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How to solve $u_t + u_x =0$ with $u_0(x) = sin(pi x)$ with Characteristics?


Solve PDE using method of characteristicsShock-wave solution for PDE $u_t+(u-1)u_x=2$How to solve 3-variable transport equation $u_t + a u_x + b u_y =0$ by the method of characteristics?$u_t + uu_x=2$ where $u(x,0)=x$. Are there any shock forming with this initial condition?Method of characteristics PDE $u_t +uu_x = x$IVP for nonlinear PDE $u_t + frac13u_x^3 = -cu$Solve IVP of $(u_t )^2 + (u_x )^2 − u^2 = 0$ using method of characteristicsSolve inviscid Burgers' equation with shockSolve initial value problem $u_t+x^2u_x=0$ by method of characteristicsSolution for $u_t+u_x=0$ using characteristics













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$begingroup$


Consider the initial-boundary value problem (IBVP) for the convection equation
beginarray l u _ t + u _ x = 0 quad x in [ a ( t ) , b ( t ) ] , t in [ 0 , T ] \ u ( x , 0 ) = u _ 0 ( x ) quad x in [ a ( 0 ) , b ( 0 ) ] endarray
with initial conditionsu $u_ 0 ( x ) = sin ( pi x )$
The boundaries $a(t)$ and $b(t)$ are determined by the
characteristics that go through $(x = 0; t = 0)$ and $(x = 2; t = 0)$, respectively.
Let $x _ j = j h$ with
$j = 0,...,20$ and $h = 0.1$ .










share|cite|improve this question











$endgroup$
















    1












    $begingroup$


    Consider the initial-boundary value problem (IBVP) for the convection equation
    beginarray l u _ t + u _ x = 0 quad x in [ a ( t ) , b ( t ) ] , t in [ 0 , T ] \ u ( x , 0 ) = u _ 0 ( x ) quad x in [ a ( 0 ) , b ( 0 ) ] endarray
    with initial conditionsu $u_ 0 ( x ) = sin ( pi x )$
    The boundaries $a(t)$ and $b(t)$ are determined by the
    characteristics that go through $(x = 0; t = 0)$ and $(x = 2; t = 0)$, respectively.
    Let $x _ j = j h$ with
    $j = 0,...,20$ and $h = 0.1$ .










    share|cite|improve this question











    $endgroup$














      1












      1








      1





      $begingroup$


      Consider the initial-boundary value problem (IBVP) for the convection equation
      beginarray l u _ t + u _ x = 0 quad x in [ a ( t ) , b ( t ) ] , t in [ 0 , T ] \ u ( x , 0 ) = u _ 0 ( x ) quad x in [ a ( 0 ) , b ( 0 ) ] endarray
      with initial conditionsu $u_ 0 ( x ) = sin ( pi x )$
      The boundaries $a(t)$ and $b(t)$ are determined by the
      characteristics that go through $(x = 0; t = 0)$ and $(x = 2; t = 0)$, respectively.
      Let $x _ j = j h$ with
      $j = 0,...,20$ and $h = 0.1$ .










      share|cite|improve this question











      $endgroup$




      Consider the initial-boundary value problem (IBVP) for the convection equation
      beginarray l u _ t + u _ x = 0 quad x in [ a ( t ) , b ( t ) ] , t in [ 0 , T ] \ u ( x , 0 ) = u _ 0 ( x ) quad x in [ a ( 0 ) , b ( 0 ) ] endarray
      with initial conditionsu $u_ 0 ( x ) = sin ( pi x )$
      The boundaries $a(t)$ and $b(t)$ are determined by the
      characteristics that go through $(x = 0; t = 0)$ and $(x = 2; t = 0)$, respectively.
      Let $x _ j = j h$ with
      $j = 0,...,20$ and $h = 0.1$ .







      pde hyperbolic-equations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 14 at 17:00









      Rebellos

      15.4k31250




      15.4k31250










      asked Mar 14 at 16:48









      Junia ShelomiJunia Shelomi

      83




      83




















          1 Answer
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          active

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          0












          $begingroup$

          I don't really understand the final sentence of your elaboration. The method of characteristics is a hand-solved method, not an iteration.



          The characteristics are :



          $$fracmathrmdt1 = fracmathrmdx1 = fracmathrmdu0$$
          Taking any of the two first with the last one, yields :
          $$fracmathrmdx1 = fracmathrmdu0 implies u_1 = u$$
          Taking the first pair, yields :
          $$fracmathrmdt1 = fracmathrmdx1 implies u_2 = x-t $$



          Since $u_2$ is straight-forwardly expressed by $u$, simply enough, the solution will be given as
          $$u_1=F(u_2) Rightarrow u(x,t) = F(x-t)$$
          where $F$ is an adequatelly smooth function.



          But $u(x,0) = u_0(x) = sin(pi x)$, which means :



          $$u(x,0) = F(x) = sin(pi x)$$



          Now, letting $x : = x-t$, we yield :
          $$u(x,t) = F(x-t) = sin[pi(x-t)]$$






          share|cite|improve this answer









          $endgroup$












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            1 Answer
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            0












            $begingroup$

            I don't really understand the final sentence of your elaboration. The method of characteristics is a hand-solved method, not an iteration.



            The characteristics are :



            $$fracmathrmdt1 = fracmathrmdx1 = fracmathrmdu0$$
            Taking any of the two first with the last one, yields :
            $$fracmathrmdx1 = fracmathrmdu0 implies u_1 = u$$
            Taking the first pair, yields :
            $$fracmathrmdt1 = fracmathrmdx1 implies u_2 = x-t $$



            Since $u_2$ is straight-forwardly expressed by $u$, simply enough, the solution will be given as
            $$u_1=F(u_2) Rightarrow u(x,t) = F(x-t)$$
            where $F$ is an adequatelly smooth function.



            But $u(x,0) = u_0(x) = sin(pi x)$, which means :



            $$u(x,0) = F(x) = sin(pi x)$$



            Now, letting $x : = x-t$, we yield :
            $$u(x,t) = F(x-t) = sin[pi(x-t)]$$






            share|cite|improve this answer









            $endgroup$

















              0












              $begingroup$

              I don't really understand the final sentence of your elaboration. The method of characteristics is a hand-solved method, not an iteration.



              The characteristics are :



              $$fracmathrmdt1 = fracmathrmdx1 = fracmathrmdu0$$
              Taking any of the two first with the last one, yields :
              $$fracmathrmdx1 = fracmathrmdu0 implies u_1 = u$$
              Taking the first pair, yields :
              $$fracmathrmdt1 = fracmathrmdx1 implies u_2 = x-t $$



              Since $u_2$ is straight-forwardly expressed by $u$, simply enough, the solution will be given as
              $$u_1=F(u_2) Rightarrow u(x,t) = F(x-t)$$
              where $F$ is an adequatelly smooth function.



              But $u(x,0) = u_0(x) = sin(pi x)$, which means :



              $$u(x,0) = F(x) = sin(pi x)$$



              Now, letting $x : = x-t$, we yield :
              $$u(x,t) = F(x-t) = sin[pi(x-t)]$$






              share|cite|improve this answer









              $endgroup$















                0












                0








                0





                $begingroup$

                I don't really understand the final sentence of your elaboration. The method of characteristics is a hand-solved method, not an iteration.



                The characteristics are :



                $$fracmathrmdt1 = fracmathrmdx1 = fracmathrmdu0$$
                Taking any of the two first with the last one, yields :
                $$fracmathrmdx1 = fracmathrmdu0 implies u_1 = u$$
                Taking the first pair, yields :
                $$fracmathrmdt1 = fracmathrmdx1 implies u_2 = x-t $$



                Since $u_2$ is straight-forwardly expressed by $u$, simply enough, the solution will be given as
                $$u_1=F(u_2) Rightarrow u(x,t) = F(x-t)$$
                where $F$ is an adequatelly smooth function.



                But $u(x,0) = u_0(x) = sin(pi x)$, which means :



                $$u(x,0) = F(x) = sin(pi x)$$



                Now, letting $x : = x-t$, we yield :
                $$u(x,t) = F(x-t) = sin[pi(x-t)]$$






                share|cite|improve this answer









                $endgroup$



                I don't really understand the final sentence of your elaboration. The method of characteristics is a hand-solved method, not an iteration.



                The characteristics are :



                $$fracmathrmdt1 = fracmathrmdx1 = fracmathrmdu0$$
                Taking any of the two first with the last one, yields :
                $$fracmathrmdx1 = fracmathrmdu0 implies u_1 = u$$
                Taking the first pair, yields :
                $$fracmathrmdt1 = fracmathrmdx1 implies u_2 = x-t $$



                Since $u_2$ is straight-forwardly expressed by $u$, simply enough, the solution will be given as
                $$u_1=F(u_2) Rightarrow u(x,t) = F(x-t)$$
                where $F$ is an adequatelly smooth function.



                But $u(x,0) = u_0(x) = sin(pi x)$, which means :



                $$u(x,0) = F(x) = sin(pi x)$$



                Now, letting $x : = x-t$, we yield :
                $$u(x,t) = F(x-t) = sin[pi(x-t)]$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Mar 14 at 16:59









                RebellosRebellos

                15.4k31250




                15.4k31250



























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