Uniqueness of solution based on characteristic curves
I have a pde
$$begin{cases} u_t − xu_x = 2u & xinmathbb{R}, t>0\
u(x, 0) = frac{1}{1+x^2}
end{cases}$$
I've solved it using method of characteristics ($u=frac{1}{1+x^2e^{2t}}e^{2t})$ and plotted charactersitic curves.
Consider the upper half-space since $t>0$.
How to argue using the drawing whether or not it is the unique solution? Thank you.
pde characteristics
edited Dec 9 at 12:28
Harry49
5,99121031
asked Nov 18 at 22:38
dxdydz
1949
add a comment |
I have a pde
$$begin{cases} u_t − xu_x = 2u & xinmathbb{R}, t>0\
u(x, 0) = frac{1}{1+x^2}
end{cases}$$
I've solved it using method of characteristics ($u=frac{1}{1+x^2e^{2t}}e^{2t})$ and plotted charactersitic curves.
Consider the upper half-space since $t>0$.
How to argue using the drawing whether or not it is the unique solution? Thank you.
pde characteristics
edited Dec 9 at 12:28
Harry49
5,99121031
asked Nov 18 at 22:38
dxdydz
1949
Are you plotting in the $(t, u)$ plane? $(x, u)$? $(t, x)$?
– Mattos
Nov 19 at 1:46
@Mattos This is the projection on the $(x,t)$-plane
– dxdydz
Nov 19 at 2:01
add a comment |
I have a pde
$$begin{cases} u_t − xu_x = 2u & xinmathbb{R}, t>0\
u(x, 0) = frac{1}{1+x^2}
end{cases}$$
I've solved it using method of characteristics ($u=frac{1}{1+x^2e^{2t}}e^{2t})$ and plotted charactersitic curves.
Consider the upper half-space since $t>0$.
How to argue using the drawing whether or not it is the unique solution? Thank you.
pde characteristics
edited Dec 9 at 12:28
Harry49
5,99121031
asked Nov 18 at 22:38
dxdydz
1949
I have a pde
$$begin{cases} u_t − xu_x = 2u & xinmathbb{R}, t>0\
u(x, 0) = frac{1}{1+x^2}
end{cases}$$
I've solved it using method of characteristics ($u=frac{1}{1+x^2e^{2t}}e^{2t})$ and plotted charactersitic curves.
Consider the upper half-space since $t>0$.
How to argue using the drawing whether or not it is the unique solution? Thank you.
pde characteristics
pde characteristics
edited Dec 9 at 12:28
Harry49
5,99121031
asked Nov 18 at 22:38
dxdydz
1949
edited Dec 9 at 12:28
Harry49
5,99121031
asked Nov 18 at 22:38
dxdydz
1949
edited Dec 9 at 12:28
Harry49
5,99121031
edited Dec 9 at 12:28
Harry49
5,99121031
edited Dec 9 at 12:28
Harry49
5,99121031
5,99121031
asked Nov 18 at 22:38
dxdydz
1949
asked Nov 18 at 22:38
dxdydz
1949
asked Nov 18 at 22:38
dxdydz
1949
1949
Are you plotting in the $(t, u)$ plane? $(x, u)$? $(t, x)$?
– Mattos
Nov 19 at 1:46
@Mattos This is the projection on the $(x,t)$-plane
– dxdydz
Nov 19 at 2:01
add a comment |
Are you plotting in the $(t, u)$ plane? $(x, u)$? $(t, x)$?
– Mattos
Nov 19 at 1:46
@Mattos This is the projection on the $(x,t)$-plane
– dxdydz
Nov 19 at 2:01
Are you plotting in the $(t, u)$ plane? $(x, u)$? $(t, x)$?
– Mattos
Nov 19 at 1:46
Are you plotting in the $(t, u)$ plane? $(x, u)$? $(t, x)$?
– Mattos
Nov 19 at 1:46
@Mattos This is the projection on the $(x,t)$-plane
– dxdydz
Nov 19 at 2:01
@Mattos This is the projection on the $(x,t)$-plane
– dxdydz
Nov 19 at 2:01
add a comment |
1 Answer
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The method of characteristics transforms the PDE into an ODE system. Therefore, existence and uniqueness is guaranteed under the assumptions of the Picard-Lindelöf theorem.
answered Dec 9 at 12:27
Harry49
5,99121031
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
The method of characteristics transforms the PDE into an ODE system. Therefore, existence and uniqueness is guaranteed under the assumptions of the Picard-Lindelöf theorem.
answered Dec 9 at 12:27
Harry49
5,99121031
add a comment |
The method of characteristics transforms the PDE into an ODE system. Therefore, existence and uniqueness is guaranteed under the assumptions of the Picard-Lindelöf theorem.
answered Dec 9 at 12:27
Harry49
5,99121031
add a comment |
The method of characteristics transforms the PDE into an ODE system. Therefore, existence and uniqueness is guaranteed under the assumptions of the Picard-Lindelöf theorem.
answered Dec 9 at 12:27
Harry49
5,99121031
The method of characteristics transforms the PDE into an ODE system. Therefore, existence and uniqueness is guaranteed under the assumptions of the Picard-Lindelöf theorem.
answered Dec 9 at 12:27
Harry49
5,99121031
answered Dec 9 at 12:27
Harry49
5,99121031
answered Dec 9 at 12:27
Harry49
5,99121031
answered Dec 9 at 12:27
Harry49
5,99121031
5,99121031
add a comment |
add a comment |
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Are you plotting in the $(t, u)$ plane? $(x, u)$? $(t, x)$?
– Mattos
Nov 19 at 1:46
@Mattos This is the projection on the $(x,t)$-plane
– dxdydz
Nov 19 at 2:01