### 数学代写|微分方程代写differential equation代考|MAT3105

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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|With Boundary Conditions

Up to this point we have not discussed much about boundary conditions, but these can be avoided no longer. As the name implies, a boundary condition is a condition on the solution at the boundary of the domain of interest. In general, one needs one condition for every derivative that appears in the equation. Thus, for example, for a differential equation that describes the time evolution of some object with an equation of the form $u_{t}=f(u, t)$, one needs one initial condition to specify $u\left(t_{0}\right)$, where $t_{0}$ is the start time. On the other hand for a differential equation that can be written as $u_{x x}=g\left(u, u_{x}, x\right)$, one needs two conditions at the boundaries in order to specify $u$ completely. Thus, for the diffusion equation on a finite domain, one needs to specify an initial condition at $t=0$ for all values of $x$ in the domain, and two boundary conditions at the ends of the spatial domain. For a one dimensional spatial domain there are four possibilities:

• Dirichlet condition is when the value of the unknown $u$ is specified at the boundary. If $u$ is a probability, the condition $u=0$ is said to be an absorbing boundary condition, because a particle that crosses the boundary disappears and cannot re-enter the domain.
• Neumann condition is when the flux of the unknown $u$ across the boundary is specified. In a biological context, the flux across a boundary is zero if the boundary is impermeable to the particles, and is often called a no-flux condition. If $u$ is a probability, $\nabla u \cdot \mathbf{n}=0$ is called a reflecting boundary condition.
• Robin condition is a weighted combination of Dirichlet and Neumann boundary conditions, typically of the form $D \nabla u \cdot \mathbf{n}+a u=b$ and is often appropriate when the diffusing species can undergo a chemical reaction at the boundary, or, as we see below, when the species can diffuse across a porous boundary.
• Periodic conditions apply when the one dimensional domain is actually a closed loop of length $L$, with the point at $x=0$ the same as the point at $x=L$. In this case, one requires that the function $u$ and its derivative be continuous at the “boundary”.

Before we move on to solve the diffusion equation with different boundary conditions, it is worthwhile to gain some exposure to these by examining what happens when a diffusion process is at steady state. Steady state means that $u$ is not changing in time, i.e., $\frac{\partial u}{\partial t}=0$, so that the process is in equilibrium, but it does not mean that nothing is happening.

To illustrate, suppose $u$ is held fixed at $u=u_{0}$ at $x=0$ and $u=u_{L}$ at $x=L$. Then, the steady state solution satisfies $u_{x x}=0$, which implies that $u$ is a linear function satisfying the boundary conditions, i.e.,
$$u(x)=u_{L} \frac{x}{L}+u_{0}\left(1-\frac{x}{L}\right) .$$
To verify that something is happening, notice that the flux is
$$J=-D u_{x}=\frac{D}{L}\left(u_{0}-u_{L}\right),$$
which is not zero, unless $u_{L}=u_{0}$.

Suppose, instead, that the boundaries at $x=0$ and $x=L$ are porous membranes, with $u=u_{0}$ and $u=u_{L}$ just outside the domain, and the species $u$ can diffuse through the boundaries and therefore must satisfy the Robin boundary conditions
$$\left.D u_{x}\right|{x=0}=\delta\left(u(0)-u{0}\right), \quad-\left.D u_{x}\right|{x=L}=\delta\left(u(L)-u{L}\right) .$$
where $\delta>0$ represents the porosity of the boundary membrane, and $u(0), u(L)$ are the values of $u$ at the membrane just inside the domain. Notice what these conditions mean in words: the term $\delta\left(u(0)-u_{0}\right)$ is the diffusive flux of u across the membrane to the outside, and $\left.D u_{x}\right|{x=0}$ is the flux of $u$ out of the domain at $x=0$. Clearly these must match. Notice also the difference in the sign for these two conditions. This is because if $u(0)>u{0}$, the flux will be out of the domain to the left, i.e., negative (and because flux is the negative spatial derivative of $u, u$ must therefore have positive slope). If $u(L)>u_{L}$, the flux will be out of the domain to the right, i.e., positive (hence $u$ must have a negative slope). As before, $u$ is a linear function in the interior of the domain, and the requirement that it satisfy the two Robin boundary conditions yields that
$$u(x)=\frac{1}{1+2 \Delta}\left(u_{L}-u_{0}\right) \frac{x}{L}+\frac{\Delta\left(u_{0}+u_{L}\right)+u_{0}}{1+2 \Delta}$$
where $\Delta=\frac{D}{\delta L}$. Once again, the flux is nontrivial, being
$$J=-D u_{x}=\frac{D}{1+2 \Delta} \frac{u_{0}-u_{L}}{L} .$$
The quantity $D_{\text {eff }}=\frac{D}{1+2 \Delta}$ is the effective diffusion coefficient for this membrane bound medium, since the species must diffuse across both membranes and well as through the interior of the medium. Notice also the identity
$$\frac{L}{D_{\text {eff }}}=\frac{1}{\delta}+\frac{L}{D}+\frac{1}{\delta}$$
(What do you suspect the answer is if the two porosities are different? Can you verify this suspicion? See Exercise 5.5.) Clearly, in the limit that the porosity of the membrane $\delta \rightarrow \infty$, the problem reduces to the Dirichlet boundary condition with solution (5.22) (see Figure 5.2).

## 数学代写|微分方程代写differential equation代考|Separation of Variables

In biological applications, the most common boundary condition is the no-flux (homogeneous Neumann) condition, when particles are trapped inside a bounded domain, and this is where we begin our study of time dependent solutions of the diffusion equation on a bounded domain.

An important feature of the no-flux boundary condition is that the total amount of the quantity $u$ is conserved; this follows immediately from the conservation law as stated in (2.1). When solving any differential equation (in time) with constant coefficients, it is reasonable to try a solution that is exponential in time. For the diffusion

equation, we try a solution of the form
$$u(x, t)=U(x) \exp (\lambda t),$$
and upon substituting into the diffusion equation (3.2), we find
$$D \frac{d^{2} U}{d x^{2}}-\lambda U=0 .$$
This equation must be solved subject to the no-flux boundary condition $U^{\prime}(0)=$ $U^{\prime}(L)=0 .$

There are an infinite number of possible solutions, but they are all of the same form, namely
$$U_{n}(x)=a_{n} \cos \left(\frac{n \pi x}{L}\right),$$
with the important restriction that
$$\lambda=\lambda_{n} \equiv-\frac{n^{2} \pi^{2} D}{L^{2}},$$
with $n=0,1,2, \ldots$
Since there are an infinite number of possible solutions, and the diffusion equation is linear, the fully general solution is an arbitrary linear combination of the possible solutions, namely
$$u(x, t)=\sum_{n=0}^{\infty} a_{n} \exp \left(-\frac{D n^{2} \pi^{2} t}{L^{2}}\right) \cos \left(\frac{n \pi x}{L}\right)$$

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

• 狄利克雷条件是当未知值在在边界处指定。如果在是概率，条件在=0被称为吸收边界条件，因为穿过边界的粒子消失并且不能重新进入域。
• 诺伊曼条件是当未知的通量在指定跨界。在生物学背景下，如果边界对粒子是不可渗透的，则跨边界的通量为零，并且通常称为无通量条件。如果在是概率，∇在⋅n=0称为反射边界条件。
• Robin 条件是 Dirichlet 和 Neumann 边界条件的加权组合，通常形式为D∇在⋅n+一个在=b当扩散物质可以在边界发生化学反应时，或者正如我们在下面看到的，当物质可以扩散穿过多孔边界时，通常是合适的。
• 当一维域实际上是一个长度的闭环时，周期性条件适用大号, 点在X=0与点相同X=大号. 在这种情况下，需要函数在并且它的导数在“边界”处是连续的。

Ĵ=−D在X=D大号(在0−在大号),

D在X|X=0=d(在(0)−在0),−D在X|X=大号=d(在(大号)−在大号).

Ĵ=−D在X=D1+2Δ在0−在大号大号.

（如果两个孔隙率不同，你怀疑答案是什么？你能证实这个怀疑吗？见习题 5.5。）显然，在膜孔隙率的极限内d→∞，问题归结为狄利克雷边界条件，解为 (5.22)（见图 5.2）。

## 数学代写|微分方程代写differential equation代考|Separation of Variables

Dd2在dX2−λ在=0.

λ=λn≡−n2圆周率2D大号2,

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## MATLAB代写

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