Uniqueness of solution based on characteristic curves
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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
asked Nov 18 at 22:38
dxdydz
1169
add a comment |
up vote
2
down vote
favorite
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
asked Nov 18 at 22:38
dxdydz
1169
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 |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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
asked Nov 18 at 22:38
dxdydz
1169
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
pde
asked Nov 18 at 22:38
dxdydz
1169
asked Nov 18 at 22:38
dxdydz
1169
asked Nov 18 at 22:38
dxdydz
1169
asked Nov 18 at 22:38
dxdydz
1169
asked Nov 18 at 22:38
dxdydz
1169
1169
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 |
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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