数学代写|Conservation of energy 数值分析代考代写 - 代考代写：100%准时可靠您的作业代写专家

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数学代写| Conservation of energy 数值分析代考

heat energy flowing across edges (at $x$ and $x+\Delta x$ )
heat energy generated inside
This describes conservation of energy.
$\left[\begin{array}{c}\text { change in } \ \text { heat energy } \ \text { per unit time }\end{array}\right]=\left[\begin{array}{c}\text { heat energy flowing } \ \text { across the boundaries } \ \text { per unit time }\end{array}\right]+\left[\begin{array}{c}\text { heat energy generated } \ \text { in the interior } \ \text { per unit time \& volume }\end{array}\right]$

The heat flux is a function $\phi(x, t)$ : the amount of thermal energy per unit time flowing to the right per unit surface area.
e.g. if $\phi(x, t)<0$ then heat flows to the left.

The net heat flow per unit time for our thin slice is
$$
\phi(x, t) A-\phi(x+\Delta x, t) A .
$$
Denote by $Q(x, t)$ internal sources of thermal energy,

Then the conservation of energy becomes
$$
\frac{\partial}{\partial t}[e(x, t) A \Delta x] \approx \phi(x, t) A-\phi(x+\Delta x, t) A+Q(x, t) A \Delta x .
$$
Divide by $A, \Delta x$, then take the limit $\Delta x \rightarrow 0$,
$$
\frac{\partial e}{\partial t}=-\frac{\partial \phi}{\partial x}+Q \text {. }
$$
Heat energy e and the temperature $u$ are related by,
$$
e(x, t)=c(x) \rho(x) u(x, t)
$$
$c(x)$ : specific heat
the amount of heat energy that must be applied to a unit mass of a substance to raise its tempurature one unit.
$c$ depends on the material, so let $c=c(x)$ as above. $c$ also depends on the temperature $u$ (i.e., $c=c(u, x)$ ) but it is approximately independent of temperature.
$\rho(x)$ is the mass density indicating that the rod can be made of non-uniform material.

流过边缘的热能（在 $x$ 和 $x+\Delta x$ 处）
2.内部产生的热能
这描述了能量守恒。
$\left[\begin{array}{c}\text { 变化 } \ \text { 热能 } \ \text { 每单位时间 }\end{array}\right]=\left[\begin{ array}{c}\text { 热能流动} \ \text { 跨越边界} \ \text { 每单位时间}\end{array}\right]+\left[\begin{array}{c} \text { 产生的热能 } \ \text { 在内部 } \ \text { 每单位时间 \& 体积 }\end{array}\right]$

那么能量守恒就变成了
$$
\frac{\partial}{\partial t}[e(x, t) A \Delta x] \approx \phi(x, t) A-\phi(x+\Delta x, t) A+Q(x, t) A \Delta x 。
$$
除以$A，\Delta x$，然后取极限$\Delta x \rightarrow 0$，
$$
\frac{\partial e}{\partial t}=-\frac{\partial \phi}{\partial x}+Q \text {. }
$$
热能 e 和温度 $u$ 相关，
$$
e(x, t)=c(x) \rho(x) u(x, t)
$$
$c(x)$ : 比热
必须施加在单位质量的物质上以将其温度升高一个单位的热量。
$c$ 取决于材料，因此如上令 $c=c(x)$。 $c$ 还取决于温度 $u$（即 $c=c(u, x)$ ），但它与温度大致无关。
$\rho(x)$ 是质量密度，表明杆可以由非均匀材料制成。