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## 数学代写|数学建模代写Mathematical Modeling代考|Pontryagin’s Maximum Principle

This principle enables us to maximize or minimize
$$P=\int_a^b \varphi_0\left(t, x_1(t), x_2(t), \ldots, x_n(t), h_1(t), \ldots, h_n(t)\right) d t$$
subject to
$$\frac{d x_i}{d t}=\varphi_i\left(t, x_1, x_2, \ldots, x_n, h_1(t), \ldots, h_n(t)\right), i=1,2, \ldots, n$$
If we know $h_1(t), h_2(t), \ldots, h_n(t)$ we can solve for $x_1(t), \ldots, x_n(t)$ from Eqn. (63) and then integrate (62) to find $P$. Thus $P$ is a function of $h_1(t), h_2(t), \ldots, h_n(t)$ and we can choose these control functions in such a manner as to maximize or minimize $P$.
According to Pontryagin’s maximum principle, we form the Hamiltonian function
$$H=\varphi_0+\sum_{i=1}^n \psi_i \varphi_i$$
where for determining the functions $\psi_1, \psi_2, \ldots, \psi_n$, we have the auxiliary equations
$$\frac{\partial H}{\partial x_i}=-\frac{d \psi_i}{d t},(i=1,2, \ldots, n)$$
$H$ is a function of $h_1, h_2, \ldots, h_n$ and we choose $h_1, h_2, \ldots, h_3$ to maximize $H$.
This gives us $n$ equations. These equations together with the $n$ equation (63) and the $n$ equation (65) give us $3 n$ equations to determine $x_i(t), \psi_i(t), h_i(t)(i=1,2, \ldots, n)$.
If $\varphi_0=1$, we get $P=t$ and this gives the solution of the time-optimal problem.

## 数学代写|数学建模代写Mathematical Modeling代考|Solution of a Simple Time-Optimal Problem

A particle starts from the point at a distance $x_0$ from the origin on the $x$-axis with a velocity $v_0$. It is acted on by a force $u(t)$ along the positive direction of $x$-axis which is at our disposal, subject to the condition that $|u(t)| \leq 1$. The particle is required to reach the origin with zero velocity. We have to determine $\mathrm{u}(\mathrm{t})$ so that the time taken in reaching the origin is minimum.
The equation of motion is
or
$$\begin{gathered} \frac{d^2 x}{d t_2}=u(t) \ \frac{d x}{d t}=v(t), \frac{d v}{d t}=u(t) \end{gathered}$$
Equations (64) and (65) then give
$$\begin{gathered} H=1+\psi_1 v(t)+\psi_2 u(t) \ \frac{\partial H}{\partial x}=-\frac{d \psi_1}{d t}, \quad \frac{\partial H}{\partial v}=-\frac{d \psi_2}{d t} \end{gathered}$$
From Eqns. (68) and (69)
$$0=-\frac{d \psi_1}{d t}, \quad \psi_1=-\frac{d \psi_2}{d t}$$
Integrating
$$\psi_1=c_1, \psi_2=c_2-c_1 t$$
Now we have to maximize $H$ as a function of $u$ when $-1 \leq u \leq 1$. This gives $u(t)=1$ whenever $\psi_2$ is positive and $u(t)=-1$ whenever $\psi_2$ is negative.

## 数学代写|数学建模代写Mathematical Modeling代考|Pontryagin’s Maximum Principle

$$P=\int_a^b \varphi_0\left(t, x_1(t), x_2(t), \ldots, x_n(t), h_1(t), \ldots, h_n(t)\right) d t$$

$$\frac{d x_i}{d t}=\varphi_i\left(t, x_1, x_2, \ldots, x_n, h_1(t), \ldots, h_n(t)\right), i=1,2, \ldots, n$$

$$H=\varphi_0+\sum_{i=1}^n \psi_i \varphi_i$$

$$\frac{\partial H}{\partial x_i}=-\frac{d \psi_i}{d t},(i=1,2, \ldots, n)$$
$H$是$h_1, h_2, \ldots, h_n$的函数，我们选择$h_1, h_2, \ldots, h_3$来最大化$H$。

## 数学代写|数学建模代写Mathematical Modeling代考|Solution of a Simple Time-Optimal Problem

$$\begin{gathered} \frac{d^2 x}{d t_2}=u(t) \ \frac{d x}{d t}=v(t), \frac{d v}{d t}=u(t) \end{gathered}$$

$$\begin{gathered} H=1+\psi_1 v(t)+\psi_2 u(t) \ \frac{\partial H}{\partial x}=-\frac{d \psi_1}{d t}, \quad \frac{\partial H}{\partial v}=-\frac{d \psi_2}{d t} \end{gathered}$$

$$0=-\frac{d \psi_1}{d t}, \quad \psi_1=-\frac{d \psi_2}{d t}$$

$$\psi_1=c_1, \psi_2=c_2-c_1 t$$

## 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。

Posted on Categories:数学代写, 数学建模

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## 数学代写|数学建模代写Mathematical Modeling代考|MATHEMATICAL MODELING IN BIOECONOMICS THROUGH CALCULUS OF VARIATIONS

Mathematical bioeconomics is an interdisciplinary subject in which we use mathematical methods to optimize the economic profits from the utilization of renewable biological resources like forests and fisheries.
Let $x(t)$ be the fish population at time $t$ and let $h(t)$ be the rate at which it is harvested, then we get the equation
$$\frac{d x}{d t}=F(x)-h(t)$$
where $F(x)$ is the natural biological rate of growth. Let $c(x)$ be the cost of harvesting a unit of fish when the population size is $x(t)$ and let $p$ be the selling price per unit fish so that the profit per unit of fish is $(p-c(x))$ and the profit in time interval $(t, t+d t)$ is $(p-c(x) h(t) d t$. If $\delta$ is the instantaneous discount rate, the present value of the total profit is
$$P=\int_0^{\infty} e^{-\delta t}(p-c(x)) h(t) d t$$
If we know $h(t)$, we can use (70) to solve for $x(t)$ and then we can use Eqn. (71) to determine $P$ so that $P$ depends on what function $h$ is of $t$. We have to determine that function $h(t)$ for which $P$ is maximum. Substituting for $h(t)$ from Eqn. (70) in Eqn. (71), we get
$$\begin{gathered} P=\int_0^{\infty} e^{-\delta t}(p-c(x))\left(F(x)-x^{\prime}\right) d t \ f\left(t, x, x^{\prime}\right)=e^{-\delta t}(p-c(x))\left(F(x)-x^{\prime}\right) \end{gathered}$$
Using Euler-Lagrange Eqn. (8),
$$\frac{\partial f}{\partial x}-\frac{d}{d t}\left(\frac{\partial f}{\partial x^{\prime}}\right)=0$$
or
$$\begin{gathered} e^{-\delta t}\left(-c^{\prime}(x)\right)\left(F(x)-x^{\prime}\right)+e^{-\delta t}(p-c(x))\left(F^{\prime}(x)-\frac{d}{d t}\left[e^{-\beta t}(c(x)-p)\right]=0\right. \ -c^{\prime}(x)\left(F(x)-x^{\prime}\right)+(p-c(x)) F^{\prime}(x)+\delta(c(x)-p)-c^{\prime}(x) x^{\prime}=0 \ -c^{\prime}(x) F(x)+(p-c(x))\left(F^{\prime}(x)-\delta\right)=0 \end{gathered}$$
which determines a constant value $x$ for $x$ and then $(70)$ gives the rate of harvesting as constant and equal to $F\left(x^\right)$.

## 数学代写|数学建模代写Mathematical Modeling代考|Mathematical Modeling in Optics Through Calculus of Variations

According to Fermat’s principle of least time, light travels from a given point $A$ to another point $B$ in such a way as to take the least possible time. If $\mu(x, y)$ is the refractive index at the point $(x, y)$, then the velocity of light at the point is $c / \mu(x, y)$ and the time taken in going from $A$ to $B$ is
$$\begin{gathered} =\int_A^B \frac{d s}{c / \mu}=\int_A^B \mu(x, y) \sqrt{1+\left(\frac{d y}{d x}\right)^2} d x \ \therefore \quad f\left(x, y, y^{\prime}\right)=\mu(x, y) \sqrt{1+y^{\prime 2}} \ \frac{\partial \mu}{\partial y} \sqrt{1+y^{\prime 2}}-\frac{d}{d x}\left[\mu \frac{y^{\prime}}{\sqrt{1+y^{\prime 2}}}\right]=0 \ \frac{\partial \mu}{\partial y}=\frac{d}{d s}(\mu \sin \psi) \end{gathered}$$
Equation (8) gives
If the $y$-axis separates two media of refractive indices $\mu_1$ and $\mu_2$, then
$$\frac{\partial \mu}{\partial y}=0$$
and so
$$\mu_1 \sin \psi_1=\mu_2 \sin \psi_2$$
which is Snell’s law of refraction.

## 数学代写|数学建模代写Mathematical Modeling代考|MATHEMATICAL MODELING IN BIOECONOMICS THROUGH CALCULUS OF VARIATIONS

$$\frac{d x}{d t}=F(x)-h(t)$$

$$P=\int_0^{\infty} e^{-\delta t}(p-c(x)) h(t) d t$$

$$\begin{gathered} P=\int_0^{\infty} e^{-\delta t}(p-c(x))\left(F(x)-x^{\prime}\right) d t \ f\left(t, x, x^{\prime}\right)=e^{-\delta t}(p-c(x))\left(F(x)-x^{\prime}\right) \end{gathered}$$

$$\frac{\partial f}{\partial x}-\frac{d}{d t}\left(\frac{\partial f}{\partial x^{\prime}}\right)=0$$

$$\begin{gathered} e^{-\delta t}\left(-c^{\prime}(x)\right)\left(F(x)-x^{\prime}\right)+e^{-\delta t}(p-c(x))\left(F^{\prime}(x)-\frac{d}{d t}\left[e^{-\beta t}(c(x)-p)\right]=0\right. \ -c^{\prime}(x)\left(F(x)-x^{\prime}\right)+(p-c(x)) F^{\prime}(x)+\delta(c(x)-p)-c^{\prime}(x) x^{\prime}=0 \ -c^{\prime}(x) F(x)+(p-c(x))\left(F^{\prime}(x)-\delta\right)=0 \end{gathered}$$

## 数学代写|数学建模代写Mathematical Modeling代考|Mathematical Modeling in Optics Through Calculus of Variations

$$\begin{gathered} =\int_A^B \frac{d s}{c / \mu}=\int_A^B \mu(x, y) \sqrt{1+\left(\frac{d y}{d x}\right)^2} d x \ \therefore \quad f\left(x, y, y^{\prime}\right)=\mu(x, y) \sqrt{1+y^{\prime 2}} \ \frac{\partial \mu}{\partial y} \sqrt{1+y^{\prime 2}}-\frac{d}{d x}\left[\mu \frac{y^{\prime}}{\sqrt{1+y^{\prime 2}}}\right]=0 \ \frac{\partial \mu}{\partial y}=\frac{d}{d s}(\mu \sin \psi) \end{gathered}$$

$$\frac{\partial \mu}{\partial y}=0$$

$$\mu_1 \sin \psi_1=\mu_2 \sin \psi_2$$

## 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。

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## 数学代写|数学建模代写Mathematical Modeling代考|Stability of Equilibrium Positions

For solving Eqn. (120), knowledge of $N(0)$ is not enough; we have to know the values of $\mathrm{N}(\mathrm{t}$ ) from time $-T$ to 0 . Even after knowing all these values, the solution of (120) is not easy.

We can however find the equilibrium or steady-state solutions by putting $d N / d t=0$ and replacing $N(t)$ and $N(t-\mathrm{T})$ by the equilibrium population size $\bar{N}$, thus getting
$$\bar{N}=\left(b_1-d_1\right) /\left(b_2+d_2\right)$$
To discuss the stability of this position, we substitute $N(t)=\bar{N}+u(t)$ in Eqn. (120) to get
$$\frac{d u}{d t}=\left(b_1-d_1\right)(\bar{N}+u(t))-b_2(\bar{N}+u(t))\left(\bar{N}+u(t-r)-d_2(\bar{N}+u(t))^2\right.$$
Neglecting squares and products of $u(t), u(t-t)$ and using Eqn. (126), we get the linear delay-differential equation
$$\frac{d u}{d t}=\left(b_1-d_1\right) u(t)-b_2 \bar{N}(u(t)+u(t-r))-2 d_2 \bar{N} u(t)$$
Trying the solution
$$u(t)=A e^{-\lambda t}$$
we get
$$\lambda=\left(b_1-d_1\right)-b_2 \bar{N}\left(1+a^{-\lambda \tau}\right)-2 d_2 \bar{N}$$
This is an equation to solve for $\lambda$, which involves both algebraic and non-algebraic (exponential) functions of $\lambda$. If all of its roots have negative real parts, the equilibrium position is stable.
We may substitute $\lambda=r+i s$ in Eqn. (130) and equate real and imaginary parts of both sides to get two equations in $r$ and $s$. By eliminating $s$ between these two equations, we can get a single equation to determine $r$. If all the roots of this equation are negative real numbers, the equilibrium position is stable.

The same method can be applied to discuss the stability of all equilibrium positions of all delay-differential equation models.

## 数学代写|数学建模代写Mathematical Modeling代考|A Model for Growth of Population Inhibited by Cumulative Effects of Pollution

The rate of growth of a population at any time is inhibited due to the metabolic products produced by populations at all earlier times. The effect of pollution produced by the population between time $t-\tau$ and $t-\tau-d \tau$ on the rate of growth of population at time $t$ may be
$$-c k(\tau) N(t) N(t-\tau) d \tau$$
where $k(\tau)$ is a decreasing function of $\tau$, so that our mathematical model is
$$\frac{d N}{d t}=a N(t)-c N(t) \int_0^{\infty} k(\tau) N(t-\tau) d \tau$$
which can also be written as
$$\frac{d N}{d t}=a N(t)-c N(t) \int_{-\infty}^t k(t-u) N(u) d u$$
Equations (132) and (133) are integro-differential equations.
Models in terms of integro-differential equations arise when our physical principles involves both a rate of change of some function as well as the sum or integral or cumulative effects on that function.

If $N_1(t), N_2(t)$ are the populations of the prey and predator species at time $t$ and the interaction effects are cumulative over time, we get the model
$$\begin{gathered} \frac{d N_1}{d t}=a N_1(t)-b N_1(t) \int_0^{\infty} k_1(\tau) N_2(t-\tau) d \tau \ z \frac{d N_2}{d t}=-p N_2(t)+q N_2(t) \int_0^{\infty} k_2(\tau) N_1(t-\tau) d \tau \end{gathered}$$
The kernel functions are usually monotonic decreasing functions of $\tau$ which can always be normalized to give
$$\int_0^{\infty} k_1(\tau) d \tau=1, \int_0^{\infty} k_2(\tau) d \tau=1$$

## 数学代写|数学建模代写Mathematical Modeling代考|Stability of Equilibrium Positions

$$\bar{N}=\left(b_1-d_1\right) /\left(b_2+d_2\right)$$

$$\frac{d u}{d t}=\left(b_1-d_1\right)(\bar{N}+u(t))-b_2(\bar{N}+u(t))\left(\bar{N}+u(t-r)-d_2(\bar{N}+u(t))^2\right.$$

$$\frac{d u}{d t}=\left(b_1-d_1\right) u(t)-b_2 \bar{N}(u(t)+u(t-r))-2 d_2 \bar{N} u(t)$$

$$u(t)=A e^{-\lambda t}$$

$$\lambda=\left(b_1-d_1\right)-b_2 \bar{N}\left(1+a^{-\lambda \tau}\right)-2 d_2 \bar{N}$$

## 数学代写|数学建模代写Mathematical Modeling代考|A Model for Growth of Population Inhibited by Cumulative Effects of Pollution

$$-c k(\tau) N(t) N(t-\tau) d \tau$$

$$\frac{d N}{d t}=a N(t)-c N(t) \int_0^{\infty} k(\tau) N(t-\tau) d \tau$$

$$\frac{d N}{d t}=a N(t)-c N(t) \int_{-\infty}^t k(t-u) N(u) d u$$

$$\begin{gathered} \frac{d N_1}{d t}=a N_1(t)-b N_1(t) \int_0^{\infty} k_1(\tau) N_2(t-\tau) d \tau \ z \frac{d N_2}{d t}=-p N_2(t)+q N_2(t) \int_0^{\infty} k_2(\tau) N_1(t-\tau) d \tau \end{gathered}$$

$$\int_0^{\infty} k_1(\tau) d \tau=1, \int_0^{\infty} k_2(\tau) d \tau=1$$

## 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。

Posted on Categories:数学代写, 数学建模

## 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！

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## 数学代写|数学建模代写Mathematical Modeling代考|A More General Two-Points Boundary Value Problem

Consider the boundary value problem
$$y^{\prime \prime}+g_1(x) y^{\prime}+g_2(x)=f(x) ; y(0)=0, y(b)=0$$
Let the solution of
$$\begin{gathered} y^{\prime \prime}+g_1(x) y^{\prime}+g_2(x)=0 \ y=A_1 y_1(x)+A_2 y_2(x)=0 \end{gathered}$$
then the solution of
$$y^{\prime \prime}+g_1(x) y^{\prime}+g_2(x) y=\delta(x-\xi) y(0)=0, y(b)=0$$
is given by
\begin{aligned} & y=c_1 y_1(x)+c_2 y_2(x), 0 \leq x \leq \xi \ & y=d_1 y_1(x)+d_2 y_2(x), \xi \leq x \leq b \end{aligned}
where the constants $c_1, c_2, d_1, d_2$ are obtained from the equations
$$\begin{gathered} c_1 y_1(0)+c_2 y_2(0)=0, d_1 y_1(b)+d_2 y_2(b)=0 \ c_1 y_1(\xi)+c_2 y_2(\xi)=d_1 y_1(\xi)+d_2 y_2(\xi) \ c_1 y_1^{\prime}(\xi)+c_2 y_2^{\prime}(\xi)-d_1 y_1^{\prime}(\xi)-d_2 y_2^{\prime}(\xi)=1 \end{gathered}$$
Knowing $c_1, c_2, d_1, d_2$ Eqns. (104) and (105) determine the Green’s function $G(x, \xi)$ for the present problem and then the solution of Eqn. (100) is
$$y(x)=\int_0^b G(x, \xi) f(\xi) d \xi$$

## 数学代写|数学建模代写Mathematical Modeling代考|Integral Equations in Population Dynamics

Knowing
$x p_0=$ Probability of a female of age zero, i.e., a female just born, surviving till age $x$.
$t p_{x-t}=$ Probability of a female of age $x-t$ surviving till age $x(x \geq t)$.
$\lambda(x) \Delta t=$ Average number of births to a female with age between $x$ and $x+\Delta x$.
$F(x, 0)=$ Initial number of females of age $x$ at time $t=0$ it is required to find.
$F(x, t) \Delta x=$ Number of females at time $t$ of ages between $x$ and $x+\Delta x$.
$B(t) \Delta t=$ Number of total female births in time interval $t, t+\Delta t$.
The previous definitions lead to the following relations:
(i) $F(x, t)=B(t-x){ }_x p_0 ; x \leq t$
This follows since $B(t-x)$ denotes the number of females born at time $t-x$ and ${ }_2 p_0$ gives the probability of their surviving for $x$ years to become of age $x$ at time $t$. Thus (110) expresses the fact that the number of females of age $x$ at time $t$ is equal to the number of females born at time $t-x$ who have survived for $x$ years.
(ii) $F(x, t)=F(x-t, 0){ }t p{x-t} x \geq t$
This expresses the fact that the number of females of age $x$ at time $t$ is equal to the number of females of age $x-t$ at time 0 who have survived for $t$ years to become of $x$ years.
(iii) $B(t) \Delta t=\int_\alpha^\beta F(x, t) \lambda(x) d x \Delta t$
where $(\alpha, \beta)$ gives the reproductive age group interval so that
$$\lambda(x)=0 \text {, when } x<\alpha \text { and when } x<\beta$$
Equation (112) expresses the fact that the total number of female births taking place during time interval $(t, t+\Delta t)$ is obtained by summing or integrating the number of female births due to females of all ages in the reproductive age group. In view of Eqn. (113), equation (112) can also be written as
\begin{aligned} B(t) & =\int_0^{\infty} F(x, t) \lambda(x) d x \ & =\int_0^t F(x, t) \lambda(x) d x+\int_t^{\infty} F(x, t) \lambda(x) d x \end{aligned}
Now using Eqns. (110) and (111), we get
\begin{aligned} B(t) & =\int_0^t B(t-x)x p_0 \lambda(x) d x+\int_t^{\infty} F(x-t, 0)_t p{x-t} \lambda(x) d x \ & =\int_0^t B(t-x)_x p_0 \lambda(x) d x+\int_t^{\infty} F(u, 0)_t p_u \lambda(t+u) d u \end{aligned}

## 数学代写|数学建模代写Mathematical Modeling代考|A More General Two-Points Boundary Value Problem

$$y^{\prime \prime}+g_1(x) y^{\prime}+g_2(x)=f(x) ; y(0)=0, y(b)=0$$

$$\begin{gathered} y^{\prime \prime}+g_1(x) y^{\prime}+g_2(x)=0 \ y=A_1 y_1(x)+A_2 y_2(x)=0 \end{gathered}$$

$$y^{\prime \prime}+g_1(x) y^{\prime}+g_2(x) y=\delta(x-\xi) y(0)=0, y(b)=0$$

\begin{aligned} & y=c_1 y_1(x)+c_2 y_2(x), 0 \leq x \leq \xi \ & y=d_1 y_1(x)+d_2 y_2(x), \xi \leq x \leq b \end{aligned}

$$\begin{gathered} c_1 y_1(0)+c_2 y_2(0)=0, d_1 y_1(b)+d_2 y_2(b)=0 \ c_1 y_1(\xi)+c_2 y_2(\xi)=d_1 y_1(\xi)+d_2 y_2(\xi) \ c_1 y_1^{\prime}(\xi)+c_2 y_2^{\prime}(\xi)-d_1 y_1^{\prime}(\xi)-d_2 y_2^{\prime}(\xi)=1 \end{gathered}$$

$$y(x)=\int_0^b G(x, \xi) f(\xi) d \xi$$

## 数学代写|数学建模代写Mathematical Modeling代考|Integral Equations in Population Dynamics

$x p_0=$ 0岁的雌性，即刚出生的雌性，存活到$x$岁的概率。
$t p_{x-t}=$年龄为$x-t$的女性存活到$x(x \geq t)$岁的概率。
$\lambda(x) \Delta t=$年龄在$x$到$x+\Delta x$之间的女性的平均生育数。
$F(x, 0)=$初始年龄女性人数$x$在时间$t=0$需要找到。
$F(x, t) \Delta x=$年龄在$x$到$x+\Delta x$之间的女性人数$t$。
$B(t) \Delta t=$时间间隔内的女性出生总数$t, t+\Delta t$。

(i) $F(x, t)=B(t-x){ }x p_0 ; x \leq t$ 这是因为$B(t-x)$表示在$t-x$时间出生的女性数量，${ }_2 p_0$表示她们在$t$时间存活$x$年达到$x$年龄的概率。因此(110)表达了这样一个事实，即在$t$时间$x$年龄的女性的数量等于在$t-x$时间出生并活了$x$岁的女性的数量。 (ii) $F(x, t)=F(x-t, 0){ }t p{x-t} x \geq t$ 这表明在时间$t$处处于$x$年龄的女性的数量等于在时间0处处于$x-t$年龄的女性从$t$岁活到$x$岁的数量。 (三)$B(t) \Delta t=\int\alpha^\beta F(x, t) \lambda(x) d x \Delta t$
$(\alpha, \beta)$在哪里给出了生育年龄组的间隔
$$\lambda(x)=0 \text {, when } x<\alpha \text { and when } x<\beta$$

\begin{aligned} B(t) & =\int_0^{\infty} F(x, t) \lambda(x) d x \ & =\int_0^t F(x, t) \lambda(x) d x+\int_t^{\infty} F(x, t) \lambda(x) d x \end{aligned}

\begin{aligned} B(t) & =\int_0^t B(t-x)x p_0 \lambda(x) d x+\int_t^{\infty} F(x-t, 0)_t p{x-t} \lambda(x) d x \ & =\int_0^t B(t-x)_x p_0 \lambda(x) d x+\int_t^{\infty} F(u, 0)_t p_u \lambda(t+u) d u \end{aligned}

## 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。

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## 数学代写|数学建模代写Mathematical Modeling代考|Weighted Digraphs and Markov Chains

A Markovian system is characterized by a transition probability matrix. Thus if the states of a system are represented by $1,2, \ldots, n$ and $p_{i j}$ gives the probability of transition from the $i$ th state to the $j$ th state, the system is characterized by the transition probability matrix (TPM)

Since $\sum_{i=1}^n p_{i j}$ represents the probability of the system going from the $i$ th state to any other state or of remaining in the same state, this sum must be equal to unity. Thus the sum of elements of every row of a TPM is unity.

Consider a set of $N$ such Markov systems where $N$ is large and suppose at any instant $N P_1$, $N P_2, \ldots, N P_n$ of these $\left(P_1+P_2+\ldots+P_n=1\right)$ are in states $1,2,3, \ldots, n$ respectively. After one step, let the proportions in these states be denoted by ${P^{\prime}}1, P_2^{\prime}, \ldots, P_n^{\prime}$, then \begin{aligned} & P_1^{\prime}=P_1 P{11}+P_2 P_{21}+P_3 P_{31}+\ldots+P_n P_{n 1} \ & P_2^{\prime}=P_2 P_{12}+P_2 P_{22}+P_3 P_{32}+\ldots+P_n P_{n 2} \ & \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ & P_n^{\prime}=P_1 P_{1 n}+P_2 P_{2 n}+P_3 P_{3 n}+\ldots+P_n P_{n n} \end{aligned}
or
$$P^{\prime}=P T$$
where $P$ and $P^{\prime}$ are row matrices representing the proportions of systems in various states before and after the step and $T$ is the TPM.

## 数学代写|数学建模代写Mathematical Modeling代考|General Communication Networks

So far we have considered communication networks in which the weight associated with a directed edge represents the probability of communication along that edge. We can however have more general networks, e.g.,
(a) for communication of messages where the directed edge represents the channel and the weight represents the capacity of the channel, say in bits per second.
(b) for communication of gas in pipelines where the weights are capacities, say in gallons per hour.
(c) for communication roads where the weights are the capacities in cars per hour.
An interesting problem is to find the maximum flow rate, of whatever is being communicated, from any vertex of the communication network to any other. Useful graph-theoretic algorithms for this have been developed by Elias, Feinstein, and Shannon, as well as by Ford and Fulkerson.

In the most general case, the weight associated with a directed edge can be positive or negative. Thus, Figure 7.21 means that a unit change at vertex 1 at time $t$ causes changes of -2 units at vertex 2 , of 2 units at vertex 4 , and of 3 units at vertex 5 at time $t+1$. Similarly a change of 1 unit at vertex 2 causes a change of -3 units at vertex 3,4 units at vertex 4,2 units at vertex 5 , and so on. Given the values at all vertices at time $\mathrm{t}$, we can find the values at time $t+1, t+2$, $t+3, \ldots$. The process of doing this systematically is known as the pulse rule. These general weighted digraphs are useful for representing energy flows, monetary flows, and changes in environmental conditions.

## 数学代写|数学建模代写Mathematical Modeling代考|Weighted Digraphs and Markov Chains

$$P^{\prime}=P T$$

## 数学代写|数学建模代写Mathematical Modeling代考|General Communication Networks

(a)用于消息通信，其中有向边表示信道，权值表示信道的容量，以比特每秒为单位。
(b)用于管道中气体的传输，其中重量是容量，例如每小时加仑。
(c)交通道路，其重量为每小时车辆的通行能力。

## 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。

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## 数学代写|数学建模代写Mathematical Modeling代考|Qualitative Relations in Applied Mathematics

It has been stated that “Applied mathematics is nothing but the solution of differential equations.” This statement is wrong on many counts: (i) Applied mathematics also deals with solutions of difference, differential-difference, integral, integro-differential, functional, and algebraic equations; (ii) applied mathematics is equally concerned with inequations of all types; (iii) applied mathematics is also concerned with mathematical modeling, in fact mathematical modeling has to precede solution of equations; and $(i v)$ applied mathematics also deals with situations which cannot be modeled in terms of equations or inequations, one such set of situations is concerned with qualitative relations.

Mathematics deals with both quantitative and qualitative relationships. Typical qualitative relations are: $y$ likes $x, y$ hates $x, y$ is superior to $x, y$ is subordinate to $x, y$ belongs to same political party as $x$, set $y$ has a non-null intersection with set $x$; point $y$ is joined to point $x$ by a road, state $y$ can be transformed into state $x$, team $y$ has defeated team $x, y$ is father of $x$, course $y$ is a prerequisite for course $x$, operation $y$ has to be done before operation $x$, species $y$ eats species $x, y$ and $x$ are connected by an airline, $y$ has a healthy influence on $x$, any increase of $y$ leads to a decrease in $x, y$ belongs to same class as $x, y$ and $x$ have different nationalities, and so on.

Such relationships are very conveniently represented by graphs where a graph consists of a set of vertices and edges joining some or all pairs of these vertices. To illustrate the typical problem situations which can be modeled through graphs, we consider the first problem so historically modeled viz. the problem of the seven bridges of Konigsberg.

## 数学代写|数学建模代写Mathematical Modeling代考|The Seven Bridges Problem

There are four land masses $A, B, C, D$ which are connected by seven bridges numbered 1 to 7 across a river (Figure 7.1). The problem is to start from any point in one of the land masses, cover each of the seven bridges once and once only, and return to the starting point.

There are two ways of attacking this problem. One method is to try to solve the problem by walking over the bridges. Hundreds of people tried to do so in their evening walks and failed to find a path satisfying the conditions of the problem. A second method is to draw a scale map of the bridges on paper and try to find a path by using a pencil.

It is at this stage that concepts of mathematical modeling are useful. It is obvious that the sizes of the land masses are unimportant, and the lengths of the bridges or whether these are straight or curved are irrelevant. What is relevant information is that $A$ and $B$ are connected by two bridges 1 and 2, $B$ and $C$ are connected by two bridges 3 and $4, B$ and $D$ are connected by one bridge number $5, A$ and $D$ are connected by bridge number 6 , and $C$ and $D$ are connected by bridge number 7 . All these facts are represented by the graph with four vertices and seven edges in Figure 7.2. If we can trace this graph in such a way that we start with any vertex and return to the same vertex and trace every edge once and once only without lifting the pencil from the paper, the problem can be solved. Again the trial and error method cannot be satisfactorily used to show that no solution is possible.

The number of edges meeting at a vertex is called the degree of that vertex. We note that the degrees of $A, B, C, D$ are $3,5,3,3$ respectively and each of these is an odd number. If we have to start from a vertex and return to it, we need an even number of edges at that vertex. Thus it is easily seen that the Konigsberg bridges problem cannot be solved.

This example also illustrates the power of mathematical modeling. We have not only disposed of the seven bridges problem, but we have discovered a technique for solving many problems of the same type.

## 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。