### 数学代写|微分方程代写differential equation代考|MATH2021

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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|Numerical Methods

The first method to numerically simulate the diffusion equation is one that we have already used, namely the method of lines. With this method, we discretize the spatial region into a grid with points at $x_{j}=j \Delta x$, $j=0,1, \ldots, N$, and then write the diffusion equation approximately as the system of ordinary differential equations
$$\frac{d u_{j}}{d t}=\frac{D}{\Delta x^{2}}\left(u_{j+1}-2 u_{j}+u_{j-1}\right) .$$
At the endpoints, take the equations to be
$$\frac{d u_{0}}{d t}=\frac{2 D}{\Delta x^{2}}\left(u_{1}-u_{0}\right), \quad \frac{d u_{N}}{d t}=\frac{2 D}{\Delta x^{2}}\left(u_{j-1}-u_{N}\right) .$$
This choice follows from the approximation to the derivative of $u,\left.\frac{\partial u}{\partial x}\right|{x=j \Delta x} \approx \frac{u{j+1}-u_{j-1}}{2 \Delta x}$, so at the boundaries the zero derivative (Neumann) boundary condition implies that $u_{-1}=u_{1}$, and $u_{N+1}=u_{N-1}$. (The points at $j=-1$ and $j=N+1$ are called ghost points and are useful for this calculation, but are never actually computed.)

This system of equations is then simulated using a numerical ordinary differential equation solver. The Matlab code for this for Neumann or Robin boundary conditions is titled Diffusion_NK_via_MUL.m and for Dirichlet boundary conditions is titled Diffusion_Dirichlet_via_MOL.m.

It is convenient for future discussions to represent $u(j \Delta x, t)$ as a vector $\mathbf{u}(t)=\left(u_{j}\right)$, and then to rewrite (5.37) using vector/matrix notation as
$$\frac{d \mathbf{u}}{d t}=\frac{D}{\Delta x^{2}} A \mathbf{u}$$
where the matrix $A$ has diagonal elements $-2$, and first upper and lower off-diagonal elements 1 , except the first element of the upper diagonal and last element of the lower diagonal are both 2, i.e.,
$$A=\left(\begin{array}{ccccc} -2 & 2 & 0 & \cdots & 0 \ 1 & -2 & 1 & 0 & \cdots \ & & \vdots & & \ 0 & \cdots & 1 & -2 & 1 \ 0 & \cdots & 0 & 2 & -2 \end{array}\right)$$

## 数学代写|微分方程代写differential equation代考|Other Boundary Conditions

Everything discussed in the previous section was for Neumann boundary conditions. However, the only difference for the numerical methods with different boundary conditions is with the definition of the corner entries of the matrix $A$. For homogeneous Robin boundary conditions (5.23), we write the approximations
$$D\left(\frac{u_{1}-u_{-1}}{2 \Delta x}\right)=\delta u_{0}, \quad-D\left(\frac{u_{N+1}-u_{N-1}}{2 \Delta x}\right)=\delta u_{N}$$
which when substituted into the finite difference approximation of the diffusion equation (5.37) yields
$$\frac{d u_{0}}{d t}=\frac{D}{\Delta x^{2}}\left(2 u_{1}-2\left(1+\frac{\delta \Delta x}{D}\right) u_{0}\right)$$
$$\frac{d u_{N}}{d t}=\frac{D}{\Delta x^{2}}\left(2\left(-1-\frac{\delta \Delta x}{D}\right) u_{N}+2 u_{N-1}\right) .$$
This implies that the matrix $A$ in (5.40) needs to be modified slightly to have first and last diagonal elements
$$A_{1,1}=-2-2 \frac{\delta \Delta x}{D}, \quad A_{N+1, N+1}=-2-2 \frac{\delta \Delta x}{D}$$
For homogeneous Dirichlet boundary conditions, the unknown variables are $u_{j}$, $j=1,2, \ldots, u_{N-1}$, (two less than for Neumann and Robin conditions) and the finite difference approximation (5.37) for $u_{1}$ and $u_{N ~}$
$$\text { (5.56) } \frac{d u_{1}}{d t}=\frac{D}{\Delta x^{2}}\left(u_{2}-2 u_{1}\right), \quad \frac{d u_{N-1}}{d t}=\frac{D}{\Delta x^{2}}\left(-2 u_{N-1}+u_{N-2}\right)$$
(since $u_{0}=u_{N}=0$ ) and consequently, the matrix $A$ (which is now an $N-1 \times N-1$ matrix) is
$$A=\left(\begin{array}{ccccc} -2 & 1 & 0 & \cdots & 0 \ 1 & -2 & 1 & 0 & \cdots \ & & \vdots & & \ 0 & \cdots & 1 & -2 & 1 \ 0 & \cdots & 0 & 1 & -2 \end{array}\right)$$
For Dirichlet boundary conditions, the Matlab codes are FEuler_diffusion_dirichlet.m, BEuler_diffusion_Dirichlet.m, and CN_diffusion_Dirichlet.m for forward Euler, backward Euler, and Crank-Nicolson methods, respectively.

## 数学代写|微分方程代写differential equation代考|Birth-Death with Diffusion

Suppose that there is some population or chemical species $U$ that diffuses on an infinite domain and experiences either decay, as in
$$U \stackrel{\alpha}{\longrightarrow} \emptyset,$$
or birth via asexual duplication as in
$$U \stackrel{\alpha}{\longrightarrow} 2 U$$
Setting $u=[U]$, the equation describing the evolution of this population is
$$\frac{\partial u}{\partial t}=D \frac{\partial^{2} u}{\partial x^{2}}+\sigma \alpha u,$$
where $\alpha>0, \sigma=-1$ for decay, and $\sigma=1$ for growth.
As an example, suppose that signaling molecules are produced at some point, and that the target of the signal is some distance away. Specifically, many copies of a transcription factor may be made when only one is needed to activate transcription of a gene. What percentage, if any, of the signaling molecules reach the target before they degrade? Roughly $10^{8}$ sperm cells are initially released to reach the oocyte in human fertilization when only one is required. What are the consequences of releasing fewer sperm cells? This could also be a model to determine the distribution of seeds falling

to the ground after release from a seed pod into the air. This seed dispersal problem is discussed more in Chapter $13 .$

Let’s begin our study of this problem of diffusion with decay with a stochastic simulation. To do this, we modify the algorithm from Chapter 4 that simulates diffusing particles to account for the possibility of degradation. At each time step, the particle can move or degrade, and if it degrades, its motion is terminated. The Matlab code to implement this is titled decay_probability.m, and the result of a simulation is shown in Figure 6.1. What we see in this figure may be a bit surprising; the distribution of decay position is clearly not a Gaussian distribution. But what is it?

## 数学代写|微分方程代写differential equation代考|Numerical Methods

d在jd吨=DΔX2(在j+1−2在j+在j−1).

d在0d吨=2DΔX2(在1−在0),d在ñd吨=2DΔX2(在j−1−在ñ).

d在d吨=DΔX2一个在

## 数学代写|微分方程代写differential equation代考|Other Boundary Conditions

D(在1−在−12ΔX)=d在0,−D(在ñ+1−在ñ−12ΔX)=d在ñ

d在0d吨=DΔX2(2在1−2(1+dΔXD)在0)

d在ñd吨=DΔX2(2(−1−dΔXD)在ñ+2在ñ−1).

(5.56) d在1d吨=DΔX2(在2−2在1),d在ñ−1d吨=DΔX2(−2在ñ−1+在ñ−2)
（自从在0=在ñ=0)，因此，矩阵一个（现在是一个ñ−1×ñ−1矩阵）是

## 数学代写|微分方程代写differential equation代考|Birth-Death with Diffusion

∂在∂吨=D∂2在∂X2+σ一个在,

## 有限元方法代写

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## MATLAB代写

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