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Principles of Mathematical Modelling: Ideas, Methods, Examples

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

Chapter 5

STUDY OF MATHEMATICAL MODELS - all with Video Answers

Educators

Section 1

Application of Similarity Methods

02:13

Problem 1

Check by a direct substitution that the equation (4) at $c=c_0, \kappa=\kappa_0 T^\sigma$ is an invariant relative transformation of expansion-compression under the condition of $\alpha \beta^{-2} \gamma^\sigma=1$.

Narayan Hari
Narayan Hari
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Problem 2

Using (6), prove that on a shock not moving along the mass of gas (tangent shock, $D=0$ ), the velocity and pressure are continuous.

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04:48

Problem 3

The shock wave for which $p_1 \gg p_0$, is called strong. Obtain from (6) the Hugoniot condition for the density of such a shock (in case of ideal gas $\varepsilon=$ $p /((\gamma-1) \rho))$, assuming for simplicity that $v_0=0$. Show that on a strong shock wave the density jump does not depend on $D$ and is given by the formula $\rho_1 / \rho_0=$ $(\gamma+1) /(\gamma-1)$.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator

Problem 4

Establish that at $\sigma \geq 1$ the first derivatives of solution (9) by $x$ and $t$ are discontinuous in the point $\bar{x}=x_{\Phi}(t)$, while at $\sigma \geq 1 / 2$ the second derivatives are discontinuous.

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Problem 5

Check that both the change of variables $\eta=\ln \xi, \eta=\xi^{2 / \sigma} \varphi(\eta), \psi=d \varphi / d \eta$ or the replacement $\varphi=-\xi f^{-1} f^\sigma d f / d \xi, \psi=-\xi f^{1-\sigma} /(d f / d \xi)$ reduces $(13)$ to a first order equation of a form $d \psi / d \varphi=\psi F(\psi, \varphi) /(\varphi \Phi(\psi, \varphi))$ (in the second case the equation is easier to investigate, since the functions $F, \Phi$ contain simple nonlinearity of a form $\psi \varphi$ ).

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01:01

Problem 6

The equation (13) at $n=1 / \sigma$ can be rewritten as $\left(f^{\sigma+1}\right)^{\prime \prime}=-(\sigma+1) n f$. Using replacements $u=f^{\sigma+1}, u^{\prime}=v$, reduce its order and integrate the obtained equation.

Raj Bala
Raj Bala
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Problem 7

Be convinced using (22), that at $n<-1 / \sigma$ for any point $m_0<\infty$ is valid $\xi\left(m_0, t\right) \rightarrow 0, t \rightarrow 0$ and $p\left(m_0, t\right) \rightarrow p(0, t) \rightarrow \infty, t \rightarrow 0$.

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