[Partial differential equation Tutorial Exercise reference solution]3.3 Characteristics and classification of first order equations

1. discriminant equation set $$\bex \sedd{\ba{ll} u_t=a (x,t) u_x-b (x,t) v_x+c_1 (x,t) \ v_t=b (x,t) u_x+a (x,t) v_x+c_2 (x,t) \ea} \eex$$ which type.

Answer: corresponding $A = (A_{ij}) $ for $$\bex a=\sex{\ba{cc}-a&b\\-b&-a \ea}\ra |\lm e-a|= (\lm +a) ^2+b^2>0, \eex$$ and $A $ no real feature Values, the original equations are elliptic.

2. Determine the following first-order equation set $$\bex \sedd{\ba{ll} \cfrac{\p u}{\p T} +a_{11}\cfrac{\p u}{\p x} +a_{12}\cfrac{\p v}{\p x}=0,\\ \cfrac{\p v} {\p T} +a_{21}\cfrac{\p u}{\p x}+a_{22}\cfrac{\p v}{\p x}=0 \ea} \eex$$ type, where $a _{11},a_{12},a_{21},a_{22}$ is constant.

Answer: With the first question, you can figure out $$\bex a=\sex{\ba{cc} a_{11}&a_{12}\\ a_{21}&a_{22} \ea}. \eex$$ therefore, if $A $ no real eigenvalue, then the equation is elliptic type; If $A $ has real eigenvalues and there are two linearly independent eigenvectors, then the equation is hyperbolic and further, if the real eigenvalues of the $A $ are different, the equations are strictly hyperbolic.

3. Proof of acoustic equation set $$\bex \sedd{\ba{ll} \cfrac{\p \rho}{\p T} +\rho_0\sum_{i=1}^3 \cfrac{\p v_i}{\p x_i}=0,\\ \cfrac{\p v_k}{\p T} +\cfrac{p ' (\rho_0)}{\rho_0}\cfrac{\p \rho}{\p x_k}=0,\ k=1,2,3 \ea} \eex$$ is a hyperbolic equation group where $\rho_0>0$, $p ' (\rho) >0$ are Constant.

Answer: Consider the $t $ as the fourth independent variable, consider $\rho$ as the fourth dependent variable, $$\bex \sedd{\ba{ll} \cfrac{p ' (\rho_0)}{\rho_0}\cfrac{\p \rho}{\p X_k} +\cfrac{\p V _k}{\p t}=0,\ k=1,2,3\\ \rho_0\sum_{i=1}^3 \cfrac{\p v_i}{\p x_i}+\cfrac{\p \rho}{\p T} =0. \ea} \eex$$ so $$\bex \varphi (\al_1,\cdots,\al_4) =\SEV{\BA{CCCC} \al_4&&&\frac{p ' (\rho_0)}{\rho_0} \al_1 \ &\al_4&&\frac{p ' (\rho_0)}{\rho_0}\al_2\\ &&\al_4&\frac{p ' (\rho_0)}{\rho_0}\al_3\\ \rho _0\al_1&\rho_0\al_2&\rho_0\al_3&\al_4 \ea} =\al_4^2[\al_4^2-p ' (\rho_0) (\al_1^2+\al_2^2+\al_3^2)]. \eex$$ thus, given the $\al_1,\al_2,\al_3$, the above-mentioned $\al_4$ polynomial has a $4$ real roots, and the equation is hyperbolic.

[Partial differential equation Tutorial Exercise reference solution]3.3 Characteristics and classification of first order equations