Transform an inhomogeneous dirichlet problem into a homogeneous one? i.e. Burgers equation
0
$begingroup$
I need to transform an inhomogeneous dirichlet problem into a homogeneous one. For exeample the burgers equation:
$$begin{cases} partial_t u(x,t) + u(x,t) partial_x u(x,t)=0\
u(0,t)=0,& forall tin [0,1]\
u(x,0)=g(x),&forall xin [0,1] end{cases}
$$
So how can I transform this problem to something with zero boundaries?
pde boundary-value-problem
asked Dec 5 '18 at 21:12
cptflintcptflint
208
$endgroup$
|
show 6 more comments
0
$begingroup$
I need to transform an inhomogeneous dirichlet problem into a homogeneous one. For exeample the burgers equation:
$$begin{cases} partial_t u(x,t) + u(x,t) partial_x u(x,t)=0\
u(0,t)=0,& forall tin [0,1]\
u(x,0)=g(x),&forall xin [0,1] end{cases}
$$
So how can I transform this problem to something with zero boundaries?
pde boundary-value-problem
asked Dec 5 '18 at 21:12
cptflintcptflint
208
$endgroup$
$begingroup$
Assuming that $g(0)=0$, you can look at $u(x,t)=v(x,t)+h(x,t)$ where $h(x,t)$ is a function that you choose for yourself with the property $h(x,0)=g(x)$ and $h(0,t)=0$. If $g(0) neq 0$ then you must be more careful in handling the corner singularity.
$endgroup$
– Ian
Dec 5 '18 at 21:15
2
$begingroup$
That said, homogeneous spatial boundary data is usually desirable but homogeneous temporal boundary data (i.e. initial or final data) is typically not a problem for most methods of solution.
$endgroup$
– Ian
Dec 5 '18 at 21:17
$begingroup$
Regarding what the equation describes, I would be happy with $g$ having compact support and $g(0)=0$. But how to choose $h$? And then I would solve $partial_t (v+h) + (v+h) partial_x (v+h)=0$?
$endgroup$
– cptflint
Dec 5 '18 at 21:18
$begingroup$
You pick $h$ to be "simple" so that the resulting inhomogeneous PDE for $v$ is not too difficult to solve. Unfortunately here the nonlinearity produces a difficulty because you cannot simply move all the terms involving $h$ to the other side as a forcing, since there are terms $h partial_x v$ and $v partial_x h$. But at least these terms are linear in $v$...
$endgroup$
– Ian
Dec 5 '18 at 21:20
$begingroup$
resulting inhomogeneous PDE? So my problem is, that I am in need of zero boundaries.
$endgroup$
– cptflint
Dec 5 '18 at 21:23
|
show 6 more comments
0
0
0
$begingroup$
I need to transform an inhomogeneous dirichlet problem into a homogeneous one. For exeample the burgers equation:
$$begin{cases} partial_t u(x,t) + u(x,t) partial_x u(x,t)=0\
u(0,t)=0,& forall tin [0,1]\
u(x,0)=g(x),&forall xin [0,1] end{cases}
$$
So how can I transform this problem to something with zero boundaries?
pde boundary-value-problem
asked Dec 5 '18 at 21:12
cptflintcptflint
208
$endgroup$
I need to transform an inhomogeneous dirichlet problem into a homogeneous one. For exeample the burgers equation:
$$begin{cases} partial_t u(x,t) + u(x,t) partial_x u(x,t)=0\
u(0,t)=0,& forall tin [0,1]\
u(x,0)=g(x),&forall xin [0,1] end{cases}
$$
So how can I transform this problem to something with zero boundaries?
pde boundary-value-problem
pde boundary-value-problem
asked Dec 5 '18 at 21:12
cptflintcptflint
208
asked Dec 5 '18 at 21:12
cptflintcptflint
208
asked Dec 5 '18 at 21:12
cptflintcptflint
208
asked Dec 5 '18 at 21:12
cptflintcptflint
208
asked Dec 5 '18 at 21:12
cptflintcptflint
208
208
$begingroup$
Assuming that $g(0)=0$, you can look at $u(x,t)=v(x,t)+h(x,t)$ where $h(x,t)$ is a function that you choose for yourself with the property $h(x,0)=g(x)$ and $h(0,t)=0$. If $g(0) neq 0$ then you must be more careful in handling the corner singularity.
$endgroup$
– Ian
Dec 5 '18 at 21:15
2
$begingroup$
That said, homogeneous spatial boundary data is usually desirable but homogeneous temporal boundary data (i.e. initial or final data) is typically not a problem for most methods of solution.
$endgroup$
– Ian
Dec 5 '18 at 21:17
$begingroup$
Regarding what the equation describes, I would be happy with $g$ having compact support and $g(0)=0$. But how to choose $h$? And then I would solve $partial_t (v+h) + (v+h) partial_x (v+h)=0$?
$endgroup$
– cptflint
Dec 5 '18 at 21:18
$begingroup$
You pick $h$ to be "simple" so that the resulting inhomogeneous PDE for $v$ is not too difficult to solve. Unfortunately here the nonlinearity produces a difficulty because you cannot simply move all the terms involving $h$ to the other side as a forcing, since there are terms $h partial_x v$ and $v partial_x h$. But at least these terms are linear in $v$...
$endgroup$
– Ian
Dec 5 '18 at 21:20
$begingroup$
resulting inhomogeneous PDE? So my problem is, that I am in need of zero boundaries.
$endgroup$
– cptflint
Dec 5 '18 at 21:23
|
show 6 more comments
$begingroup$
Assuming that $g(0)=0$, you can look at $u(x,t)=v(x,t)+h(x,t)$ where $h(x,t)$ is a function that you choose for yourself with the property $h(x,0)=g(x)$ and $h(0,t)=0$. If $g(0) neq 0$ then you must be more careful in handling the corner singularity.
$endgroup$
– Ian
Dec 5 '18 at 21:15
2
$begingroup$
That said, homogeneous spatial boundary data is usually desirable but homogeneous temporal boundary data (i.e. initial or final data) is typically not a problem for most methods of solution.
$endgroup$
– Ian
Dec 5 '18 at 21:17
$begingroup$
Regarding what the equation describes, I would be happy with $g$ having compact support and $g(0)=0$. But how to choose $h$? And then I would solve $partial_t (v+h) + (v+h) partial_x (v+h)=0$?
$endgroup$
– cptflint
Dec 5 '18 at 21:18
$begingroup$
You pick $h$ to be "simple" so that the resulting inhomogeneous PDE for $v$ is not too difficult to solve. Unfortunately here the nonlinearity produces a difficulty because you cannot simply move all the terms involving $h$ to the other side as a forcing, since there are terms $h partial_x v$ and $v partial_x h$. But at least these terms are linear in $v$...
$endgroup$
– Ian
Dec 5 '18 at 21:20
$begingroup$
resulting inhomogeneous PDE? So my problem is, that I am in need of zero boundaries.
$endgroup$
– cptflint
Dec 5 '18 at 21:23
$begingroup$
Assuming that $g(0)=0$, you can look at $u(x,t)=v(x,t)+h(x,t)$ where $h(x,t)$ is a function that you choose for yourself with the property $h(x,0)=g(x)$ and $h(0,t)=0$. If $g(0) neq 0$ then you must be more careful in handling the corner singularity.
$endgroup$
– Ian
Dec 5 '18 at 21:15
$begingroup$
Assuming that $g(0)=0$, you can look at $u(x,t)=v(x,t)+h(x,t)$ where $h(x,t)$ is a function that you choose for yourself with the property $h(x,0)=g(x)$ and $h(0,t)=0$. If $g(0) neq 0$ then you must be more careful in handling the corner singularity.
$endgroup$
– Ian
Dec 5 '18 at 21:15
2
2
$begingroup$
That said, homogeneous spatial boundary data is usually desirable but homogeneous temporal boundary data (i.e. initial or final data) is typically not a problem for most methods of solution.
$endgroup$
– Ian
Dec 5 '18 at 21:17
$begingroup$
That said, homogeneous spatial boundary data is usually desirable but homogeneous temporal boundary data (i.e. initial or final data) is typically not a problem for most methods of solution.
$endgroup$
– Ian
Dec 5 '18 at 21:17
$begingroup$
Regarding what the equation describes, I would be happy with $g$ having compact support and $g(0)=0$. But how to choose $h$? And then I would solve $partial_t (v+h) + (v+h) partial_x (v+h)=0$?
$endgroup$
– cptflint
Dec 5 '18 at 21:18
$begingroup$
Regarding what the equation describes, I would be happy with $g$ having compact support and $g(0)=0$. But how to choose $h$? And then I would solve $partial_t (v+h) + (v+h) partial_x (v+h)=0$?
$endgroup$
– cptflint
Dec 5 '18 at 21:18
$begingroup$
You pick $h$ to be "simple" so that the resulting inhomogeneous PDE for $v$ is not too difficult to solve. Unfortunately here the nonlinearity produces a difficulty because you cannot simply move all the terms involving $h$ to the other side as a forcing, since there are terms $h partial_x v$ and $v partial_x h$. But at least these terms are linear in $v$...
$endgroup$
– Ian
Dec 5 '18 at 21:20
$begingroup$
You pick $h$ to be "simple" so that the resulting inhomogeneous PDE for $v$ is not too difficult to solve. Unfortunately here the nonlinearity produces a difficulty because you cannot simply move all the terms involving $h$ to the other side as a forcing, since there are terms $h partial_x v$ and $v partial_x h$. But at least these terms are linear in $v$...
$endgroup$
– Ian
Dec 5 '18 at 21:20
$begingroup$
resulting inhomogeneous PDE? So my problem is, that I am in need of zero boundaries.
$endgroup$
– cptflint
Dec 5 '18 at 21:23
$begingroup$
resulting inhomogeneous PDE? So my problem is, that I am in need of zero boundaries.
$endgroup$
– cptflint
Dec 5 '18 at 21:23
|
show 6 more comments
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$begingroup$
Assuming that $g(0)=0$, you can look at $u(x,t)=v(x,t)+h(x,t)$ where $h(x,t)$ is a function that you choose for yourself with the property $h(x,0)=g(x)$ and $h(0,t)=0$. If $g(0) neq 0$ then you must be more careful in handling the corner singularity.
$endgroup$
– Ian
Dec 5 '18 at 21:15
2
$begingroup$
That said, homogeneous spatial boundary data is usually desirable but homogeneous temporal boundary data (i.e. initial or final data) is typically not a problem for most methods of solution.
$endgroup$
– Ian
Dec 5 '18 at 21:17
$begingroup$
Regarding what the equation describes, I would be happy with $g$ having compact support and $g(0)=0$. But how to choose $h$? And then I would solve $partial_t (v+h) + (v+h) partial_x (v+h)=0$?
$endgroup$
– cptflint
Dec 5 '18 at 21:18
$begingroup$
You pick $h$ to be "simple" so that the resulting inhomogeneous PDE for $v$ is not too difficult to solve. Unfortunately here the nonlinearity produces a difficulty because you cannot simply move all the terms involving $h$ to the other side as a forcing, since there are terms $h partial_x v$ and $v partial_x h$. But at least these terms are linear in $v$...
$endgroup$
– Ian
Dec 5 '18 at 21:20
$begingroup$
resulting inhomogeneous PDE? So my problem is, that I am in need of zero boundaries.
$endgroup$
– cptflint
Dec 5 '18 at 21:23