Plot the solutions of the following PDEs at the specified times












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I am studying for my PDEs final and this is one of the practice questions from Strauss and I don't understand how to solve it.



Plot $u$ vs $x$ for the following problem on the half-line
$$u_{tt} = c^2 u_{xx}$$
$$u_x(0,t) = 0$$
$$u(x,0) = 0$$
$$u_t(x,0) = begin{cases} V & xin(a, 2a) \ 0 & x notin (a,2a)end{cases}$$



for $t = 0, a/c, 3a/2c, 2a/c, 3a/c$



The method that I was told was good to follow was to split the $x$ vs $t$ plot into regions or zones and use that plot to find the limits of integration for the general solution.



enter image description here



But I don't know how to obtain the lines the divide the plane or where to put my solution for some specific $t$. Say for $t = a/c$ we can mark a horizontal line that will give us our solution, but I don't know how to find $a/c$ on the vertical $t$ axis with relation to the other lines that divide the plane. I don't understand the solutions online. Any help is appreciated and sorry for the bad drawing.










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    1












    $begingroup$


    I am studying for my PDEs final and this is one of the practice questions from Strauss and I don't understand how to solve it.



    Plot $u$ vs $x$ for the following problem on the half-line
    $$u_{tt} = c^2 u_{xx}$$
    $$u_x(0,t) = 0$$
    $$u(x,0) = 0$$
    $$u_t(x,0) = begin{cases} V & xin(a, 2a) \ 0 & x notin (a,2a)end{cases}$$



    for $t = 0, a/c, 3a/2c, 2a/c, 3a/c$



    The method that I was told was good to follow was to split the $x$ vs $t$ plot into regions or zones and use that plot to find the limits of integration for the general solution.



    enter image description here



    But I don't know how to obtain the lines the divide the plane or where to put my solution for some specific $t$. Say for $t = a/c$ we can mark a horizontal line that will give us our solution, but I don't know how to find $a/c$ on the vertical $t$ axis with relation to the other lines that divide the plane. I don't understand the solutions online. Any help is appreciated and sorry for the bad drawing.










    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I am studying for my PDEs final and this is one of the practice questions from Strauss and I don't understand how to solve it.



      Plot $u$ vs $x$ for the following problem on the half-line
      $$u_{tt} = c^2 u_{xx}$$
      $$u_x(0,t) = 0$$
      $$u(x,0) = 0$$
      $$u_t(x,0) = begin{cases} V & xin(a, 2a) \ 0 & x notin (a,2a)end{cases}$$



      for $t = 0, a/c, 3a/2c, 2a/c, 3a/c$



      The method that I was told was good to follow was to split the $x$ vs $t$ plot into regions or zones and use that plot to find the limits of integration for the general solution.



      enter image description here



      But I don't know how to obtain the lines the divide the plane or where to put my solution for some specific $t$. Say for $t = a/c$ we can mark a horizontal line that will give us our solution, but I don't know how to find $a/c$ on the vertical $t$ axis with relation to the other lines that divide the plane. I don't understand the solutions online. Any help is appreciated and sorry for the bad drawing.










      share|cite|improve this question











      $endgroup$




      I am studying for my PDEs final and this is one of the practice questions from Strauss and I don't understand how to solve it.



      Plot $u$ vs $x$ for the following problem on the half-line
      $$u_{tt} = c^2 u_{xx}$$
      $$u_x(0,t) = 0$$
      $$u(x,0) = 0$$
      $$u_t(x,0) = begin{cases} V & xin(a, 2a) \ 0 & x notin (a,2a)end{cases}$$



      for $t = 0, a/c, 3a/2c, 2a/c, 3a/c$



      The method that I was told was good to follow was to split the $x$ vs $t$ plot into regions or zones and use that plot to find the limits of integration for the general solution.



      enter image description here



      But I don't know how to obtain the lines the divide the plane or where to put my solution for some specific $t$. Say for $t = a/c$ we can mark a horizontal line that will give us our solution, but I don't know how to find $a/c$ on the vertical $t$ axis with relation to the other lines that divide the plane. I don't understand the solutions online. Any help is appreciated and sorry for the bad drawing.







      pde wave-equation






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      edited Dec 14 '18 at 8:31







      The Bosco

















      asked Dec 14 '18 at 7:48









      The BoscoThe Bosco

      613212




      613212






















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