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# 数学代写|数学建模代写Mathematical Modeling代考|TMA4195 Use of analogies in the construction of models

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## 数学代写|数学建模代写Mathematical Modeling代考|Use of analogies in the construction of models

Use of analogies in the construction of models. In plenty of cases where one is attempting to construct a model of a given object it is either impossible to specify directly the sought fundamental laws or variational principles, or, from the point of view of our present knowledge, there is no confidence in the existence of such laws admitting mathematical formulation. One of the fruitful approaches to such objects is to use analogies with already investigated phenomena. Indeed, what can be common between radioactive decay and the dynamics of populations, in particular, the change in the population of our planet? Even at the elementary level such an analogy is quite visible, as it is clear for one of the simplest models of population – the Malthus model. It is based on the simple assumption that the speed of change of the population in time $t$ is proportional to its current number $N(t)$, multiplied on the sum of factors of the birth $\alpha(t) \geq 0$ and the death rate $\beta(t) \leq 0$. As a result one comes to the equation
$$\frac{d N(t)}{d t}=[\alpha(t)-\beta(t)] N(t),$$
which is rather similar to the equation of radioactive decay and coinciding with it at $\alpha<\beta$ (if $\alpha$ and $\beta$ are constants). It is not surprising, since identical assumptions were made for their derivation. The integration of the equation (10) gives
$$N(t)=N(0) \exp \left(\int_{t_0}^t[\alpha(t)-\beta(t)] d t\right),$$
where $N(0)=N\left(t=t_0\right)$ is the initial population.

## 数学代写|数学建模代写Mathematical Modeling代考|Hierarchical approach to the construction of models

1. Hierarchical approach to the construction of models. Only in rare cases it is convenient and justified to construct complete mathematical models at once, even of quite simple objects, in view of all the factors essential for their behavior. Therefore it is natural to proceed in accordance to the principle “from the simple to the complex”, when the following step is made after the detailed study of models which are not too complex. Then, a chain (hierarchy) of more and more complete models is appearing, each of which generalizes the previous ones, including the former as a particular case.

Let us construct such a hierarchical chain on an example of a model of a multistage rocket. As was established at the end of Section 1, a real onestage rocket is unable to develop the first space speed. The reason is due to the amount of fuel to be used for the speeding up of the unnecessary parts of the structural mass of the rocket. Hence, with a movement of a rocket it is necessary to periodically get rid of a ballast. In terms of practical design it means that the rocket consists of several stages, which are discarded in the process of their use.

Let $m_i$ be the total mass of $i$-th stage, $\lambda m_i$ be the corresponding structural mass (so that the fuel mass is $\left.(1-\lambda) m_i\right), m_p$ be the mass of the useful loading. The value of $\lambda$ and speed of the escape of gases $u$ are the same for all stages. Consider for clarity the number of stages $n=3$. The initial mass of such a rocket is equal
$$m_0=m_p+m_1+m_2+m_3 .$$
Consider the moment when all the fuel of the first stage is spent and the mass of the rocket is equal
$$m_p+\lambda m_1+m_2+m_3 .$$
Then by the formula (6) of the initial model, the speed of the rocket equals
$$v_1=u \ln \left(\frac{m_0}{m_p+\lambda m_1+m_2+m_3}\right) .$$

## 数学代写|数学建模代写Mathematical Modeling代考|Use of analogies in the construction of models

$$\frac{d N(t)}{d t}=[\alpha(t)-\beta(t)] N(t)$$

$$N(t)=N(0) \exp \left(\int_{t_0}^t[\alpha(t)-\beta(t)] d t\right),$$

## 数学代写|数学建模代写Mathematical Modeling代考|Hierarchical approach to the construction of models

$$m_0=m_p+m_1+m_2+m_3$$

$$m_p+\lambda m_1+m_2+m_3$$

$$v_1=u \ln \left(\frac{m_0}{m_p+\lambda m_1+m_2+m_3}\right) .$$

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