population of particles is with an agent-based (or individual-based) approach, in which particles are tracked separately, but simultaneously.

Suppose, for example, we want to simulate a population (say, several hundred) of one dimensional run and tumble organisms. In the last section, we followed the motion of individuals one at a time, using a next reaction time algorithm to determine changes of state and position. This will not work for following several particles at once, since the transitions and movement are not synchronous.

So how does an agent-based approach work? We suppose that there are $N$ particles with position $x_{n}, n=1,2, \ldots, N$, each in state $s_{n}, n=1,2, \ldots, N$, where $s_{j}$ is one of the $K$ possible states $1,2, \ldots, K$. Now, there are rules for how a particle in state $k$ moves, say, with velocity $v(k)=v_{k}$, and there are rates for transitioning between states, say $\lambda_{j k}$ is the rate of transitioning from state $k$ to state $j$. We discretize time with a fixed time step $\Delta t$, and with each time step, let $\lambda_{j k} \Delta t$ be the probability of changing from the state $k$ to state $j$. The algorithm proceeds by first moving each particle by the amount $v\left(s_{n}\right) \Delta t$ and then modifying the states based on the probabilities $\lambda_{j k} \Delta t$ and $N$ uniformly distributed random numbers $R_{n}$.

To be specific, for the one dimensional run and tumble model, there are three states, say, $s={1,2,3}$ corresponding to leftward, resting, and rightward motion. The velocities in these three states are $-v, 0, v$, respectively. The rates of transition are $\lambda_{21}=\lambda_{23}=k_{\text {off }}$ and $\lambda_{12}=\lambda_{32}=\frac{k_{\text {on }}}{2}$.

The Matlab code that simulates this agent-based particle movement for run and tumble particles in one dimension is titled agent_based_run_and_tumble.m.

A reason that agent-based modeling is both useful and popular is that the rules for movement and change of state can be diverse and can be easily simulated, even though a partial differential equation description of the dynamics may not be known. We use agent-based modeling throughout this book, especially in Chapter 14 on Collective Behavior, where we discuss swarming behaviors of things like flying birds.

数学代写|微分方程代写differential equation代考|On an Infinite Domain

If the domain is the infinite line, and the initial data are concentrated at the origin, a solution is the normal distribution $\mathcal{N}(0,2 D t)$, found in Chapter 3 and given by
$$u(x, t)=\frac{1}{\sqrt{4 \pi D t}} \exp \left(-\frac{x^{2}}{4 D t}\right)$$
If the domain is the two dimensional plane, we look for radially symmetric solutions, and therefore need a solution of the equation
$$\frac{\partial u}{\partial t}=\frac{D}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right),$$
where $r$ is the radius. We guess a solution of the form
$$u(r, t)=\frac{1}{a(t)} \exp \left(\frac{-r^{2}}{b(t)}\right),$$
and find it must be that
$$a\left(\frac{d b}{d t}-4 D\right) r^{2}+4 a b D-b^{2} \frac{d a}{d t}=0$$
for all $r$. This is a quadratic polynomial in $r$ which can be identically zero for all $r$ only if the individual coefficients of powers of $r$ are zero, or, that
$$\frac{d b}{d t}=4 D, \quad \frac{d a}{d t}=4 D \frac{a}{b}$$

so that $b(t)=4 D t, a(t)=a_{0} t$. Consequently, the solution is
$$u(r, t)=\frac{1}{4 \pi D t} \exp \left(\frac{-r^{2}}{4 D t}\right),$$
and this solution has the property
$$2 \pi \int_{0}^{\infty} u(r, t) r d r=1$$
for all time. Furthermore, the percentage of the population contained within a circle of radius $R$ is given by
$$2 \pi \int_{0}^{R} u(r, t) r d r=\int_{0}^{R} \frac{1}{2 D t} \exp \left(\frac{-r^{2}}{4 D t}\right) r d r=1-\exp \left(\frac{-R^{2}}{4 D t}\right)$$
confirming what we observed in the particle diffusion simulation in the last chapter. (See Figures $4.5$ and 4.6.)

数学代写|微分方程代写differential equation代考|On the Semi-infinite Line

Suppose that a long capillary, open at one end, with uniform cross-sectional area $A$ and filled with water, is inserted into a solution of known chemical concentration $u_{0}$, and the chemical species is free to diffuse into the capillary through the open end. Since the concentration of the chemical species depends only on the distance along the tube and time, it is governed by the diffusion equation (3.2), and for convenience we assume that the capillary is infinitely long, so that $0<x<\infty$. Because the solute bath in which the capillary sits is large, it is reasonable to assume that the chemical concentration at the tip is fixed at $u(0, t)=u_{0}$, and since the tube is initially filled with pure water, $u(x, 0)=0$ for all $x, 0<x<\infty$.

There are (at least) two ways to find the solution of this problem. One is to use the Fourier-Sine transform, a technique which is beyond the scope of this text (but you can learn about it in [34]). The second is to make a lucky (or semi-informed) guess. Here, we make the guess that the solution should be of the form $u(x, t)=f(\xi)$, where $\xi=\frac{x}{\sqrt{2 D t}}$. Substitute this guess into the diffusion equation and find
$$f^{\prime} \xi+f^{\prime}=0 .$$
This is a separable equation for $f^{\prime}$ and can be written as
$$\frac{d f^{\prime}}{f^{\prime}}=-\xi d \xi$$
so that
$$\frac{d f}{d \xi}=a \exp \left(-\frac{\xi^{2}}{2}\right)$$
where $a$ is a yet to be determined constant. From this we determine that a solution of the diffusion equation is given by
$$u(x, t)=b+a \int_{0}^{z} \exp \left(-\frac{s^{2}}{2}\right) d s, \quad z=\frac{x}{\sqrt{2 D t}},$$
with constants $a$ and $b$ determined from boundary and initial data. Setting $x=z=0$, and requiring $u(0, t)=u_{0}$ determines that $b=u_{0}$. Setting $t=0$, i.e., $z=\infty$, and requiring $u(x, 0)=0$ implies that $a=-u_{0} \sqrt{\frac{2}{\pi}}$, and consequently,
$$u(x, t)=u_{0}\left(1-\sqrt{\frac{2}{\pi}} \int_{0}^{z} \exp \left(-\frac{s^{2}}{2}\right) d s\right), \quad z=\frac{x}{\sqrt{2 D t}}$$
Plots of this solution plotted as a function of $z$ (a surrogate for $x$ ), and as a function of $z^{-\frac{1}{2}}$ (a surrogate for $t$ ) are shown in Figure $5.1$ and were made using Matlab code tube_diffusion.m.

数学代写|微分方程代写differential equation代考|On an Infinite Domain

∂在∂吨=Dr∂∂r(r∂在∂r),

dbd吨=4D,d一个d吨=4D一个b

2圆周率∫0∞在(r,吨)rdr=1

2圆周率∫0R在(r,吨)rdr=∫0R12D吨经验⁡(−r24D吨)rdr=1−经验⁡(−R24D吨)

F′X+F′=0.

dF′F′=−XdX

dFdX=一个经验⁡(−X22)

有限元方法代写

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MATLAB代写

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