QQ:541278835|Wechat:TUTOR233

MTH2040 Mathematical modelling 数学建模 莫纳什大学

The mathematical modelling of physical systems is based upon differential equations and linear algebra. This unit will introduce fundamental techniques for studying linear systems and differential equations, focusing on applications to physical systems. The topics in linear algebra to be considered include eigenvalues and eigenvectors, diagonalisation of square matrices, matrix functions, LU-decomposition, applications. The topics in optimisation include Lagrange multipliers, the method of least-squares, linear programming, applications. Finally, the topics in differential equations include matrix solutions of constant coefficient systems of ordinary differential equations, conservative systems and phase-planes of simple non-linear ordinary differential equations. Solutions of first order partial differential equations will be analysed and applied to real world problems. You will be introduced to the Mathematica computer package, and learn how to use it for analytical and numerical calculations and graphics. Mathematica will be integrated into most activities.

Content: Mathematical modelling is the process of mathematically representing real-world scenarios to make predictions and insights. Applying mathematical formulae is different from actually creating mathematical relationships.

MTH2040 Mathematical modelling 数学建模 蒙纳士大学
问题 1.

Theorem 1. Around the equilibrium point $E_1\left(\frac{\Lambda}{\mu}, 0,0,0\right)$ the system (1) is locally asymptotically stable if the inequalities $\frac{p \beta \Lambda}{\mu}<\phi_1+\phi_2$ and $\phi_1 \phi_2>\frac{p \beta \Lambda \phi_1}{\mu}+(1-p) \frac{\alpha \beta \Lambda}{\mu}$ hold simultaneously.

Proof. Around the disease-free equilibrium point $E_1\left(\frac{\Lambda}{\mu}, 0,0,0\right)$, we investigate the eigenvalues of the following Jacobian matrix
$$
J\left(\frac{\Lambda}{\mu}, 0,0,0\right)=\left[\begin{array}{cccc}
-\mu & 0 & -\beta \bar{S} & 0 \
0 & -\phi_1 & (1-p) \beta \bar{S} & 0 \
0 & \alpha & p \beta \bar{S}-\phi_2 & 0 \
0 & \varepsilon_1 & \varepsilon_2 & -\mu
\end{array}\right] .
$$
Around the disease-free equilibrium point $E_1$, the Jacobian matrix (10) takes the following form,
$$
J\left(E_1\right)=\left[\begin{array}{cccc}
-\mu & 0 & -\beta \frac{\Lambda}{\mu} & 0 \
0 & -\phi_1 & (1-p) \beta \frac{\Lambda}{\mu} & 0 \
0 & \alpha & p \beta \frac{\Lambda}{\mu}-\phi_2 & 0 \
0 & \varepsilon_1 & \varepsilon_2 & -\mu
\end{array}\right] .
$$

问题 2.

Let $p_t=$ price of a commodity in the year $t$, and $q_t=$ amount of the commodity available in the market in year $t$. Then we make the following assumptions:

(i) Amount of the commodity produced this year and available for sale is a linear function of the price of the commodity in the last year, i.e.,
$$
q_t=\alpha+\beta p_{t-1}
$$
where $\beta>0$ since if the last year’s price was high, the amount available this year would also be high.
(ii) The price of the commodity this year is a linear function of the amount available this year, i.e.,
$$
p_t=\gamma+\delta q_t
$$
where $\delta<0$, since if $q_t$ is large, the price would be low. From (68) and (69)
$$
p_t-\beta \delta p_{t-1}=\gamma+\alpha \delta
$$
which has the solution
$$
\left(p_t-\frac{\alpha \delta+\gamma}{1-\beta \delta}\right)=\left(p_0-\frac{\alpha \delta+\gamma}{1-\beta \delta}\right)(\beta \delta)^t
$$
so that
$$
\left(p_t-\frac{\alpha \delta+\gamma}{1-\beta \delta}\right)=\left(p_{t-1}-\frac{\alpha \delta+\gamma}{1-\beta \delta}\right)(\beta \delta)
$$

学习资料推荐

针对 Mathematical modelling 推荐三本教材:

✅Mathematical Modelling of Industrial Processes

✅Mathematical modelling of an activated sludge treatment plant

✅ Modelling financial derivatives with Mathematica: mathematical models and benchmark algorithms

数学建模预测介绍:

  • 在研讨会上,来自美国国家环境健康科学研究所的克里斯托弗-波蒂尔强调,人类与环境之间复杂的相互作用要求采用系统方法来理解环境健康和实施环境健康决策。基本需求、住所因素和内源因素等环境因素相互作用,决定了一个人的健康状况。他认为,关于风险评估的一般假设–很容易找出问题的主要原因并制定解决方案–是一种过时的方法。此外,他还指出,关于人体系统以及科学如何解决其暴露于危害的问题,目前正在进行大量的研究和测试,从人口和临床层面到分子层面。虽然所有这些科学都有助于了解环境对健康的影响,但大多数风险评估都是以毒理学和流行病学证据为基础,而不是以遗传学和毒物基因组学等新兴科学为基础。因此,科学家和决策者需要关注新兴科学领域,找到将这些研究纳入环境健康决策过程的方法。