Finding the characteristic curves and Riemann Invariants.
I am trying to find the characteristics and Riemann Invariants for the following system:
$$
u_t+uv_x=0;
\
v_t+vu_x=0;
$$
where $u=u(x,t)$ and $v=v(x,t)$
So far I have done the following. We can rewrite this in the following form, $U_t+AU_x=0$ for $U=(u,v)^T$. By doing this we can find the eigenvalues s.t.
$det(A-lambda I)=lambda^2-uv implies lambda=pm sqrt {uv}$. From here we find the eigenvectors as $(1,pm frac{sqrt{uv}}{u})$ with characteristics,
$$
frac{du}{dt}pmfrac{sqrt{uv}}{u}frac{dv}{dt}=0 text{ along } frac{dx(t)}{dt}=frac{sqrt{uv}}{u}
$$
My question here is: Does the $pm frac{sqrt{uv}}{u}$ need to be the inverse in the above? My notes do not make this clear, as it is done on some occasions and note others.
Then to find the Riemann Invariants we integrate w.r.t t to find,
$$
upm left[v^{1.5}u^{0.5}-int frac{v^{0.5}v'}{2u^{0.5}}-frac{v^{1.5}u'}{2u^{1.5}}right]=const.
$$
However, I am very unsure whether this right due to the presence of the integral.
Any tips would be appreciated.
linear-algebra differential-equations pde dynamical-systems characteristics
asked 2 days ago
KieranSQ
748
add a comment |
I am trying to find the characteristics and Riemann Invariants for the following system:
$$
u_t+uv_x=0;
\
v_t+vu_x=0;
$$
where $u=u(x,t)$ and $v=v(x,t)$
So far I have done the following. We can rewrite this in the following form, $U_t+AU_x=0$ for $U=(u,v)^T$. By doing this we can find the eigenvalues s.t.
$det(A-lambda I)=lambda^2-uv implies lambda=pm sqrt {uv}$. From here we find the eigenvectors as $(1,pm frac{sqrt{uv}}{u})$ with characteristics,
$$
frac{du}{dt}pmfrac{sqrt{uv}}{u}frac{dv}{dt}=0 text{ along } frac{dx(t)}{dt}=frac{sqrt{uv}}{u}
$$
My question here is: Does the $pm frac{sqrt{uv}}{u}$ need to be the inverse in the above? My notes do not make this clear, as it is done on some occasions and note others.
Then to find the Riemann Invariants we integrate w.r.t t to find,
$$
upm left[v^{1.5}u^{0.5}-int frac{v^{0.5}v'}{2u^{0.5}}-frac{v^{1.5}u'}{2u^{1.5}}right]=const.
$$
However, I am very unsure whether this right due to the presence of the integral.
Any tips would be appreciated.
linear-algebra differential-equations pde dynamical-systems characteristics
asked 2 days ago
KieranSQ
748
add a comment |
I am trying to find the characteristics and Riemann Invariants for the following system:
$$
u_t+uv_x=0;
\
v_t+vu_x=0;
$$
where $u=u(x,t)$ and $v=v(x,t)$
So far I have done the following. We can rewrite this in the following form, $U_t+AU_x=0$ for $U=(u,v)^T$. By doing this we can find the eigenvalues s.t.
$det(A-lambda I)=lambda^2-uv implies lambda=pm sqrt {uv}$. From here we find the eigenvectors as $(1,pm frac{sqrt{uv}}{u})$ with characteristics,
$$
frac{du}{dt}pmfrac{sqrt{uv}}{u}frac{dv}{dt}=0 text{ along } frac{dx(t)}{dt}=frac{sqrt{uv}}{u}
$$
My question here is: Does the $pm frac{sqrt{uv}}{u}$ need to be the inverse in the above? My notes do not make this clear, as it is done on some occasions and note others.
Then to find the Riemann Invariants we integrate w.r.t t to find,
$$
upm left[v^{1.5}u^{0.5}-int frac{v^{0.5}v'}{2u^{0.5}}-frac{v^{1.5}u'}{2u^{1.5}}right]=const.
$$
However, I am very unsure whether this right due to the presence of the integral.
Any tips would be appreciated.
linear-algebra differential-equations pde dynamical-systems characteristics
asked 2 days ago
KieranSQ
748
I am trying to find the characteristics and Riemann Invariants for the following system:
$$
u_t+uv_x=0;
\
v_t+vu_x=0;
$$
where $u=u(x,t)$ and $v=v(x,t)$
So far I have done the following. We can rewrite this in the following form, $U_t+AU_x=0$ for $U=(u,v)^T$. By doing this we can find the eigenvalues s.t.
$det(A-lambda I)=lambda^2-uv implies lambda=pm sqrt {uv}$. From here we find the eigenvectors as $(1,pm frac{sqrt{uv}}{u})$ with characteristics,
$$
frac{du}{dt}pmfrac{sqrt{uv}}{u}frac{dv}{dt}=0 text{ along } frac{dx(t)}{dt}=frac{sqrt{uv}}{u}
$$
My question here is: Does the $pm frac{sqrt{uv}}{u}$ need to be the inverse in the above? My notes do not make this clear, as it is done on some occasions and note others.
Then to find the Riemann Invariants we integrate w.r.t t to find,
$$
upm left[v^{1.5}u^{0.5}-int frac{v^{0.5}v'}{2u^{0.5}}-frac{v^{1.5}u'}{2u^{1.5}}right]=const.
$$
However, I am very unsure whether this right due to the presence of the integral.
Any tips would be appreciated.
linear-algebra differential-equations pde dynamical-systems characteristics
linear-algebra differential-equations pde dynamical-systems characteristics
asked 2 days ago
KieranSQ
748
asked 2 days ago
KieranSQ
748
edited yesterday
asked 2 days ago
KieranSQ
748
asked 2 days ago
KieranSQ
748
asked 2 days ago
KieranSQ
748
748
add a comment |
add a comment |
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