Coupled first order PDEs using characteristics
$begingroup$
I am trying to solve a system like this:
begin{equation}
frac{partial u}{partial t} + alpha frac{partial u}{partial x} = f(x,t)u + h(x,t)v
end{equation}
begin{equation}
frac{partial v}{partial t} + beta frac{partial v}{partial x} = g(x,t)v + h(x,t)u
end{equation}
Using the method of characteristics. I have found it is not possible to intgrate both equations along different characteristic lines, so I am considering the special case where $alpha = beta$, which leads to the following set of equations
begin{equation}
frac{dt(eta,xi)}{dxi} = 1,quad frac{dx(eta,xi)}{dxi} = alpha, quad frac{du(eta,xi)}{dxi}=f(eta,xi)u + h(eta,xi)v,quad frac{du(eta,xi)}{dxi}=f(eta,xi)u + h(eta,xi)v
end{equation}
by choosing $xi(t=0)=0$ and $eta = x - xi$, we arrive at $xi = alpha t$ and $eta = x-alpha t$. My questions is, can the coupled system of ODEs in terms of $xi$ be solved using standard methods (e.g. matrix methods)?
pde
asked Dec 30 '18 at 23:18
OscarNievesOscarNieves
464
$endgroup$
add a comment |
$begingroup$
I am trying to solve a system like this:
begin{equation}
frac{partial u}{partial t} + alpha frac{partial u}{partial x} = f(x,t)u + h(x,t)v
end{equation}
begin{equation}
frac{partial v}{partial t} + beta frac{partial v}{partial x} = g(x,t)v + h(x,t)u
end{equation}
Using the method of characteristics. I have found it is not possible to intgrate both equations along different characteristic lines, so I am considering the special case where $alpha = beta$, which leads to the following set of equations
begin{equation}
frac{dt(eta,xi)}{dxi} = 1,quad frac{dx(eta,xi)}{dxi} = alpha, quad frac{du(eta,xi)}{dxi}=f(eta,xi)u + h(eta,xi)v,quad frac{du(eta,xi)}{dxi}=f(eta,xi)u + h(eta,xi)v
end{equation}
by choosing $xi(t=0)=0$ and $eta = x - xi$, we arrive at $xi = alpha t$ and $eta = x-alpha t$. My questions is, can the coupled system of ODEs in terms of $xi$ be solved using standard methods (e.g. matrix methods)?
pde
asked Dec 30 '18 at 23:18
OscarNievesOscarNieves
464
$endgroup$
$begingroup$
I don't think that you can solve that by characterisrics method. If one of them is free then you may can solve it, otherwise you can find a solution in function of resolvent but it is quite difficult.
$endgroup$
– Gustave
Dec 30 '18 at 23:35
add a comment |
$begingroup$
I am trying to solve a system like this:
begin{equation}
frac{partial u}{partial t} + alpha frac{partial u}{partial x} = f(x,t)u + h(x,t)v
end{equation}
begin{equation}
frac{partial v}{partial t} + beta frac{partial v}{partial x} = g(x,t)v + h(x,t)u
end{equation}
Using the method of characteristics. I have found it is not possible to intgrate both equations along different characteristic lines, so I am considering the special case where $alpha = beta$, which leads to the following set of equations
begin{equation}
frac{dt(eta,xi)}{dxi} = 1,quad frac{dx(eta,xi)}{dxi} = alpha, quad frac{du(eta,xi)}{dxi}=f(eta,xi)u + h(eta,xi)v,quad frac{du(eta,xi)}{dxi}=f(eta,xi)u + h(eta,xi)v
end{equation}
by choosing $xi(t=0)=0$ and $eta = x - xi$, we arrive at $xi = alpha t$ and $eta = x-alpha t$. My questions is, can the coupled system of ODEs in terms of $xi$ be solved using standard methods (e.g. matrix methods)?
pde
asked Dec 30 '18 at 23:18
OscarNievesOscarNieves
464
$endgroup$
I am trying to solve a system like this:
begin{equation}
frac{partial u}{partial t} + alpha frac{partial u}{partial x} = f(x,t)u + h(x,t)v
end{equation}
begin{equation}
frac{partial v}{partial t} + beta frac{partial v}{partial x} = g(x,t)v + h(x,t)u
end{equation}
Using the method of characteristics. I have found it is not possible to intgrate both equations along different characteristic lines, so I am considering the special case where $alpha = beta$, which leads to the following set of equations
begin{equation}
frac{dt(eta,xi)}{dxi} = 1,quad frac{dx(eta,xi)}{dxi} = alpha, quad frac{du(eta,xi)}{dxi}=f(eta,xi)u + h(eta,xi)v,quad frac{du(eta,xi)}{dxi}=f(eta,xi)u + h(eta,xi)v
end{equation}
by choosing $xi(t=0)=0$ and $eta = x - xi$, we arrive at $xi = alpha t$ and $eta = x-alpha t$. My questions is, can the coupled system of ODEs in terms of $xi$ be solved using standard methods (e.g. matrix methods)?
pde
pde
asked Dec 30 '18 at 23:18
OscarNievesOscarNieves
464
asked Dec 30 '18 at 23:18
OscarNievesOscarNieves
464
asked Dec 30 '18 at 23:18
OscarNievesOscarNieves
464
asked Dec 30 '18 at 23:18
OscarNievesOscarNieves
464
asked Dec 30 '18 at 23:18
OscarNievesOscarNieves
464
464
$begingroup$
I don't think that you can solve that by characterisrics method. If one of them is free then you may can solve it, otherwise you can find a solution in function of resolvent but it is quite difficult.
$endgroup$
– Gustave
Dec 30 '18 at 23:35
add a comment |
$begingroup$
I don't think that you can solve that by characterisrics method. If one of them is free then you may can solve it, otherwise you can find a solution in function of resolvent but it is quite difficult.
$endgroup$
– Gustave
Dec 30 '18 at 23:35
$begingroup$
I don't think that you can solve that by characterisrics method. If one of them is free then you may can solve it, otherwise you can find a solution in function of resolvent but it is quite difficult.
$endgroup$
– Gustave
Dec 30 '18 at 23:35
$begingroup$
I don't think that you can solve that by characterisrics method. If one of them is free then you may can solve it, otherwise you can find a solution in function of resolvent but it is quite difficult.
$endgroup$
– Gustave
Dec 30 '18 at 23:35
add a comment |
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$begingroup$
I don't think that you can solve that by characterisrics method. If one of them is free then you may can solve it, otherwise you can find a solution in function of resolvent but it is quite difficult.
$endgroup$
– Gustave
Dec 30 '18 at 23:35