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Numerical integration with boundary conditions
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I have $N$ equally spaced samples (hundreds or thousands) of the second derivative of a function. I do not have derivatives of any other order. The second derivative has occasional rapid changes (almost-jumps).
The samples are noisy. A repeated Newton-Cotes integration therefore tends to produce offsets between the beginning and end, as errors tend to accumulate quadratically.
I can choose my sample interval such that the desired function is effectively a constant zero both at the beginning and end.
Is there a numerical integration method that can take these boundary conditions into account? Ideally I wouldn’t mind doing the whole double integration in one step, but I’ll settle for a method that does a single integration (I can then repeat it).
numerical-methods
asked Jul 23 at 9:30
user3139990
132
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up vote
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down vote
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I have $N$ equally spaced samples (hundreds or thousands) of the second derivative of a function. I do not have derivatives of any other order. The second derivative has occasional rapid changes (almost-jumps).
The samples are noisy. A repeated Newton-Cotes integration therefore tends to produce offsets between the beginning and end, as errors tend to accumulate quadratically.
I can choose my sample interval such that the desired function is effectively a constant zero both at the beginning and end.
Is there a numerical integration method that can take these boundary conditions into account? Ideally I wouldn’t mind doing the whole double integration in one step, but I’ll settle for a method that does a single integration (I can then repeat it).
numerical-methods
asked Jul 23 at 9:30
user3139990
132
1
Integrate once and you get the first derivative plus an unknown constant $c_1$. Integrate again and you get the function plus an unknown linear term $c_1x+c_2$. Solve for $c_1$ and $c_2$ using the two boundary conditions.
– Rahul
Jul 23 at 9:45
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have $N$ equally spaced samples (hundreds or thousands) of the second derivative of a function. I do not have derivatives of any other order. The second derivative has occasional rapid changes (almost-jumps).
The samples are noisy. A repeated Newton-Cotes integration therefore tends to produce offsets between the beginning and end, as errors tend to accumulate quadratically.
I can choose my sample interval such that the desired function is effectively a constant zero both at the beginning and end.
Is there a numerical integration method that can take these boundary conditions into account? Ideally I wouldn’t mind doing the whole double integration in one step, but I’ll settle for a method that does a single integration (I can then repeat it).
numerical-methods
asked Jul 23 at 9:30
user3139990
132
I have $N$ equally spaced samples (hundreds or thousands) of the second derivative of a function. I do not have derivatives of any other order. The second derivative has occasional rapid changes (almost-jumps).
The samples are noisy. A repeated Newton-Cotes integration therefore tends to produce offsets between the beginning and end, as errors tend to accumulate quadratically.
I can choose my sample interval such that the desired function is effectively a constant zero both at the beginning and end.
Is there a numerical integration method that can take these boundary conditions into account? Ideally I wouldn’t mind doing the whole double integration in one step, but I’ll settle for a method that does a single integration (I can then repeat it).
numerical-methods
asked Jul 23 at 9:30
user3139990
132
asked Jul 23 at 9:30
user3139990
132
asked Jul 23 at 9:30
user3139990
132
asked Jul 23 at 9:30
user3139990
132
132
1
Integrate once and you get the first derivative plus an unknown constant $c_1$. Integrate again and you get the function plus an unknown linear term $c_1x+c_2$. Solve for $c_1$ and $c_2$ using the two boundary conditions.
– Rahul
Jul 23 at 9:45
add a comment |Â
1
Integrate once and you get the first derivative plus an unknown constant $c_1$. Integrate again and you get the function plus an unknown linear term $c_1x+c_2$. Solve for $c_1$ and $c_2$ using the two boundary conditions.
– Rahul
Jul 23 at 9:45
1
1
Integrate once and you get the first derivative plus an unknown constant $c_1$. Integrate again and you get the function plus an unknown linear term $c_1x+c_2$. Solve for $c_1$ and $c_2$ using the two boundary conditions.
– Rahul
Jul 23 at 9:45
Integrate once and you get the first derivative plus an unknown constant $c_1$. Integrate again and you get the function plus an unknown linear term $c_1x+c_2$. Solve for $c_1$ and $c_2$ using the two boundary conditions.
– Rahul
Jul 23 at 9:45
add a comment |Â
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1
Integrate once and you get the first derivative plus an unknown constant $c_1$. Integrate again and you get the function plus an unknown linear term $c_1x+c_2$. Solve for $c_1$ and $c_2$ using the two boundary conditions.
– Rahul
Jul 23 at 9:45