Mathematical Transformation in a paper












2












$begingroup$


I am currently reading this paper: https://ieeexplore.ieee.org/abstract/document/1707778
but I don't understand how they made the following mathematical transformation.



They have the following function:



begin{equation*} w^{IV}(x)-N(w)w^{prime prime}(x)=F_{e}(w(x)) tag{7} end{equation*}
whith
begin{equation*} N(w)=N_{1} int_{0}^{1}(w^{prime})^{2}dx+N_{0} end{equation*}



Then they define:begin{equation*} w(x)=zetabar{w}(x). tag{15} end{equation*}



And derive:



begin{equation*} (k_{0}+k_{1})zeta+k_{2}zeta^{3}=mu f_{e}(zeta) tag{16} end{equation*}



Don't bother about the right side of the equation, I am only interested to know how they derive the left side.



In the paper, they write:




The governing equation for ζ is derived by multiplying both sides of
(7) by w¯, integrating over the beam span, and substituting into the
resulting equation the approximate solution begin{equation*} w(x)=zetabar{w}(x). tag{15} end{equation*} After integrating by
parts, the governing equation for ζ becomes











share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    I am currently reading this paper: https://ieeexplore.ieee.org/abstract/document/1707778
    but I don't understand how they made the following mathematical transformation.



    They have the following function:



    begin{equation*} w^{IV}(x)-N(w)w^{prime prime}(x)=F_{e}(w(x)) tag{7} end{equation*}
    whith
    begin{equation*} N(w)=N_{1} int_{0}^{1}(w^{prime})^{2}dx+N_{0} end{equation*}



    Then they define:begin{equation*} w(x)=zetabar{w}(x). tag{15} end{equation*}



    And derive:



    begin{equation*} (k_{0}+k_{1})zeta+k_{2}zeta^{3}=mu f_{e}(zeta) tag{16} end{equation*}



    Don't bother about the right side of the equation, I am only interested to know how they derive the left side.



    In the paper, they write:




    The governing equation for ζ is derived by multiplying both sides of
    (7) by w¯, integrating over the beam span, and substituting into the
    resulting equation the approximate solution begin{equation*} w(x)=zetabar{w}(x). tag{15} end{equation*} After integrating by
    parts, the governing equation for ζ becomes











    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      I am currently reading this paper: https://ieeexplore.ieee.org/abstract/document/1707778
      but I don't understand how they made the following mathematical transformation.



      They have the following function:



      begin{equation*} w^{IV}(x)-N(w)w^{prime prime}(x)=F_{e}(w(x)) tag{7} end{equation*}
      whith
      begin{equation*} N(w)=N_{1} int_{0}^{1}(w^{prime})^{2}dx+N_{0} end{equation*}



      Then they define:begin{equation*} w(x)=zetabar{w}(x). tag{15} end{equation*}



      And derive:



      begin{equation*} (k_{0}+k_{1})zeta+k_{2}zeta^{3}=mu f_{e}(zeta) tag{16} end{equation*}



      Don't bother about the right side of the equation, I am only interested to know how they derive the left side.



      In the paper, they write:




      The governing equation for ζ is derived by multiplying both sides of
      (7) by w¯, integrating over the beam span, and substituting into the
      resulting equation the approximate solution begin{equation*} w(x)=zetabar{w}(x). tag{15} end{equation*} After integrating by
      parts, the governing equation for ζ becomes











      share|cite|improve this question









      $endgroup$




      I am currently reading this paper: https://ieeexplore.ieee.org/abstract/document/1707778
      but I don't understand how they made the following mathematical transformation.



      They have the following function:



      begin{equation*} w^{IV}(x)-N(w)w^{prime prime}(x)=F_{e}(w(x)) tag{7} end{equation*}
      whith
      begin{equation*} N(w)=N_{1} int_{0}^{1}(w^{prime})^{2}dx+N_{0} end{equation*}



      Then they define:begin{equation*} w(x)=zetabar{w}(x). tag{15} end{equation*}



      And derive:



      begin{equation*} (k_{0}+k_{1})zeta+k_{2}zeta^{3}=mu f_{e}(zeta) tag{16} end{equation*}



      Don't bother about the right side of the equation, I am only interested to know how they derive the left side.



      In the paper, they write:




      The governing equation for ζ is derived by multiplying both sides of
      (7) by w¯, integrating over the beam span, and substituting into the
      resulting equation the approximate solution begin{equation*} w(x)=zetabar{w}(x). tag{15} end{equation*} After integrating by
      parts, the governing equation for ζ becomes








      differential-equations






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 1 '18 at 15:44









      jamesjames

      1519




      1519






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          The equation is
          $$
          w^{IV}-N w''=F_e(w).
          $$

          Multiplying both sides by $bar{w}$
          $$
          bar{w} w^{IV}-N bar{w} w''=F_e(w) bar{w},
          $$

          and integrating in $0leq x leq 1$,
          $$
          int_0^1 bar{w} w^{IV} dx - N int_0^1 bar{w} w'' dx= int_0^1 F_e(w) bar{w} dx.
          $$

          Now we substitute
          $$
          w=zeta bar{w},
          $$

          leading to
          $$
          zeta int_0^1 bar{w} bar{w}^{IV} dx - N zeta int_0^1 bar{w} bar{w}'' dx= int_0^1 F_e(zetabar{w}) bar{w} dx,
          $$

          with $N$ now given by
          $$
          N = N_1 zeta^2 int_0^1 (bar{w}')^2 dx + N_0.
          $$



          See that
          $$int_0^1 bar{w} bar{w}'' dx$$
          can be integrated by parts using $u=bar{w}$ and $dv=bar{w}'' dx$, leading to
          $$
          int_0^1 bar{w} bar{w}'' dx = left.bar{w} bar{w}'right|_0^1 - int_0^1 (bar{w}')^2 dx.
          $$

          If the boundary conditions $(2)$ or $(3)$ holds, the first term in the RHS is $0$, then
          $$
          int_0^1 bar{w} bar{w}'' dx = - int_0^1 (bar{w}')^2 dx.
          $$

          The integral
          $$int_0^1 bar{w} bar{w}^{IV} dx $$
          can be evaluated in the same way as
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = left. bar{w} bar{w}^{III} right|_0^1 - int_0^1 bar{w}' bar{w}^{III} dx.
          $$

          Assuming, for the sake of simplicity, that boundary conditions $(3)$ holds, the first term in RHS vanishes and the integral in RHS can be evaluated again with integration by parts, leading to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = -left.bar{w}'bar{w}'' right|_0^1 + int_0^1 (bar{w}'')^2 dx.
          $$

          The first term in RHS vanishes with any of the boundary conditions, leading finally to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = int_0^1 (bar{w}'')^2 dx.
          $$



          The equation is now
          $$
          zeta int_0^1 (bar{w}'')^2 dx + N_1 zeta^3 left(int_0^1 (bar{w}')^2 dx right)^2 + N_0 zeta int_0^1 (bar{w}')^2 dx= int_0^1 F_e bar{w} dx.
          $$



          Then, defining
          $$
          k_0 = int_0^1 (bar{w}'')^2 dx,
          $$

          $$
          k_1 = N_0 int_0^1 (bar{w}')^2dx,
          $$

          $$
          k_2 = N_1 left(int_0^1 (bar{w}')^2 dxright)^2,
          $$

          and
          $$
          f_e(zeta) = frac{1}{mu} int_0^1 F_e (zeta bar{w}) bar{w} dx,
          $$

          the equation can be written as
          $$
          (k_0+k_1) zeta + k_2 zeta^3 = mu f_e(zeta),
          $$

          which is equation $(16)$ in the paper.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Wow ! Thank you so much for your derivation! It is definitely not a straight-forward one, so I wonder why they just chose to summarize it quickly in one sentence, when the rest of the paper is quite detailed...
            $endgroup$
            – james
            Dec 1 '18 at 16:45












          • $begingroup$
            @james You are welcome! Maybe the authors thought that showing only the expressions for $k_1$ etc. would be sufficient to "see" the integration by parts between the lines. Furthermore, sometimes such derivations take precious space from the manuscript...
            $endgroup$
            – rafa11111
            Dec 1 '18 at 16:50











          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%2f3021487%2fmathematical-transformation-in-a-paper%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









          1












          $begingroup$

          The equation is
          $$
          w^{IV}-N w''=F_e(w).
          $$

          Multiplying both sides by $bar{w}$
          $$
          bar{w} w^{IV}-N bar{w} w''=F_e(w) bar{w},
          $$

          and integrating in $0leq x leq 1$,
          $$
          int_0^1 bar{w} w^{IV} dx - N int_0^1 bar{w} w'' dx= int_0^1 F_e(w) bar{w} dx.
          $$

          Now we substitute
          $$
          w=zeta bar{w},
          $$

          leading to
          $$
          zeta int_0^1 bar{w} bar{w}^{IV} dx - N zeta int_0^1 bar{w} bar{w}'' dx= int_0^1 F_e(zetabar{w}) bar{w} dx,
          $$

          with $N$ now given by
          $$
          N = N_1 zeta^2 int_0^1 (bar{w}')^2 dx + N_0.
          $$



          See that
          $$int_0^1 bar{w} bar{w}'' dx$$
          can be integrated by parts using $u=bar{w}$ and $dv=bar{w}'' dx$, leading to
          $$
          int_0^1 bar{w} bar{w}'' dx = left.bar{w} bar{w}'right|_0^1 - int_0^1 (bar{w}')^2 dx.
          $$

          If the boundary conditions $(2)$ or $(3)$ holds, the first term in the RHS is $0$, then
          $$
          int_0^1 bar{w} bar{w}'' dx = - int_0^1 (bar{w}')^2 dx.
          $$

          The integral
          $$int_0^1 bar{w} bar{w}^{IV} dx $$
          can be evaluated in the same way as
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = left. bar{w} bar{w}^{III} right|_0^1 - int_0^1 bar{w}' bar{w}^{III} dx.
          $$

          Assuming, for the sake of simplicity, that boundary conditions $(3)$ holds, the first term in RHS vanishes and the integral in RHS can be evaluated again with integration by parts, leading to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = -left.bar{w}'bar{w}'' right|_0^1 + int_0^1 (bar{w}'')^2 dx.
          $$

          The first term in RHS vanishes with any of the boundary conditions, leading finally to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = int_0^1 (bar{w}'')^2 dx.
          $$



          The equation is now
          $$
          zeta int_0^1 (bar{w}'')^2 dx + N_1 zeta^3 left(int_0^1 (bar{w}')^2 dx right)^2 + N_0 zeta int_0^1 (bar{w}')^2 dx= int_0^1 F_e bar{w} dx.
          $$



          Then, defining
          $$
          k_0 = int_0^1 (bar{w}'')^2 dx,
          $$

          $$
          k_1 = N_0 int_0^1 (bar{w}')^2dx,
          $$

          $$
          k_2 = N_1 left(int_0^1 (bar{w}')^2 dxright)^2,
          $$

          and
          $$
          f_e(zeta) = frac{1}{mu} int_0^1 F_e (zeta bar{w}) bar{w} dx,
          $$

          the equation can be written as
          $$
          (k_0+k_1) zeta + k_2 zeta^3 = mu f_e(zeta),
          $$

          which is equation $(16)$ in the paper.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Wow ! Thank you so much for your derivation! It is definitely not a straight-forward one, so I wonder why they just chose to summarize it quickly in one sentence, when the rest of the paper is quite detailed...
            $endgroup$
            – james
            Dec 1 '18 at 16:45












          • $begingroup$
            @james You are welcome! Maybe the authors thought that showing only the expressions for $k_1$ etc. would be sufficient to "see" the integration by parts between the lines. Furthermore, sometimes such derivations take precious space from the manuscript...
            $endgroup$
            – rafa11111
            Dec 1 '18 at 16:50
















          1












          $begingroup$

          The equation is
          $$
          w^{IV}-N w''=F_e(w).
          $$

          Multiplying both sides by $bar{w}$
          $$
          bar{w} w^{IV}-N bar{w} w''=F_e(w) bar{w},
          $$

          and integrating in $0leq x leq 1$,
          $$
          int_0^1 bar{w} w^{IV} dx - N int_0^1 bar{w} w'' dx= int_0^1 F_e(w) bar{w} dx.
          $$

          Now we substitute
          $$
          w=zeta bar{w},
          $$

          leading to
          $$
          zeta int_0^1 bar{w} bar{w}^{IV} dx - N zeta int_0^1 bar{w} bar{w}'' dx= int_0^1 F_e(zetabar{w}) bar{w} dx,
          $$

          with $N$ now given by
          $$
          N = N_1 zeta^2 int_0^1 (bar{w}')^2 dx + N_0.
          $$



          See that
          $$int_0^1 bar{w} bar{w}'' dx$$
          can be integrated by parts using $u=bar{w}$ and $dv=bar{w}'' dx$, leading to
          $$
          int_0^1 bar{w} bar{w}'' dx = left.bar{w} bar{w}'right|_0^1 - int_0^1 (bar{w}')^2 dx.
          $$

          If the boundary conditions $(2)$ or $(3)$ holds, the first term in the RHS is $0$, then
          $$
          int_0^1 bar{w} bar{w}'' dx = - int_0^1 (bar{w}')^2 dx.
          $$

          The integral
          $$int_0^1 bar{w} bar{w}^{IV} dx $$
          can be evaluated in the same way as
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = left. bar{w} bar{w}^{III} right|_0^1 - int_0^1 bar{w}' bar{w}^{III} dx.
          $$

          Assuming, for the sake of simplicity, that boundary conditions $(3)$ holds, the first term in RHS vanishes and the integral in RHS can be evaluated again with integration by parts, leading to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = -left.bar{w}'bar{w}'' right|_0^1 + int_0^1 (bar{w}'')^2 dx.
          $$

          The first term in RHS vanishes with any of the boundary conditions, leading finally to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = int_0^1 (bar{w}'')^2 dx.
          $$



          The equation is now
          $$
          zeta int_0^1 (bar{w}'')^2 dx + N_1 zeta^3 left(int_0^1 (bar{w}')^2 dx right)^2 + N_0 zeta int_0^1 (bar{w}')^2 dx= int_0^1 F_e bar{w} dx.
          $$



          Then, defining
          $$
          k_0 = int_0^1 (bar{w}'')^2 dx,
          $$

          $$
          k_1 = N_0 int_0^1 (bar{w}')^2dx,
          $$

          $$
          k_2 = N_1 left(int_0^1 (bar{w}')^2 dxright)^2,
          $$

          and
          $$
          f_e(zeta) = frac{1}{mu} int_0^1 F_e (zeta bar{w}) bar{w} dx,
          $$

          the equation can be written as
          $$
          (k_0+k_1) zeta + k_2 zeta^3 = mu f_e(zeta),
          $$

          which is equation $(16)$ in the paper.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Wow ! Thank you so much for your derivation! It is definitely not a straight-forward one, so I wonder why they just chose to summarize it quickly in one sentence, when the rest of the paper is quite detailed...
            $endgroup$
            – james
            Dec 1 '18 at 16:45












          • $begingroup$
            @james You are welcome! Maybe the authors thought that showing only the expressions for $k_1$ etc. would be sufficient to "see" the integration by parts between the lines. Furthermore, sometimes such derivations take precious space from the manuscript...
            $endgroup$
            – rafa11111
            Dec 1 '18 at 16:50














          1












          1








          1





          $begingroup$

          The equation is
          $$
          w^{IV}-N w''=F_e(w).
          $$

          Multiplying both sides by $bar{w}$
          $$
          bar{w} w^{IV}-N bar{w} w''=F_e(w) bar{w},
          $$

          and integrating in $0leq x leq 1$,
          $$
          int_0^1 bar{w} w^{IV} dx - N int_0^1 bar{w} w'' dx= int_0^1 F_e(w) bar{w} dx.
          $$

          Now we substitute
          $$
          w=zeta bar{w},
          $$

          leading to
          $$
          zeta int_0^1 bar{w} bar{w}^{IV} dx - N zeta int_0^1 bar{w} bar{w}'' dx= int_0^1 F_e(zetabar{w}) bar{w} dx,
          $$

          with $N$ now given by
          $$
          N = N_1 zeta^2 int_0^1 (bar{w}')^2 dx + N_0.
          $$



          See that
          $$int_0^1 bar{w} bar{w}'' dx$$
          can be integrated by parts using $u=bar{w}$ and $dv=bar{w}'' dx$, leading to
          $$
          int_0^1 bar{w} bar{w}'' dx = left.bar{w} bar{w}'right|_0^1 - int_0^1 (bar{w}')^2 dx.
          $$

          If the boundary conditions $(2)$ or $(3)$ holds, the first term in the RHS is $0$, then
          $$
          int_0^1 bar{w} bar{w}'' dx = - int_0^1 (bar{w}')^2 dx.
          $$

          The integral
          $$int_0^1 bar{w} bar{w}^{IV} dx $$
          can be evaluated in the same way as
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = left. bar{w} bar{w}^{III} right|_0^1 - int_0^1 bar{w}' bar{w}^{III} dx.
          $$

          Assuming, for the sake of simplicity, that boundary conditions $(3)$ holds, the first term in RHS vanishes and the integral in RHS can be evaluated again with integration by parts, leading to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = -left.bar{w}'bar{w}'' right|_0^1 + int_0^1 (bar{w}'')^2 dx.
          $$

          The first term in RHS vanishes with any of the boundary conditions, leading finally to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = int_0^1 (bar{w}'')^2 dx.
          $$



          The equation is now
          $$
          zeta int_0^1 (bar{w}'')^2 dx + N_1 zeta^3 left(int_0^1 (bar{w}')^2 dx right)^2 + N_0 zeta int_0^1 (bar{w}')^2 dx= int_0^1 F_e bar{w} dx.
          $$



          Then, defining
          $$
          k_0 = int_0^1 (bar{w}'')^2 dx,
          $$

          $$
          k_1 = N_0 int_0^1 (bar{w}')^2dx,
          $$

          $$
          k_2 = N_1 left(int_0^1 (bar{w}')^2 dxright)^2,
          $$

          and
          $$
          f_e(zeta) = frac{1}{mu} int_0^1 F_e (zeta bar{w}) bar{w} dx,
          $$

          the equation can be written as
          $$
          (k_0+k_1) zeta + k_2 zeta^3 = mu f_e(zeta),
          $$

          which is equation $(16)$ in the paper.






          share|cite|improve this answer









          $endgroup$



          The equation is
          $$
          w^{IV}-N w''=F_e(w).
          $$

          Multiplying both sides by $bar{w}$
          $$
          bar{w} w^{IV}-N bar{w} w''=F_e(w) bar{w},
          $$

          and integrating in $0leq x leq 1$,
          $$
          int_0^1 bar{w} w^{IV} dx - N int_0^1 bar{w} w'' dx= int_0^1 F_e(w) bar{w} dx.
          $$

          Now we substitute
          $$
          w=zeta bar{w},
          $$

          leading to
          $$
          zeta int_0^1 bar{w} bar{w}^{IV} dx - N zeta int_0^1 bar{w} bar{w}'' dx= int_0^1 F_e(zetabar{w}) bar{w} dx,
          $$

          with $N$ now given by
          $$
          N = N_1 zeta^2 int_0^1 (bar{w}')^2 dx + N_0.
          $$



          See that
          $$int_0^1 bar{w} bar{w}'' dx$$
          can be integrated by parts using $u=bar{w}$ and $dv=bar{w}'' dx$, leading to
          $$
          int_0^1 bar{w} bar{w}'' dx = left.bar{w} bar{w}'right|_0^1 - int_0^1 (bar{w}')^2 dx.
          $$

          If the boundary conditions $(2)$ or $(3)$ holds, the first term in the RHS is $0$, then
          $$
          int_0^1 bar{w} bar{w}'' dx = - int_0^1 (bar{w}')^2 dx.
          $$

          The integral
          $$int_0^1 bar{w} bar{w}^{IV} dx $$
          can be evaluated in the same way as
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = left. bar{w} bar{w}^{III} right|_0^1 - int_0^1 bar{w}' bar{w}^{III} dx.
          $$

          Assuming, for the sake of simplicity, that boundary conditions $(3)$ holds, the first term in RHS vanishes and the integral in RHS can be evaluated again with integration by parts, leading to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = -left.bar{w}'bar{w}'' right|_0^1 + int_0^1 (bar{w}'')^2 dx.
          $$

          The first term in RHS vanishes with any of the boundary conditions, leading finally to
          $$
          int_0^1 bar{w} bar{w}^{IV} dx = int_0^1 (bar{w}'')^2 dx.
          $$



          The equation is now
          $$
          zeta int_0^1 (bar{w}'')^2 dx + N_1 zeta^3 left(int_0^1 (bar{w}')^2 dx right)^2 + N_0 zeta int_0^1 (bar{w}')^2 dx= int_0^1 F_e bar{w} dx.
          $$



          Then, defining
          $$
          k_0 = int_0^1 (bar{w}'')^2 dx,
          $$

          $$
          k_1 = N_0 int_0^1 (bar{w}')^2dx,
          $$

          $$
          k_2 = N_1 left(int_0^1 (bar{w}')^2 dxright)^2,
          $$

          and
          $$
          f_e(zeta) = frac{1}{mu} int_0^1 F_e (zeta bar{w}) bar{w} dx,
          $$

          the equation can be written as
          $$
          (k_0+k_1) zeta + k_2 zeta^3 = mu f_e(zeta),
          $$

          which is equation $(16)$ in the paper.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 1 '18 at 16:23









          rafa11111rafa11111

          1,106417




          1,106417








          • 1




            $begingroup$
            Wow ! Thank you so much for your derivation! It is definitely not a straight-forward one, so I wonder why they just chose to summarize it quickly in one sentence, when the rest of the paper is quite detailed...
            $endgroup$
            – james
            Dec 1 '18 at 16:45












          • $begingroup$
            @james You are welcome! Maybe the authors thought that showing only the expressions for $k_1$ etc. would be sufficient to "see" the integration by parts between the lines. Furthermore, sometimes such derivations take precious space from the manuscript...
            $endgroup$
            – rafa11111
            Dec 1 '18 at 16:50














          • 1




            $begingroup$
            Wow ! Thank you so much for your derivation! It is definitely not a straight-forward one, so I wonder why they just chose to summarize it quickly in one sentence, when the rest of the paper is quite detailed...
            $endgroup$
            – james
            Dec 1 '18 at 16:45












          • $begingroup$
            @james You are welcome! Maybe the authors thought that showing only the expressions for $k_1$ etc. would be sufficient to "see" the integration by parts between the lines. Furthermore, sometimes such derivations take precious space from the manuscript...
            $endgroup$
            – rafa11111
            Dec 1 '18 at 16:50








          1




          1




          $begingroup$
          Wow ! Thank you so much for your derivation! It is definitely not a straight-forward one, so I wonder why they just chose to summarize it quickly in one sentence, when the rest of the paper is quite detailed...
          $endgroup$
          – james
          Dec 1 '18 at 16:45






          $begingroup$
          Wow ! Thank you so much for your derivation! It is definitely not a straight-forward one, so I wonder why they just chose to summarize it quickly in one sentence, when the rest of the paper is quite detailed...
          $endgroup$
          – james
          Dec 1 '18 at 16:45














          $begingroup$
          @james You are welcome! Maybe the authors thought that showing only the expressions for $k_1$ etc. would be sufficient to "see" the integration by parts between the lines. Furthermore, sometimes such derivations take precious space from the manuscript...
          $endgroup$
          – rafa11111
          Dec 1 '18 at 16:50




          $begingroup$
          @james You are welcome! Maybe the authors thought that showing only the expressions for $k_1$ etc. would be sufficient to "see" the integration by parts between the lines. Furthermore, sometimes such derivations take precious space from the manuscript...
          $endgroup$
          – rafa11111
          Dec 1 '18 at 16:50


















          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%2f3021487%2fmathematical-transformation-in-a-paper%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

          Probability when a professor distributes a quiz and homework assignment to a class of n students.

          Aardman Animations

          Are they similar matrix