A generalised Cauchy problem with Burgers' differential equation












0












$begingroup$


Consider the following:
$$u_t+uu_x=0, ~~ t>0$$



and the initial data:
$$
u(x,0)=begin{cases}
1,&text{ if }xin[0,1]text{ and }
\
0,&text{ otherwise. }
end{cases}$$



I have found a solution to the above problem like the following picture but is valid only for $t<2$ ...I do not know how to extend this solution for all $t>0$. Any hint please?
enter image description here



Remark: I have found this weak solution drawing the characteristic lines of the problem and applying a shock and fan wave where the characteristics intersect and where a "gap" is created, respectively. I confirmed this solution using Runkine - Hugionot theorem.






ordinary-differential-equations pde







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You should add some remarks on how you computed the shock front and its meeting the dispersion front. The image shows this, but that is not enough.
    $endgroup$
    – LutzL
    Dec 9 '18 at 11:13








  • 1




    $begingroup$
    See Burgers' equation after rarefaction wave catches up with the shock for a very similar question with a complete answer. Also related Rarefaction and shock waves colliding in Burgers' equation for a more complex example (with illustrations).
    $endgroup$
    – LutzL
    Dec 9 '18 at 11:32
















0












$begingroup$


Consider the following:
$$u_t+uu_x=0, ~~ t>0$$



and the initial data:
$$
u(x,0)=begin{cases}
1,&text{ if }xin[0,1]text{ and }
\
0,&text{ otherwise. }
end{cases}$$



I have found a solution to the above problem like the following picture but is valid only for $t<2$ ...I do not know how to extend this solution for all $t>0$. Any hint please?
enter image description here



Remark: I have found this weak solution drawing the characteristic lines of the problem and applying a shock and fan wave where the characteristics intersect and where a "gap" is created, respectively. I confirmed this solution using Runkine - Hugionot theorem.






ordinary-differential-equations pde







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    You should add some remarks on how you computed the shock front and its meeting the dispersion front. The image shows this, but that is not enough.
    $endgroup$
    – LutzL
    Dec 9 '18 at 11:13








  • 1




    $begingroup$
    See Burgers' equation after rarefaction wave catches up with the shock for a very similar question with a complete answer. Also related Rarefaction and shock waves colliding in Burgers' equation for a more complex example (with illustrations).
    $endgroup$
    – LutzL
    Dec 9 '18 at 11:32














0












0








0





$begingroup$


Consider the following:
$$u_t+uu_x=0, ~~ t>0$$



and the initial data:
$$
u(x,0)=begin{cases}
1,&text{ if }xin[0,1]text{ and }
\
0,&text{ otherwise. }
end{cases}$$



I have found a solution to the above problem like the following picture but is valid only for $t<2$ ...I do not know how to extend this solution for all $t>0$. Any hint please?
enter image description here



Remark: I have found this weak solution drawing the characteristic lines of the problem and applying a shock and fan wave where the characteristics intersect and where a "gap" is created, respectively. I confirmed this solution using Runkine - Hugionot theorem.






ordinary-differential-equations pde







share|cite|improve this question











$endgroup$




Consider the following:
$$u_t+uu_x=0, ~~ t>0$$



and the initial data:
$$
u(x,0)=begin{cases}
1,&text{ if }xin[0,1]text{ and }
\
0,&text{ otherwise. }
end{cases}$$



I have found a solution to the above problem like the following picture but is valid only for $t<2$ ...I do not know how to extend this solution for all $t>0$. Any hint please?
enter image description here



Remark: I have found this weak solution drawing the characteristic lines of the problem and applying a shock and fan wave where the characteristics intersect and where a "gap" is created, respectively. I confirmed this solution using Runkine - Hugionot theorem.






ordinary-differential-equations pde




ordinary-differential-equations pde






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 12:16









LutzL

57.3k42054




57.3k42054










asked Dec 9 '18 at 10:41









dmtridmtri

1,4522521




1,4522521








  • 1




    $begingroup$
    You should add some remarks on how you computed the shock front and its meeting the dispersion front. The image shows this, but that is not enough.
    $endgroup$
    – LutzL
    Dec 9 '18 at 11:13








  • 1




    $begingroup$
    See Burgers' equation after rarefaction wave catches up with the shock for a very similar question with a complete answer. Also related Rarefaction and shock waves colliding in Burgers' equation for a more complex example (with illustrations).
    $endgroup$
    – LutzL
    Dec 9 '18 at 11:32














  • 1




    $begingroup$
    You should add some remarks on how you computed the shock front and its meeting the dispersion front. The image shows this, but that is not enough.
    $endgroup$
    – LutzL
    Dec 9 '18 at 11:13








  • 1




    $begingroup$
    See Burgers' equation after rarefaction wave catches up with the shock for a very similar question with a complete answer. Also related Rarefaction and shock waves colliding in Burgers' equation for a more complex example (with illustrations).
    $endgroup$
    – LutzL
    Dec 9 '18 at 11:32








1




1




$begingroup$
You should add some remarks on how you computed the shock front and its meeting the dispersion front. The image shows this, but that is not enough.
$endgroup$
– LutzL
Dec 9 '18 at 11:13






$begingroup$
You should add some remarks on how you computed the shock front and its meeting the dispersion front. The image shows this, but that is not enough.
$endgroup$
– LutzL
Dec 9 '18 at 11:13






1




1




$begingroup$
See Burgers' equation after rarefaction wave catches up with the shock for a very similar question with a complete answer. Also related Rarefaction and shock waves colliding in Burgers' equation for a more complex example (with illustrations).
$endgroup$
– LutzL
Dec 9 '18 at 11:32




$begingroup$
See Burgers' equation after rarefaction wave catches up with the shock for a very similar question with a complete answer. Also related Rarefaction and shock waves colliding in Burgers' equation for a more complex example (with illustrations).
$endgroup$
– LutzL
Dec 9 '18 at 11:32










1 Answer
1






active

oldest

votes


















1












$begingroup$

After $t=2$ you have that the rarefaction segment $0<x<a(t)$ where $u(x,t)=x/t$ is directly followed by the "unchanged" segment with $u(x,t)=0$ on $a(t)<x<infty$. The change of the phase boundary is again governed by the Runkine-Hugionot condition, that is
$$
dot a(t)=frac{a(t)/t+0}{2}implies a(t)=csqrt{t}
$$
and from the initial condition $a(2)=2$ it follows that $c=sqrt2$, $a(t)=sqrt{2t}$.



See Burgers' equation after rarefaction wave catches up with the shock for a more extensive discussion of this situation.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    @dmtri The full solution is also provided in this post
    $endgroup$
    – Harry49
    Dec 10 '18 at 10:03










  • $begingroup$
    Great help, thanks again! It is also weird, at least for me, that the first shock appears in zero time, @Harry49
    $endgroup$
    – dmtri
    Dec 10 '18 at 13:57






  • 1




    $begingroup$
    @dmtri : A shock is a discontinuity in the solution. In the Burger's equation, a shock is associated to a downward jump discontinuity. You have the jump already in the initial condition.
    $endgroup$
    – LutzL
    Dec 10 '18 at 14:05











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1 Answer
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1 Answer
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1












$begingroup$

After $t=2$ you have that the rarefaction segment $0<x<a(t)$ where $u(x,t)=x/t$ is directly followed by the "unchanged" segment with $u(x,t)=0$ on $a(t)<x<infty$. The change of the phase boundary is again governed by the Runkine-Hugionot condition, that is
$$
dot a(t)=frac{a(t)/t+0}{2}implies a(t)=csqrt{t}
$$
and from the initial condition $a(2)=2$ it follows that $c=sqrt2$, $a(t)=sqrt{2t}$.



See Burgers' equation after rarefaction wave catches up with the shock for a more extensive discussion of this situation.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    @dmtri The full solution is also provided in this post
    $endgroup$
    – Harry49
    Dec 10 '18 at 10:03










  • $begingroup$
    Great help, thanks again! It is also weird, at least for me, that the first shock appears in zero time, @Harry49
    $endgroup$
    – dmtri
    Dec 10 '18 at 13:57






  • 1




    $begingroup$
    @dmtri : A shock is a discontinuity in the solution. In the Burger's equation, a shock is associated to a downward jump discontinuity. You have the jump already in the initial condition.
    $endgroup$
    – LutzL
    Dec 10 '18 at 14:05
















1












$begingroup$

After $t=2$ you have that the rarefaction segment $0<x<a(t)$ where $u(x,t)=x/t$ is directly followed by the "unchanged" segment with $u(x,t)=0$ on $a(t)<x<infty$. The change of the phase boundary is again governed by the Runkine-Hugionot condition, that is
$$
dot a(t)=frac{a(t)/t+0}{2}implies a(t)=csqrt{t}
$$
and from the initial condition $a(2)=2$ it follows that $c=sqrt2$, $a(t)=sqrt{2t}$.



See Burgers' equation after rarefaction wave catches up with the shock for a more extensive discussion of this situation.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    @dmtri The full solution is also provided in this post
    $endgroup$
    – Harry49
    Dec 10 '18 at 10:03










  • $begingroup$
    Great help, thanks again! It is also weird, at least for me, that the first shock appears in zero time, @Harry49
    $endgroup$
    – dmtri
    Dec 10 '18 at 13:57






  • 1




    $begingroup$
    @dmtri : A shock is a discontinuity in the solution. In the Burger's equation, a shock is associated to a downward jump discontinuity. You have the jump already in the initial condition.
    $endgroup$
    – LutzL
    Dec 10 '18 at 14:05














1












1








1





$begingroup$

After $t=2$ you have that the rarefaction segment $0<x<a(t)$ where $u(x,t)=x/t$ is directly followed by the "unchanged" segment with $u(x,t)=0$ on $a(t)<x<infty$. The change of the phase boundary is again governed by the Runkine-Hugionot condition, that is
$$
dot a(t)=frac{a(t)/t+0}{2}implies a(t)=csqrt{t}
$$
and from the initial condition $a(2)=2$ it follows that $c=sqrt2$, $a(t)=sqrt{2t}$.



See Burgers' equation after rarefaction wave catches up with the shock for a more extensive discussion of this situation.






share|cite|improve this answer









$endgroup$



After $t=2$ you have that the rarefaction segment $0<x<a(t)$ where $u(x,t)=x/t$ is directly followed by the "unchanged" segment with $u(x,t)=0$ on $a(t)<x<infty$. The change of the phase boundary is again governed by the Runkine-Hugionot condition, that is
$$
dot a(t)=frac{a(t)/t+0}{2}implies a(t)=csqrt{t}
$$
and from the initial condition $a(2)=2$ it follows that $c=sqrt2$, $a(t)=sqrt{2t}$.



See Burgers' equation after rarefaction wave catches up with the shock for a more extensive discussion of this situation.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 9 '18 at 16:23









LutzLLutzL

57.3k42054




57.3k42054








  • 1




    $begingroup$
    @dmtri The full solution is also provided in this post
    $endgroup$
    – Harry49
    Dec 10 '18 at 10:03










  • $begingroup$
    Great help, thanks again! It is also weird, at least for me, that the first shock appears in zero time, @Harry49
    $endgroup$
    – dmtri
    Dec 10 '18 at 13:57






  • 1




    $begingroup$
    @dmtri : A shock is a discontinuity in the solution. In the Burger's equation, a shock is associated to a downward jump discontinuity. You have the jump already in the initial condition.
    $endgroup$
    – LutzL
    Dec 10 '18 at 14:05














  • 1




    $begingroup$
    @dmtri The full solution is also provided in this post
    $endgroup$
    – Harry49
    Dec 10 '18 at 10:03










  • $begingroup$
    Great help, thanks again! It is also weird, at least for me, that the first shock appears in zero time, @Harry49
    $endgroup$
    – dmtri
    Dec 10 '18 at 13:57






  • 1




    $begingroup$
    @dmtri : A shock is a discontinuity in the solution. In the Burger's equation, a shock is associated to a downward jump discontinuity. You have the jump already in the initial condition.
    $endgroup$
    – LutzL
    Dec 10 '18 at 14:05








1




1




$begingroup$
@dmtri The full solution is also provided in this post
$endgroup$
– Harry49
Dec 10 '18 at 10:03




$begingroup$
@dmtri The full solution is also provided in this post
$endgroup$
– Harry49
Dec 10 '18 at 10:03












$begingroup$
Great help, thanks again! It is also weird, at least for me, that the first shock appears in zero time, @Harry49
$endgroup$
– dmtri
Dec 10 '18 at 13:57




$begingroup$
Great help, thanks again! It is also weird, at least for me, that the first shock appears in zero time, @Harry49
$endgroup$
– dmtri
Dec 10 '18 at 13:57




1




1




$begingroup$
@dmtri : A shock is a discontinuity in the solution. In the Burger's equation, a shock is associated to a downward jump discontinuity. You have the jump already in the initial condition.
$endgroup$
– LutzL
Dec 10 '18 at 14:05




$begingroup$
@dmtri : A shock is a discontinuity in the solution. In the Burger's equation, a shock is associated to a downward jump discontinuity. You have the jump already in the initial condition.
$endgroup$
– LutzL
Dec 10 '18 at 14:05


















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