ODE numerical method that produces a region containing the integral curve
$begingroup$
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
756612
$endgroup$
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
756612
$endgroup$
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
756612
$endgroup$
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
756612
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
756612
edited Jan 8 at 9:05
Gregory Nisbet
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
756612
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
756612
asked Jan 8 at 8:11
Gregory NisbetGregory Nisbet
756612
756612
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3065915%2fode-numerical-method-that-produces-a-region-containing-the-integral-curve%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3065915%2fode-numerical-method-that-produces-a-region-containing-the-integral-curve%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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