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  • THE ELEMENTARY MATHEMATICAL MODELS AND BASIC CONCEPTS OF MATHEMATICAL MODELING

Principles of Mathematical Modelling: Ideas, Methods, Examples

Alexander A. Samarskii (Author); Alexander P. Mikhailov

Chapter 1

THE ELEMENTARY MATHEMATICAL MODELS AND BASIC CONCEPTS OF MATHEMATICAL MODELING - all with Video Answers

Educators

Section 1

Elementary Mathematical Models

08:40

Problem 1

In the first problem of Subsection 1a to obtain the value of $V$ (speed of the system "a bullet-load" immediately after the collision) apply the law of conservation of momentum instead of the law of conservation of energy. Be sure, that for the speed of the bullet $v$ one gets a value smaller by a factor $((M+m) / m)^{1 / 2}$, than predicted by formula (1).

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 2

Let the power of the laser drilling a material (Subsection 1a), depends on time: $W=W(t)$. How will the formula (3) be changed? Will the conclusion that the depth of the hole is proportional to the spent energy remain valid?

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06:35

Problem 3

Find the moment in time when the last atom of a radioactive substance (Subsection 1b) will decay. In the model (5), why the matter is disintegrated completely only at $t \rightarrow \infty$ ?

Nicholas Majtenyi
Nicholas Majtenyi
Numerade Educator

Problem 4

Assume that in Subsection 1c an "ideal" one-stage rocket is considered, which is dropping the unnecessary fraction of its structural mass, (at the moment of complete combustion of the fuel $m_s=0$ ). Using the law of conservation of momentum, show that the maximal speed of such a rocket is determined by the formula $v=(1-\lambda) u \ln \left(m_0 / m_p\right)$. Compare it with the formula (6). Why can the ideal rocket reach any speed?

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00:25

Problem 5

Check, with the use of (8), that the condition (7) is a condition for minimal value of $t(a)$. From Fig. 5 determine which speed is more $-v_a$ or $v_b$ ? Using the formula (9), find out at what angles the light beam does not penetrate from medium $a$ into medium $b$, i.e. when the effect "of complete internal reflection of light" used in a number of technical devices is realized.

Mayukh Banik
Mayukh Banik
Numerade Educator
08:04

Problem 6

Determine the behavior of $\tau(t)=\alpha(t)-\beta(t)>0$ at large $t$ in Malthus model (10), in order the keep the population finite at $t \rightarrow \infty$.

Amy Jiang
Amy Jiang
Numerade Educator

Problem 7

In Eq. (11) for the multistage rocket take the limit $n \rightarrow \infty$, be convinced that its limiting speed is determined via the formula for an ideal rocket from exercise 4. Why do their results coincide?

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Problem 8

In logistic model (12) consider small deviations from the equilibrium, i.e. when the solution is $N(t)=N_p+\delta N(t)$, where $|\delta N(t)| \ll N p$. Show that for $\delta N(t)$ in the first approximation the linear Malthus model (10) is valid.

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