ODE numerical method that produces a region containing the integral curve
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Are there numerical methods for ODEs that are capable of producing regions (linearly bounded or otherwise) that definitely contain a portion of a specified integral curve?
I'm wondering if there are numerical methods for solving ODEs that are roughly analogous to interval arithmetic for real-valued functions.
Suppose we have two unknowns $y(x)$ and $x$ where $y' = F(x, y)$. Suppose that $(x_0, y_0)$ is our initial value and suppose that $-1 le x le 1$ is the range of values of $x$ that we're interested in. In other words, $(x_0, y_0)$ is how we pick out a particular integral curve and $S stackrel{text{def}}{=} left{ (x,y) : x, y in mathbb{R} land -1 le x le 1 right}$ is the region of the $xy$ plane that we want to pick a subset of.
Are there any methods that produce a subset of $S$ that the integral curve passing through $(x_0, y_0)$ must reside in that is smaller than $S$ itself, preferably one where the area covered by the estimate region can be made arbitrarily small by picking a step size or something similar?
ordinary-differential-equations numerical-methods
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
697512
$endgroup$
add a comment |
$begingroup$
Are there numerical methods for ODEs that are capable of producing regions (linearly bounded or otherwise) that definitely contain a portion of a specified integral curve?
I'm wondering if there are numerical methods for solving ODEs that are roughly analogous to interval arithmetic for real-valued functions.
Suppose we have two unknowns $y(x)$ and $x$ where $y' = F(x, y)$. Suppose that $(x_0, y_0)$ is our initial value and suppose that $-1 le x le 1$ is the range of values of $x$ that we're interested in. In other words, $(x_0, y_0)$ is how we pick out a particular integral curve and $S stackrel{text{def}}{=} left{ (x,y) : x, y in mathbb{R} land -1 le x le 1 right}$ is the region of the $xy$ plane that we want to pick a subset of.
Are there any methods that produce a subset of $S$ that the integral curve passing through $(x_0, y_0)$ must reside in that is smaller than $S$ itself, preferably one where the area covered by the estimate region can be made arbitrarily small by picking a step size or something similar?
ordinary-differential-equations numerical-methods
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
697512
$endgroup$
1
$begingroup$
Definitely yes (and based on interval arithmetic), although it's a somewhat tricky problem. Try searching for "validated numerics ODE".
$endgroup$
– Hans Lundmark
Jan 8 at 9:01
add a comment |
$begingroup$
Are there numerical methods for ODEs that are capable of producing regions (linearly bounded or otherwise) that definitely contain a portion of a specified integral curve?
I'm wondering if there are numerical methods for solving ODEs that are roughly analogous to interval arithmetic for real-valued functions.
Suppose we have two unknowns $y(x)$ and $x$ where $y' = F(x, y)$. Suppose that $(x_0, y_0)$ is our initial value and suppose that $-1 le x le 1$ is the range of values of $x$ that we're interested in. In other words, $(x_0, y_0)$ is how we pick out a particular integral curve and $S stackrel{text{def}}{=} left{ (x,y) : x, y in mathbb{R} land -1 le x le 1 right}$ is the region of the $xy$ plane that we want to pick a subset of.
Are there any methods that produce a subset of $S$ that the integral curve passing through $(x_0, y_0)$ must reside in that is smaller than $S$ itself, preferably one where the area covered by the estimate region can be made arbitrarily small by picking a step size or something similar?
ordinary-differential-equations numerical-methods
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
697512
$endgroup$
Are there numerical methods for ODEs that are capable of producing regions (linearly bounded or otherwise) that definitely contain a portion of a specified integral curve?
I'm wondering if there are numerical methods for solving ODEs that are roughly analogous to interval arithmetic for real-valued functions.
Suppose we have two unknowns $y(x)$ and $x$ where $y' = F(x, y)$. Suppose that $(x_0, y_0)$ is our initial value and suppose that $-1 le x le 1$ is the range of values of $x$ that we're interested in. In other words, $(x_0, y_0)$ is how we pick out a particular integral curve and $S stackrel{text{def}}{=} left{ (x,y) : x, y in mathbb{R} land -1 le x le 1 right}$ is the region of the $xy$ plane that we want to pick a subset of.
Are there any methods that produce a subset of $S$ that the integral curve passing through $(x_0, y_0)$ must reside in that is smaller than $S$ itself, preferably one where the area covered by the estimate region can be made arbitrarily small by picking a step size or something similar?
ordinary-differential-equations numerical-methods
ordinary-differential-equations numerical-methods
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
697512
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
697512
edited Jan 8 at 9:05
Gregory Nisbet
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
697512
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
697512
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
697512
697512
1
$begingroup$
Definitely yes (and based on interval arithmetic), although it's a somewhat tricky problem. Try searching for "validated numerics ODE".
$endgroup$
– Hans Lundmark
Jan 8 at 9:01
add a comment |
1
$begingroup$
Definitely yes (and based on interval arithmetic), although it's a somewhat tricky problem. Try searching for "validated numerics ODE".
$endgroup$
– Hans Lundmark
Jan 8 at 9:01
1
1
$begingroup$
Definitely yes (and based on interval arithmetic), although it's a somewhat tricky problem. Try searching for "validated numerics ODE".
$endgroup$
– Hans Lundmark
Jan 8 at 9:01
$begingroup$
Definitely yes (and based on interval arithmetic), although it's a somewhat tricky problem. Try searching for "validated numerics ODE".
$endgroup$
– Hans Lundmark
Jan 8 at 9:01
add a comment |
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Definitely yes (and based on interval arithmetic), although it's a somewhat tricky problem. Try searching for "validated numerics ODE".
$endgroup$
– Hans Lundmark
Jan 8 at 9:01