## 数学代写|偏微分方程代写partial difference equations代考|MATH365

2023年2月2日

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## 数学代写|偏微分方程代写partial difference equations代考|Robin boundary conditions

Next, we wish to consider Robin boundary conditions. Let $\beta_0, \beta_1 \geq 0, \lambda>0$ and $f \in L_2(a, b)$ be given.

Theorem $5.19$ (Robin boundary conditions) There exists a unique $u \in H^2(a, b)$ such that
\begin{aligned} & \lambda u-u^{\prime \prime}=f \quad \text { in }(a, b), \ & -u^{\prime}(a)+\beta_0 u(a)=0, \quad u^{\prime}(b)+\beta_1 u(b)=0 . \ & \end{aligned}
The condition (5.15b) is called a Robin boundary condition, or sometimes a boundary condition of the third kind. Since $H^2(a, b) \subset C^1([a, b])$, this condition makes sense for $u \in H^2(a, b)$. In the special case $\beta_0=\beta_1=0$ we recover Neumann boundary conditions.

Proof (of Theorem 5.19) Suppose that $u \in H^2(a, b)$ is a solution of (5.15). We then have, for $v \in H^1(a, b)$,
\begin{aligned} & \int_a^b f(x) v(x) d x=\int_a^b \lambda u(x) v(x) d x-\int_a^b u^{\prime \prime}(x) v(x) d x \ &=\int_a^b \lambda u(x) v(x) d x+\int_a^b u^{\prime}(x) v^{\prime}(x) d x-u^{\prime}(b) v(b)+u^{\prime}(a) v(a) \ &=\lambda \int_a^b u(x) v(x) d x+\int_a^b u^{\prime}(x) v^{\prime}(x) d x+\beta_1 u(b) v(b)+\beta_0 u(a) v(a) . \end{aligned}
This observation leads us to set $V:=H^1(a, b)$ and define $a: V \times V \rightarrow \mathbb{R}$ by
$$a(u, v):=\lambda \int_a^b u(x) v(x) d x+\int_a^b u^{\prime}(x) v^{\prime}(x) d x+\beta_1 u(b) v(b)+\beta_0 u(a) v(a) .$$

## 数学代写|偏微分方程代写partial difference equations代考|Mixed and periodic boundary conditions

We will now consider a somewhat more complicated differential operator. In Chapter 1 we saw how one can interpret spatial derivatives physically for problems involving time and space variables. More precisely, we know that second derivatives describe diffusion, first derivatives transport processes (convection) and terms of zeroth order (i.e., no derivatives) reactive processes. The study of solutions which are constant in time is an essential step towards understanding equations in space and time. We thus wish to consider stationary equations in which terms of zeroth, first and second order appear. Such general elliptic differential equations are referred to as stationary diffusion-convection-reaction equations. As an example we will study mixed boundary conditions, that is, Dirichlet boundary conditions at one endpoint and Neumann boundary conditions at the other endpoint of the interval.

Theorem 5.20 (Mixed boundary conditions) Let $p \in C^1([a, b])$ and $r \in C([a, b])$ with $r \geq 0$. Suppose that there exists $\alpha>0$ such that $p(x) \geq \alpha$ for all $x \in[a, b]$, and let $f \in L_2(a, b)$. Then there exists a unique solution $u \in H^2(a, b)$ of
\begin{aligned} -\left(p u^{\prime}\right)^{\prime}+r u & =f \text { almost everywhere in }(a, b), \ u(a)=0, u^{\prime}(b) & =0 . \end{aligned}
If $f \in C([a, b])$, then $u \in C^2([a, b])$.
One should observe that, by Theorem $5.13$ (the product rule), $p u^{\prime} \in H^1(a, b)$ if $u \in H^2(a, b)$, meaning that (5.16a) is well defined.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Robin boundary conditions

$$\lambda u-u^{\prime \prime}=f \quad \text { in }(a, b), \quad-u^{\prime}(a)+\beta_0 u(a)=0,$$

$$\int_a^b f(x) v(x) d x=\int_a^b \lambda u(x) v(x) d x-\int_a^b u^{\prime \prime}(x) v(x)$$

\begin{aligned} & a: V \times V \rightarrow \mathbb{R} \text { 经过 } \ & a(u, v):=\lambda \int_a^b u(x) v(x) d x+\int_a^b u^{\prime}(x) v^{\prime}(x) d x+\beta_1 \end{aligned}

## 数学代写|偏微分方程代写partial difference equations代考|Mixed and periodic boundary conditions

$-\left(p u^{\prime}\right)^{\prime}+r u=f$ almost everywhere in $(a, b), u(a)$

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