Picard iteration for second order ODE












2












$begingroup$


Any help for the following two questions will be much appreciated:



Question 1:



Consder the ODE with IVP



$y''(t)+g(t,y)=0, y(0)=y_0, y'(0)=z_0$



where g is continues on some region $D$, $(0,y_0)in D$. I want to show that this is equivalent to the integral equation



$ begin{align}
y(t)=y_0+z_0t-int_0^t (t-s)g(s,y(s))ds
end{align}
\
$



This is what I have done so far to solve this: I integra either side and get



$ int_0^t y''(s) ds =-int_0^t g(s,f(s)) ds$ I calculate this, integrate again and get



$y(s)=y_0+z_0t-int_0^t int_0^t g(s,f(s))ds$. My problem is that I don't know what to do with this last double integral.



Question 2:



Consider the same integral equation where $g(y,t)$, $frac{partial g}{ partial y} (t,y)$ are continuous on the rectangle $D={ (t,y)in mathbb{R}^2 : mid t mid leq a, mid y-y_0 mid leq b}$. Let $mid g(t,y) mid leq M$, $mid frac{partial g}{ partial y} (t,y) mid leq K$ for all $(t,y) in D$. We define $phi_0(t)=y_0$ and



$
phi_n(t)=y_0+z_0t-int_0^t (t-s)g(s,y(s))ds
$



I want to show that $phi_n(t)$ is well defiend for $mid tmid leq min {a,frac{2b}{2mid z_0mid+2M}}$. Here I am quite lost. To show it is well-defined I need to show that the integral is never infinity. But I don't know how to prove this.






real-analysis ordinary-differential-equations







share|cite|improve this question











$endgroup$












  • $begingroup$
    How come you have not got an answer yet!
    $endgroup$
    – Mhenni Benghorbal
    Feb 13 '16 at 23:32












  • $begingroup$
    I don't know, hopefully some kind soul will come by soon.
    $endgroup$
    – Soren123
    Feb 14 '16 at 8:00
















2












$begingroup$


Any help for the following two questions will be much appreciated:



Question 1:



Consder the ODE with IVP



$y''(t)+g(t,y)=0, y(0)=y_0, y'(0)=z_0$



where g is continues on some region $D$, $(0,y_0)in D$. I want to show that this is equivalent to the integral equation



$ begin{align}
y(t)=y_0+z_0t-int_0^t (t-s)g(s,y(s))ds
end{align}
\
$



This is what I have done so far to solve this: I integra either side and get



$ int_0^t y''(s) ds =-int_0^t g(s,f(s)) ds$ I calculate this, integrate again and get



$y(s)=y_0+z_0t-int_0^t int_0^t g(s,f(s))ds$. My problem is that I don't know what to do with this last double integral.



Question 2:



Consider the same integral equation where $g(y,t)$, $frac{partial g}{ partial y} (t,y)$ are continuous on the rectangle $D={ (t,y)in mathbb{R}^2 : mid t mid leq a, mid y-y_0 mid leq b}$. Let $mid g(t,y) mid leq M$, $mid frac{partial g}{ partial y} (t,y) mid leq K$ for all $(t,y) in D$. We define $phi_0(t)=y_0$ and



$
phi_n(t)=y_0+z_0t-int_0^t (t-s)g(s,y(s))ds
$



I want to show that $phi_n(t)$ is well defiend for $mid tmid leq min {a,frac{2b}{2mid z_0mid+2M}}$. Here I am quite lost. To show it is well-defined I need to show that the integral is never infinity. But I don't know how to prove this.






real-analysis ordinary-differential-equations







share|cite|improve this question











$endgroup$












  • $begingroup$
    How come you have not got an answer yet!
    $endgroup$
    – Mhenni Benghorbal
    Feb 13 '16 at 23:32












  • $begingroup$
    I don't know, hopefully some kind soul will come by soon.
    $endgroup$
    – Soren123
    Feb 14 '16 at 8:00














2












2








2


1



$begingroup$


Any help for the following two questions will be much appreciated:



Question 1:



Consder the ODE with IVP



$y''(t)+g(t,y)=0, y(0)=y_0, y'(0)=z_0$



where g is continues on some region $D$, $(0,y_0)in D$. I want to show that this is equivalent to the integral equation



$ begin{align}
y(t)=y_0+z_0t-int_0^t (t-s)g(s,y(s))ds
end{align}
\
$



This is what I have done so far to solve this: I integra either side and get



$ int_0^t y''(s) ds =-int_0^t g(s,f(s)) ds$ I calculate this, integrate again and get



$y(s)=y_0+z_0t-int_0^t int_0^t g(s,f(s))ds$. My problem is that I don't know what to do with this last double integral.



Question 2:



Consider the same integral equation where $g(y,t)$, $frac{partial g}{ partial y} (t,y)$ are continuous on the rectangle $D={ (t,y)in mathbb{R}^2 : mid t mid leq a, mid y-y_0 mid leq b}$. Let $mid g(t,y) mid leq M$, $mid frac{partial g}{ partial y} (t,y) mid leq K$ for all $(t,y) in D$. We define $phi_0(t)=y_0$ and



$
phi_n(t)=y_0+z_0t-int_0^t (t-s)g(s,y(s))ds
$



I want to show that $phi_n(t)$ is well defiend for $mid tmid leq min {a,frac{2b}{2mid z_0mid+2M}}$. Here I am quite lost. To show it is well-defined I need to show that the integral is never infinity. But I don't know how to prove this.






real-analysis ordinary-differential-equations







share|cite|improve this question











$endgroup$




Any help for the following two questions will be much appreciated:



Question 1:



Consder the ODE with IVP



$y''(t)+g(t,y)=0, y(0)=y_0, y'(0)=z_0$



where g is continues on some region $D$, $(0,y_0)in D$. I want to show that this is equivalent to the integral equation



$ begin{align}
y(t)=y_0+z_0t-int_0^t (t-s)g(s,y(s))ds
end{align}
\
$



This is what I have done so far to solve this: I integra either side and get



$ int_0^t y''(s) ds =-int_0^t g(s,f(s)) ds$ I calculate this, integrate again and get



$y(s)=y_0+z_0t-int_0^t int_0^t g(s,f(s))ds$. My problem is that I don't know what to do with this last double integral.



Question 2:



Consider the same integral equation where $g(y,t)$, $frac{partial g}{ partial y} (t,y)$ are continuous on the rectangle $D={ (t,y)in mathbb{R}^2 : mid t mid leq a, mid y-y_0 mid leq b}$. Let $mid g(t,y) mid leq M$, $mid frac{partial g}{ partial y} (t,y) mid leq K$ for all $(t,y) in D$. We define $phi_0(t)=y_0$ and



$
phi_n(t)=y_0+z_0t-int_0^t (t-s)g(s,y(s))ds
$



I want to show that $phi_n(t)$ is well defiend for $mid tmid leq min {a,frac{2b}{2mid z_0mid+2M}}$. Here I am quite lost. To show it is well-defined I need to show that the integral is never infinity. But I don't know how to prove this.






real-analysis ordinary-differential-equations




real-analysis ordinary-differential-equations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Feb 14 '16 at 7:59







Soren123

















asked Feb 13 '16 at 20:27









Soren123Soren123

9311




9311












  • $begingroup$
    How come you have not got an answer yet!
    $endgroup$
    – Mhenni Benghorbal
    Feb 13 '16 at 23:32












  • $begingroup$
    I don't know, hopefully some kind soul will come by soon.
    $endgroup$
    – Soren123
    Feb 14 '16 at 8:00


















  • $begingroup$
    How come you have not got an answer yet!
    $endgroup$
    – Mhenni Benghorbal
    Feb 13 '16 at 23:32












  • $begingroup$
    I don't know, hopefully some kind soul will come by soon.
    $endgroup$
    – Soren123
    Feb 14 '16 at 8:00
















$begingroup$
How come you have not got an answer yet!
$endgroup$
– Mhenni Benghorbal
Feb 13 '16 at 23:32






$begingroup$
How come you have not got an answer yet!
$endgroup$
– Mhenni Benghorbal
Feb 13 '16 at 23:32














$begingroup$
I don't know, hopefully some kind soul will come by soon.
$endgroup$
– Soren123
Feb 14 '16 at 8:00




$begingroup$
I don't know, hopefully some kind soul will come by soon.
$endgroup$
– Soren123
Feb 14 '16 at 8:00










1 Answer
1






active

oldest

votes


















0












$begingroup$

Try showing $r_j(t)=|phi_{j+1}(t)-phi_j(t)| leq frac{M(|z_0|+frac{Ma}{2})K^{j}t^{2j+1}}{(2j+1)!}$.



If K<1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})t^{2j+1}}{(2j+1)!}$
If K>1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})K^{2j+1}t^{2j+1}}{(2j+1)!}$ which are the terms of hyperbolic sin function, which converges.






share|cite|improve this answer









$endgroup$













    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
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1653640%2fpicard-iteration-for-second-order-ode%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    Try showing $r_j(t)=|phi_{j+1}(t)-phi_j(t)| leq frac{M(|z_0|+frac{Ma}{2})K^{j}t^{2j+1}}{(2j+1)!}$.



    If K<1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})t^{2j+1}}{(2j+1)!}$
    If K>1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})K^{2j+1}t^{2j+1}}{(2j+1)!}$ which are the terms of hyperbolic sin function, which converges.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Try showing $r_j(t)=|phi_{j+1}(t)-phi_j(t)| leq frac{M(|z_0|+frac{Ma}{2})K^{j}t^{2j+1}}{(2j+1)!}$.



      If K<1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})t^{2j+1}}{(2j+1)!}$
      If K>1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})K^{2j+1}t^{2j+1}}{(2j+1)!}$ which are the terms of hyperbolic sin function, which converges.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Try showing $r_j(t)=|phi_{j+1}(t)-phi_j(t)| leq frac{M(|z_0|+frac{Ma}{2})K^{j}t^{2j+1}}{(2j+1)!}$.



        If K<1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})t^{2j+1}}{(2j+1)!}$
        If K>1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})K^{2j+1}t^{2j+1}}{(2j+1)!}$ which are the terms of hyperbolic sin function, which converges.






        share|cite|improve this answer









        $endgroup$



        Try showing $r_j(t)=|phi_{j+1}(t)-phi_j(t)| leq frac{M(|z_0|+frac{Ma}{2})K^{j}t^{2j+1}}{(2j+1)!}$.



        If K<1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})t^{2j+1}}{(2j+1)!}$
        If K>1, then $r_j(t) leq frac{M(|z_0|+frac{Ma}{2})K^{2j+1}t^{2j+1}}{(2j+1)!}$ which are the terms of hyperbolic sin function, which converges.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 9 '18 at 0:42









        JungleshrimpJungleshrimp

        314111




        314111






























            draft saved

            draft discarded




















































            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1653640%2fpicard-iteration-for-second-order-ode%23new-answer', 'question_page');
            }
            );

            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







            -

            Popular posts from this blog

            How do I know what Microsoft account the skydrive app is syncing to?

            Image
            up vote 4 down vote favorite 1 Some time ago, I set up skydrive for my wife on her mac. But now she wants to access the files online, I cannot remember what Microsoft account I setup and linked her skydive app to and nor can I see anything in the skydrive app settings to tell me either? How can I see the name of the Microsoft account the skydive app is currently linked to? I would like to know this for skydrive app for Mac and PC onedrive share | improve this question asked Feb 25 '13 at 4:52 Gary Barrett 260 2 5 12
            Read more

            When does type information flow backwards in C++?

            Image
            up vote 62 down vote favorite 19 I just watched Stephan T. Lavavej talk at CppCon 2018 on "Class Template Argument Deduction", where at some point he incidentally says: In C++ type information almost never flows backwards ... I had to say "almost" because there's one or two cases, possibly more but very few . Despite trying to figure out which cases he might be referring to, I couldn't come up with anything. Hence the question: In which cases the C++17 standard mandates that type information propagate backwards? c++ language-lawyer c++17 share | improve this question edited 7 hours ago scohe0
            Read more

            Grease: Live!

            Image
            A tradução deste artigo está abaixo da qualidade média aceitável. É possível que tenha sido feita por um tradutor automático ou por alguém que não conhece bem o português ou a língua original do texto. Caso queira colaborar com a Wikipédia, tente encontrar a página original e melhore este verbete conforme o guia de tradução. Grease: Live! Grease: Live!  (PRT/BRA)  Estados Unidos 2016 •   cor •   130 min. min   Direção Thomas Kail Alex Rudzinski Produção Marc Platt Adam Siegel Roteiro Allan Carr Robert Cary Warren Casey Jim Jacobs Jonathan Tolins Bronte Woodard Baseado em Grease (musical) Grease (filme) Elenco Julianne Hough Aaron Tveit Vanessa Hudgens Carlos Pena Jr. Keke Palmer Carly Rae Jepsen Kether Donohue Jordan Fisher David Del Rio Andrew Call Gênero Musical Música Tom Kitt Figurino William Ivey Long Companhia(s) produtora(s) Warner Bros. Studios Distribuição Fox Lançamento 31 de Janei
            Read more