Viscosity Solution for Hamilton Jacobi equation
$begingroup$
I am trying to do a numerical approximation of the viscosity solution for the Hamilton-Jacobi problem
$hspace{5cm}displaystyle frac{du}{dt}+frac{|u_x|^2}{2}=0 hspace{0.5cm} x in mathbb{R},hspace{0.3cm}t in [0,+infty[$
$hspace{5cm}displaystyle u(x,0)=sin(x)hspace{0.5cm}x in mathbb{R}$
So I need to calculate the explicit solution to compare them. I read about Lax-Hopf formula applied to the general problem and the solution should be
$$u(x,t)=inf_{y in mathbb{R}} left { sin(y)+frac{|x-y|^2}{2t}right }$$
Can someone give me some help/ hint to find a explicit form for this infimus? Greetings
pde hamilton-jacobi-equation viscosity-solutions
edited Dec 11 '18 at 11:36
Harry49
7,26831240
asked Oct 9 '17 at 9:23
Mat128Mat128
1204
$endgroup$
add a comment |
$begingroup$
I am trying to do a numerical approximation of the viscosity solution for the Hamilton-Jacobi problem
$hspace{5cm}displaystyle frac{du}{dt}+frac{|u_x|^2}{2}=0 hspace{0.5cm} x in mathbb{R},hspace{0.3cm}t in [0,+infty[$
$hspace{5cm}displaystyle u(x,0)=sin(x)hspace{0.5cm}x in mathbb{R}$
So I need to calculate the explicit solution to compare them. I read about Lax-Hopf formula applied to the general problem and the solution should be
$$u(x,t)=inf_{y in mathbb{R}} left { sin(y)+frac{|x-y|^2}{2t}right }$$
Can someone give me some help/ hint to find a explicit form for this infimus? Greetings
pde hamilton-jacobi-equation viscosity-solutions
edited Dec 11 '18 at 11:36
Harry49
7,26831240
asked Oct 9 '17 at 9:23
Mat128Mat128
1204
$endgroup$
$begingroup$
The minimum is attained when $$cos(y) + (y-x)/t=0.$$ Such equations generally can not be solved explicitly. You might try the method of characteristics.
$endgroup$
– Jeff
Oct 11 '17 at 15:00
$begingroup$
I should add, you can always compute the inf using a root finding algorithm, like Newton's method, up to machine accuracy to compare your numerical solution against.
$endgroup$
– Jeff
Oct 11 '17 at 15:02
$begingroup$
Thx, ill try to compute this.
$endgroup$
– Mat128
Oct 12 '17 at 9:14
add a comment |
$begingroup$
I am trying to do a numerical approximation of the viscosity solution for the Hamilton-Jacobi problem
$hspace{5cm}displaystyle frac{du}{dt}+frac{|u_x|^2}{2}=0 hspace{0.5cm} x in mathbb{R},hspace{0.3cm}t in [0,+infty[$
$hspace{5cm}displaystyle u(x,0)=sin(x)hspace{0.5cm}x in mathbb{R}$
So I need to calculate the explicit solution to compare them. I read about Lax-Hopf formula applied to the general problem and the solution should be
$$u(x,t)=inf_{y in mathbb{R}} left { sin(y)+frac{|x-y|^2}{2t}right }$$
Can someone give me some help/ hint to find a explicit form for this infimus? Greetings
pde hamilton-jacobi-equation viscosity-solutions
edited Dec 11 '18 at 11:36
Harry49
7,26831240
asked Oct 9 '17 at 9:23
Mat128Mat128
1204
$endgroup$
I am trying to do a numerical approximation of the viscosity solution for the Hamilton-Jacobi problem
$hspace{5cm}displaystyle frac{du}{dt}+frac{|u_x|^2}{2}=0 hspace{0.5cm} x in mathbb{R},hspace{0.3cm}t in [0,+infty[$
$hspace{5cm}displaystyle u(x,0)=sin(x)hspace{0.5cm}x in mathbb{R}$
So I need to calculate the explicit solution to compare them. I read about Lax-Hopf formula applied to the general problem and the solution should be
$$u(x,t)=inf_{y in mathbb{R}} left { sin(y)+frac{|x-y|^2}{2t}right }$$
Can someone give me some help/ hint to find a explicit form for this infimus? Greetings
pde hamilton-jacobi-equation viscosity-solutions
pde hamilton-jacobi-equation viscosity-solutions
edited Dec 11 '18 at 11:36
Harry49
7,26831240
asked Oct 9 '17 at 9:23
Mat128Mat128
1204
edited Dec 11 '18 at 11:36
Harry49
7,26831240
asked Oct 9 '17 at 9:23
Mat128Mat128
1204
edited Dec 11 '18 at 11:36
Harry49
7,26831240
edited Dec 11 '18 at 11:36
Harry49
7,26831240
edited Dec 11 '18 at 11:36
Harry49
7,26831240
7,26831240
asked Oct 9 '17 at 9:23
Mat128Mat128
1204
asked Oct 9 '17 at 9:23
Mat128Mat128
1204
asked Oct 9 '17 at 9:23
Mat128Mat128
1204
1204
$begingroup$
The minimum is attained when $$cos(y) + (y-x)/t=0.$$ Such equations generally can not be solved explicitly. You might try the method of characteristics.
$endgroup$
– Jeff
Oct 11 '17 at 15:00
$begingroup$
I should add, you can always compute the inf using a root finding algorithm, like Newton's method, up to machine accuracy to compare your numerical solution against.
$endgroup$
– Jeff
Oct 11 '17 at 15:02
$begingroup$
Thx, ill try to compute this.
$endgroup$
– Mat128
Oct 12 '17 at 9:14
add a comment |
$begingroup$
The minimum is attained when $$cos(y) + (y-x)/t=0.$$ Such equations generally can not be solved explicitly. You might try the method of characteristics.
$endgroup$
– Jeff
Oct 11 '17 at 15:00
$begingroup$
I should add, you can always compute the inf using a root finding algorithm, like Newton's method, up to machine accuracy to compare your numerical solution against.
$endgroup$
– Jeff
Oct 11 '17 at 15:02
$begingroup$
Thx, ill try to compute this.
$endgroup$
– Mat128
Oct 12 '17 at 9:14
$begingroup$
The minimum is attained when $$cos(y) + (y-x)/t=0.$$ Such equations generally can not be solved explicitly. You might try the method of characteristics.
$endgroup$
– Jeff
Oct 11 '17 at 15:00
$begingroup$
The minimum is attained when $$cos(y) + (y-x)/t=0.$$ Such equations generally can not be solved explicitly. You might try the method of characteristics.
$endgroup$
– Jeff
Oct 11 '17 at 15:00
$begingroup$
I should add, you can always compute the inf using a root finding algorithm, like Newton's method, up to machine accuracy to compare your numerical solution against.
$endgroup$
– Jeff
Oct 11 '17 at 15:02
$begingroup$
I should add, you can always compute the inf using a root finding algorithm, like Newton's method, up to machine accuracy to compare your numerical solution against.
$endgroup$
– Jeff
Oct 11 '17 at 15:02
$begingroup$
Thx, ill try to compute this.
$endgroup$
– Mat128
Oct 12 '17 at 9:14
$begingroup$
Thx, ill try to compute this.
$endgroup$
– Mat128
Oct 12 '17 at 9:14
add a comment |
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$begingroup$
The minimum is attained when $$cos(y) + (y-x)/t=0.$$ Such equations generally can not be solved explicitly. You might try the method of characteristics.
$endgroup$
– Jeff
Oct 11 '17 at 15:00
$begingroup$
I should add, you can always compute the inf using a root finding algorithm, like Newton's method, up to machine accuracy to compare your numerical solution against.
$endgroup$
– Jeff
Oct 11 '17 at 15:02
$begingroup$
Thx, ill try to compute this.
$endgroup$
– Mat128
Oct 12 '17 at 9:14