Mixing
Solving the Dirichlet problem using conformal maps
$begingroup$
I have been tasked with using conformal maps to find solutions to the Dirichlet problem on various regions given the solution $u(x,y) = y$ on the strip ${x + iy mid 0 leq y leq 1}$ satisfying $u(x, 0) = 0, u(x, 1) = 1$.
The first two (of five) regions are:
$U = {zmid r_1 leq |z| leq r_2}, u(z)=0$ when $|z| = r_1, u(z) = 1$ when $|z| = r_2$,
$U={zmid Im(z) geq 0}, u(x,0)=0$ when $x > 0, u(x,0)=1$ when $x <0$.
How do I proceed to solve this sort of question? I cannot even see where to begin. I feel as though I'm somehow meant to find conformal maps from the various $U$'s to the given strip, but I cannot figure out how I'm meant to find those.
I've deliberately not asked about all the parts, because I do want to do this question. I just feel as though I'm missing an obvious trick that's stopping me from getting anywhere.
complex-analysis
edited Jan 5 at 1:45
amWhy
1
asked Dec 27 '16 at 12:59
NethesisNethesis
1,8011823
$endgroup$
add a comment |
$begingroup$
I have been tasked with using conformal maps to find solutions to the Dirichlet problem on various regions given the solution $u(x,y) = y$ on the strip ${x + iy mid 0 leq y leq 1}$ satisfying $u(x, 0) = 0, u(x, 1) = 1$.
The first two (of five) regions are:
$U = {zmid r_1 leq |z| leq r_2}, u(z)=0$ when $|z| = r_1, u(z) = 1$ when $|z| = r_2$,
$U={zmid Im(z) geq 0}, u(x,0)=0$ when $x > 0, u(x,0)=1$ when $x <0$.
How do I proceed to solve this sort of question? I cannot even see where to begin. I feel as though I'm somehow meant to find conformal maps from the various $U$'s to the given strip, but I cannot figure out how I'm meant to find those.
I've deliberately not asked about all the parts, because I do want to do this question. I just feel as though I'm missing an obvious trick that's stopping me from getting anywhere.
complex-analysis
edited Jan 5 at 1:45
amWhy
1
asked Dec 27 '16 at 12:59
NethesisNethesis
1,8011823
$endgroup$
add a comment |
$begingroup$
I have been tasked with using conformal maps to find solutions to the Dirichlet problem on various regions given the solution $u(x,y) = y$ on the strip ${x + iy mid 0 leq y leq 1}$ satisfying $u(x, 0) = 0, u(x, 1) = 1$.
The first two (of five) regions are:
$U = {zmid r_1 leq |z| leq r_2}, u(z)=0$ when $|z| = r_1, u(z) = 1$ when $|z| = r_2$,
$U={zmid Im(z) geq 0}, u(x,0)=0$ when $x > 0, u(x,0)=1$ when $x <0$.
How do I proceed to solve this sort of question? I cannot even see where to begin. I feel as though I'm somehow meant to find conformal maps from the various $U$'s to the given strip, but I cannot figure out how I'm meant to find those.
I've deliberately not asked about all the parts, because I do want to do this question. I just feel as though I'm missing an obvious trick that's stopping me from getting anywhere.
complex-analysis
edited Jan 5 at 1:45
amWhy
1
asked Dec 27 '16 at 12:59
NethesisNethesis
1,8011823
$endgroup$
I have been tasked with using conformal maps to find solutions to the Dirichlet problem on various regions given the solution $u(x,y) = y$ on the strip ${x + iy mid 0 leq y leq 1}$ satisfying $u(x, 0) = 0, u(x, 1) = 1$.
The first two (of five) regions are:
$U = {zmid r_1 leq |z| leq r_2}, u(z)=0$ when $|z| = r_1, u(z) = 1$ when $|z| = r_2$,
$U={zmid Im(z) geq 0}, u(x,0)=0$ when $x > 0, u(x,0)=1$ when $x <0$.
How do I proceed to solve this sort of question? I cannot even see where to begin. I feel as though I'm somehow meant to find conformal maps from the various $U$'s to the given strip, but I cannot figure out how I'm meant to find those.
I've deliberately not asked about all the parts, because I do want to do this question. I just feel as though I'm missing an obvious trick that's stopping me from getting anywhere.
complex-analysis
complex-analysis
edited Jan 5 at 1:45
amWhy
1
asked Dec 27 '16 at 12:59
NethesisNethesis
1,8011823
edited Jan 5 at 1:45
amWhy
1
asked Dec 27 '16 at 12:59
NethesisNethesis
1,8011823
edited Jan 5 at 1:45
amWhy
1
edited Jan 5 at 1:45
amWhy
1
edited Jan 5 at 1:45
amWhy
1
1
asked Dec 27 '16 at 12:59
NethesisNethesis
1,8011823
asked Dec 27 '16 at 12:59
NethesisNethesis
1,8011823
asked Dec 27 '16 at 12:59
NethesisNethesis
1,8011823
1,8011823
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%2f2073523%2fsolving-the-dirichlet-problem-using-conformal-maps%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%2f2073523%2fsolving-the-dirichlet-problem-using-conformal-maps%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
