Poisson Equation with Boundary Value Problem
I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$:
$$
begin{cases}
dfrac{mathrm{d}^2}{mathrm{d}x^2}u(x)=-g(x) \
u(0)= a \
u(1)= b
end{cases}
$$
where $g(x)=−6pi cos(3pi x)+9pi^2x sin(3pi x)$ and $a=b=0$.
Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.
differential-equations boundary-value-problem poissons-equation
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enes
245
add a comment |
I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$:
$$
begin{cases}
dfrac{mathrm{d}^2}{mathrm{d}x^2}u(x)=-g(x) \
u(0)= a \
u(1)= b
end{cases}
$$
where $g(x)=−6pi cos(3pi x)+9pi^2x sin(3pi x)$ and $a=b=0$.
Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.
differential-equations boundary-value-problem poissons-equation
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enes
245
add a comment |
I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$:
$$
begin{cases}
dfrac{mathrm{d}^2}{mathrm{d}x^2}u(x)=-g(x) \
u(0)= a \
u(1)= b
end{cases}
$$
where $g(x)=−6pi cos(3pi x)+9pi^2x sin(3pi x)$ and $a=b=0$.
Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.
differential-equations boundary-value-problem poissons-equation
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enes
245
I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$:
$$
begin{cases}
dfrac{mathrm{d}^2}{mathrm{d}x^2}u(x)=-g(x) \
u(0)= a \
u(1)= b
end{cases}
$$
where $g(x)=−6pi cos(3pi x)+9pi^2x sin(3pi x)$ and $a=b=0$.
Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.
differential-equations boundary-value-problem poissons-equation
differential-equations boundary-value-problem poissons-equation
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enes
245
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
asked Oct 31 '18 at 20:34
enes
245
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
edited Nov 27 '18 at 20:13
Dylan
12.4k31026
12.4k31026
asked Oct 31 '18 at 20:34
enes
245
asked Oct 31 '18 at 20:34
enes
245
asked Oct 31 '18 at 20:34
enes
245
245
add a comment |
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%2f2979625%2fpoisson-equation-with-boundary-value-problem%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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f2979625%2fpoisson-equation-with-boundary-value-problem%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
