Solving for integrand from integrated quantities.
$begingroup$
Given equations of the form:
$A(r) = int_{t_{1}}^{t_{2}}F(r,t)dt$
$B(t) = int_a^b F(r,t)r^2dr$
where $A(r)$, $B(t)$, and all of the limits on the integrals are known, is there enough information to solve for F(r,t)? If so how would one do this?
For more context this is a scenario where there is a quantity, $F(r,t)$, that varies in space and time but is measured as only a function of time and a function of space separately. I am trying to figure out if the full space and time dependence can be reconstructed from these two measurements alone.
EDIT:
Perhaps a better way of stating the question.
Is $F(r,t)$ uniquely constrained given $A(r)$ and $B(t)$?
integral-equations constraints
asked Dec 8 '18 at 16:47
JJR4JJR4
12
$endgroup$
add a comment |
$begingroup$
Given equations of the form:
$A(r) = int_{t_{1}}^{t_{2}}F(r,t)dt$
$B(t) = int_a^b F(r,t)r^2dr$
where $A(r)$, $B(t)$, and all of the limits on the integrals are known, is there enough information to solve for F(r,t)? If so how would one do this?
For more context this is a scenario where there is a quantity, $F(r,t)$, that varies in space and time but is measured as only a function of time and a function of space separately. I am trying to figure out if the full space and time dependence can be reconstructed from these two measurements alone.
EDIT:
Perhaps a better way of stating the question.
Is $F(r,t)$ uniquely constrained given $A(r)$ and $B(t)$?
integral-equations constraints
asked Dec 8 '18 at 16:47
JJR4JJR4
12
$endgroup$
$begingroup$
Do you have the actual function $F$? Looks like something to do with the second moment of mass/area...
$endgroup$
– Karn Watcharasupat
Dec 8 '18 at 16:58
$begingroup$
I do not have the function F. It is actually an x-ray flux of a plasma with unknown temperature and density distributions (which is what would set F).
$endgroup$
– JJR4
Dec 8 '18 at 17:01
add a comment |
$begingroup$
Given equations of the form:
$A(r) = int_{t_{1}}^{t_{2}}F(r,t)dt$
$B(t) = int_a^b F(r,t)r^2dr$
where $A(r)$, $B(t)$, and all of the limits on the integrals are known, is there enough information to solve for F(r,t)? If so how would one do this?
For more context this is a scenario where there is a quantity, $F(r,t)$, that varies in space and time but is measured as only a function of time and a function of space separately. I am trying to figure out if the full space and time dependence can be reconstructed from these two measurements alone.
EDIT:
Perhaps a better way of stating the question.
Is $F(r,t)$ uniquely constrained given $A(r)$ and $B(t)$?
integral-equations constraints
asked Dec 8 '18 at 16:47
JJR4JJR4
12
$endgroup$
Given equations of the form:
$A(r) = int_{t_{1}}^{t_{2}}F(r,t)dt$
$B(t) = int_a^b F(r,t)r^2dr$
where $A(r)$, $B(t)$, and all of the limits on the integrals are known, is there enough information to solve for F(r,t)? If so how would one do this?
For more context this is a scenario where there is a quantity, $F(r,t)$, that varies in space and time but is measured as only a function of time and a function of space separately. I am trying to figure out if the full space and time dependence can be reconstructed from these two measurements alone.
EDIT:
Perhaps a better way of stating the question.
Is $F(r,t)$ uniquely constrained given $A(r)$ and $B(t)$?
integral-equations constraints
integral-equations constraints
asked Dec 8 '18 at 16:47
JJR4JJR4
12
asked Dec 8 '18 at 16:47
JJR4JJR4
12
edited Jan 7 at 15:40
JJR4
asked Dec 8 '18 at 16:47
JJR4JJR4
12
asked Dec 8 '18 at 16:47
JJR4JJR4
12
asked Dec 8 '18 at 16:47
JJR4JJR4
12
12
$begingroup$
Do you have the actual function $F$? Looks like something to do with the second moment of mass/area...
$endgroup$
– Karn Watcharasupat
Dec 8 '18 at 16:58
$begingroup$
I do not have the function F. It is actually an x-ray flux of a plasma with unknown temperature and density distributions (which is what would set F).
$endgroup$
– JJR4
Dec 8 '18 at 17:01
add a comment |
$begingroup$
Do you have the actual function $F$? Looks like something to do with the second moment of mass/area...
$endgroup$
– Karn Watcharasupat
Dec 8 '18 at 16:58
$begingroup$
I do not have the function F. It is actually an x-ray flux of a plasma with unknown temperature and density distributions (which is what would set F).
$endgroup$
– JJR4
Dec 8 '18 at 17:01
$begingroup$
Do you have the actual function $F$? Looks like something to do with the second moment of mass/area...
$endgroup$
– Karn Watcharasupat
Dec 8 '18 at 16:58
$begingroup$
Do you have the actual function $F$? Looks like something to do with the second moment of mass/area...
$endgroup$
– Karn Watcharasupat
Dec 8 '18 at 16:58
$begingroup$
I do not have the function F. It is actually an x-ray flux of a plasma with unknown temperature and density distributions (which is what would set F).
$endgroup$
– JJR4
Dec 8 '18 at 17:01
$begingroup$
I do not have the function F. It is actually an x-ray flux of a plasma with unknown temperature and density distributions (which is what would set F).
$endgroup$
– JJR4
Dec 8 '18 at 17:01
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
What I would do is to find
$$A(r,t) = int_{t_1}^t F(r,t) dt$$
so that
$$A'_t(r,t) = F(r,t)$$
Hence,
$$A'_t(r,t_1)=F(r,t_1), A'_t(r,t_2)=F(r,t_2), A'_t(r,t_3)=F(r,t_3), dots$$
which means you should be able reconstruct numerically the function $F$ by finding $A$ as a function of $r$ and different time $t_i$ endpoints if you have enough data taken and are able to find a time gradient of $A$. We can also apply an analogous trick to $B$ and make sure the two reconstructions match.
answered Dec 8 '18 at 17:09
Karn WatcharasupatKarn Watcharasupat
3,9642526
$endgroup$
$begingroup$
Although this is a good suggestion unfortunately the measurement is such that it is not a tenable option.
$endgroup$
– JJR4
Dec 10 '18 at 13:39
add a comment |
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1 Answer
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active
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votes
1 Answer
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active
oldest
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votes
$begingroup$
What I would do is to find
$$A(r,t) = int_{t_1}^t F(r,t) dt$$
so that
$$A'_t(r,t) = F(r,t)$$
Hence,
$$A'_t(r,t_1)=F(r,t_1), A'_t(r,t_2)=F(r,t_2), A'_t(r,t_3)=F(r,t_3), dots$$
which means you should be able reconstruct numerically the function $F$ by finding $A$ as a function of $r$ and different time $t_i$ endpoints if you have enough data taken and are able to find a time gradient of $A$. We can also apply an analogous trick to $B$ and make sure the two reconstructions match.
answered Dec 8 '18 at 17:09
Karn WatcharasupatKarn Watcharasupat
3,9642526
$endgroup$
$begingroup$
Although this is a good suggestion unfortunately the measurement is such that it is not a tenable option.
$endgroup$
– JJR4
Dec 10 '18 at 13:39
add a comment |
$begingroup$
What I would do is to find
$$A(r,t) = int_{t_1}^t F(r,t) dt$$
so that
$$A'_t(r,t) = F(r,t)$$
Hence,
$$A'_t(r,t_1)=F(r,t_1), A'_t(r,t_2)=F(r,t_2), A'_t(r,t_3)=F(r,t_3), dots$$
which means you should be able reconstruct numerically the function $F$ by finding $A$ as a function of $r$ and different time $t_i$ endpoints if you have enough data taken and are able to find a time gradient of $A$. We can also apply an analogous trick to $B$ and make sure the two reconstructions match.
answered Dec 8 '18 at 17:09
Karn WatcharasupatKarn Watcharasupat
3,9642526
$endgroup$
$begingroup$
Although this is a good suggestion unfortunately the measurement is such that it is not a tenable option.
$endgroup$
– JJR4
Dec 10 '18 at 13:39
add a comment |
$begingroup$
What I would do is to find
$$A(r,t) = int_{t_1}^t F(r,t) dt$$
so that
$$A'_t(r,t) = F(r,t)$$
Hence,
$$A'_t(r,t_1)=F(r,t_1), A'_t(r,t_2)=F(r,t_2), A'_t(r,t_3)=F(r,t_3), dots$$
which means you should be able reconstruct numerically the function $F$ by finding $A$ as a function of $r$ and different time $t_i$ endpoints if you have enough data taken and are able to find a time gradient of $A$. We can also apply an analogous trick to $B$ and make sure the two reconstructions match.
answered Dec 8 '18 at 17:09
Karn WatcharasupatKarn Watcharasupat
3,9642526
$endgroup$
What I would do is to find
$$A(r,t) = int_{t_1}^t F(r,t) dt$$
so that
$$A'_t(r,t) = F(r,t)$$
Hence,
$$A'_t(r,t_1)=F(r,t_1), A'_t(r,t_2)=F(r,t_2), A'_t(r,t_3)=F(r,t_3), dots$$
which means you should be able reconstruct numerically the function $F$ by finding $A$ as a function of $r$ and different time $t_i$ endpoints if you have enough data taken and are able to find a time gradient of $A$. We can also apply an analogous trick to $B$ and make sure the two reconstructions match.
answered Dec 8 '18 at 17:09
Karn WatcharasupatKarn Watcharasupat
3,9642526
answered Dec 8 '18 at 17:09
Karn WatcharasupatKarn Watcharasupat
3,9642526
answered Dec 8 '18 at 17:09
Karn WatcharasupatKarn Watcharasupat
3,9642526
answered Dec 8 '18 at 17:09
Karn WatcharasupatKarn Watcharasupat
3,9642526
3,9642526
$begingroup$
Although this is a good suggestion unfortunately the measurement is such that it is not a tenable option.
$endgroup$
– JJR4
Dec 10 '18 at 13:39
add a comment |
$begingroup$
Although this is a good suggestion unfortunately the measurement is such that it is not a tenable option.
$endgroup$
– JJR4
Dec 10 '18 at 13:39
$begingroup$
Although this is a good suggestion unfortunately the measurement is such that it is not a tenable option.
$endgroup$
– JJR4
Dec 10 '18 at 13:39
$begingroup$
Although this is a good suggestion unfortunately the measurement is such that it is not a tenable option.
$endgroup$
– JJR4
Dec 10 '18 at 13:39
add a comment |
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$begingroup$
Do you have the actual function $F$? Looks like something to do with the second moment of mass/area...
$endgroup$
– Karn Watcharasupat
Dec 8 '18 at 16:58
$begingroup$
I do not have the function F. It is actually an x-ray flux of a plasma with unknown temperature and density distributions (which is what would set F).
$endgroup$
– JJR4
Dec 8 '18 at 17:01