Posted on Categories:数学代写, 数学建模

# 数学代写|数学建模代写Mathematical Modeling代考|SITUATIONS GIVING RISE TO PARTIAL DIFFERENTIAL EQUATION MODELS

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|数学建模代写Mathematical Modeling代考|SITUATIONS GIVING RISE TO PARTIAL DIFFERENTIAL EQUATION MODELS

Partial differential equation (PDE) models arise when the variables of interest are functions of more than one independent variable and all the dependent and independent variables are continuous. Thus in fluid dynamics, the velocity components $u, v, w$ and the pressure $p$ at any point $x, y, z$ and at any time $t$ are functions of $x, y, z, t$, and in general $u(x, y, z, \mathrm{t}), v(x, y, z, t)$, $w(x, y, z, t), p(x, y, z, t)$ are continuous functions, with continuous first and second order partial derivatives, of the continuous independent variables $x, y, z, t$. Similarly the electric field intensity vector $\vec{E}(x, y, z, t)$, the magnetic field intensity vector $\vec{H}(x, y, z, t)$, the electric current density vector $\vec{J}(x, y, z, t)$, the temperature $T(x, y, z, t)$, and the displacement vector $\vec{D}(x, y, z, t)$ of an elastic substance are in general continuous vector or scalar functions with continuous derivatives. One object of mathematical modeling is to translate the physical laws governing these functions into partial differential equations whose solution, subject to appropriate initial and boundary conditions, should determine the values of these functions at any point $x, y, z$ at any time $t$. For this purpose, we consider an elementary volume element and apply to it the principles of continuity and heat, momentum, energy balance, etc.
According to the principle of mass balance, the amount of the substance flowing across the surface of the volume element in a small time $\Delta t$ is equal to the decrease in the mass of the substance inside the volume in that time. The amount of the mass flowing across the surface can be expressed as a surface integral and the change of mass inside the volume can be expressed as a volume integral. However, the surface integral can also be converted into a volume integral by using the Gauss divergence theorem so that finally the mass balance principle requires the vanishing of a volume integral for all arbitrary volume elements.
This can happen only if the integrand vanishes identically. The vanishing of the integrand gives rise to a partial differential equation. We shall discuss this method of deriving partial differential equations in Section 6.2.

## 数学代写|数学建模代写Mathematical Modeling代考|Equation of Continuity in Fluid Dynamics

If $V_n$ is the normal component of the velocity of the fluid at any point of the surface of our conceptual volume element (Figure 6.1), the mass of the fluid flowing out in time $\Delta t$ across the surface
\begin{aligned} & =\Delta t \iint_S \rho V_n d S=\Delta t \iint_S \overrightarrow{\rho V} \overrightarrow{d S} \ & =\Delta t \iiint_T \operatorname{div}(\rho \vec{V}) d x d y d z \end{aligned}
on using Gauss’s divergence theorem. The change of mass of fluid in the volume element in the time $\Delta t$ is given by
$$-\Delta t \frac{\partial}{\partial t} \iiint_T \rho d x d y d z=-\Delta t \iiint_T \frac{\partial \rho}{\partial t} d x d y d z$$
Using Eqns. (2) and (3), the principle of mass-balance gives
$$\iiint_T\left[\frac{\partial \rho}{\partial t}-\operatorname{div}(\rho \vec{V})\right] d x d y d z=0$$
Since Eqn. (4) is to be true for all arbitrary volume elements, we get
$$\frac{\partial \rho}{\partial t}+\operatorname{div}(\rho \vec{V})=0$$
or
$$\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)+\frac{\partial}{\partial y}(\rho w)=0$$
If the fluid is incompressible, $\rho$ is constant and Eqns. (5), (6) give
$$\operatorname{div}(\vec{V})=0 \text { or } \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$$

## 数学代写|数学建模代写Mathematical Modeling代考|Equation of Continuity in Fluid Dynamics

\begin{aligned} & =\Delta t \iint_S \rho V_n d S=\Delta t \iint_S \overrightarrow{\rho V} \overrightarrow{d S} \ & =\Delta t \iiint_T \operatorname{div}(\rho \vec{V}) d x d y d z \end{aligned}

$$-\Delta t \frac{\partial}{\partial t} \iiint_T \rho d x d y d z=-\Delta t \iiint_T \frac{\partial \rho}{\partial t} d x d y d z$$

$$\iiint_T\left[\frac{\partial \rho}{\partial t}-\operatorname{div}(\rho \vec{V})\right] d x d y d z=0$$

$$\frac{\partial \rho}{\partial t}+\operatorname{div}(\rho \vec{V})=0$$

$$\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)+\frac{\partial}{\partial y}(\rho w)=0$$

$$\operatorname{div}(\vec{V})=0 \text { or } \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。