Underdetermined IBVP's: constructing boundary condition during evolution
$begingroup$
For $(t,x)in[0,T]times[0,X]$ let there be a partial differential equation (PDE) of the generic form
$$partial_ty=f(t,x,y,partial_xy)$$
along with some initial condition (i.c.)
$$y_i(x):=y(0,x)$$
but no boundary condition (b.c.).
The above initial-boundary value problem (IBVP) is profoundly underdetermined.
However one may come to think that for some infinitesimal $h$ there are two distinct approximations for the $x$-partial derivative:
$$partial_xyapproxfrac{y(t,x+h)-f(t,x)}{h}qquadqquadpartial_xyapproxfrac{y(t,x)-f(t,x-h)}{h}$$
Based on these approximations one can
- evaluate $partial_xy(0,x)$ for any $xin[0,X]$ and then
- evaluate $partial_ty(0,x)$ for any $xin[0,X]$ and then
- evaluate $y(0+dt,x)=y(0,x)+dtcdot partial_ty(0,x)$
In this way one is able to construct a unique solution $y(t,x)$ although the above IBVP is underdetermined. It is as if the missing boundary condition $y(t,X)$ is constructed in parallel with the evolution of initial data.
I would appreciate any recommendations of bibliography or ideas concerning this kind of integration.
ordinary-differential-equations pde numerical-methods partial-derivative boundary-value-problem
asked Dec 29 '18 at 10:51
dkstackdkstack
204
$endgroup$
add a comment |
$begingroup$
For $(t,x)in[0,T]times[0,X]$ let there be a partial differential equation (PDE) of the generic form
$$partial_ty=f(t,x,y,partial_xy)$$
along with some initial condition (i.c.)
$$y_i(x):=y(0,x)$$
but no boundary condition (b.c.).
The above initial-boundary value problem (IBVP) is profoundly underdetermined.
However one may come to think that for some infinitesimal $h$ there are two distinct approximations for the $x$-partial derivative:
$$partial_xyapproxfrac{y(t,x+h)-f(t,x)}{h}qquadqquadpartial_xyapproxfrac{y(t,x)-f(t,x-h)}{h}$$
Based on these approximations one can
- evaluate $partial_xy(0,x)$ for any $xin[0,X]$ and then
- evaluate $partial_ty(0,x)$ for any $xin[0,X]$ and then
- evaluate $y(0+dt,x)=y(0,x)+dtcdot partial_ty(0,x)$
In this way one is able to construct a unique solution $y(t,x)$ although the above IBVP is underdetermined. It is as if the missing boundary condition $y(t,X)$ is constructed in parallel with the evolution of initial data.
I would appreciate any recommendations of bibliography or ideas concerning this kind of integration.
ordinary-differential-equations pde numerical-methods partial-derivative boundary-value-problem
asked Dec 29 '18 at 10:51
dkstackdkstack
204
$endgroup$
add a comment |
$begingroup$
For $(t,x)in[0,T]times[0,X]$ let there be a partial differential equation (PDE) of the generic form
$$partial_ty=f(t,x,y,partial_xy)$$
along with some initial condition (i.c.)
$$y_i(x):=y(0,x)$$
but no boundary condition (b.c.).
The above initial-boundary value problem (IBVP) is profoundly underdetermined.
However one may come to think that for some infinitesimal $h$ there are two distinct approximations for the $x$-partial derivative:
$$partial_xyapproxfrac{y(t,x+h)-f(t,x)}{h}qquadqquadpartial_xyapproxfrac{y(t,x)-f(t,x-h)}{h}$$
Based on these approximations one can
- evaluate $partial_xy(0,x)$ for any $xin[0,X]$ and then
- evaluate $partial_ty(0,x)$ for any $xin[0,X]$ and then
- evaluate $y(0+dt,x)=y(0,x)+dtcdot partial_ty(0,x)$
In this way one is able to construct a unique solution $y(t,x)$ although the above IBVP is underdetermined. It is as if the missing boundary condition $y(t,X)$ is constructed in parallel with the evolution of initial data.
I would appreciate any recommendations of bibliography or ideas concerning this kind of integration.
ordinary-differential-equations pde numerical-methods partial-derivative boundary-value-problem
asked Dec 29 '18 at 10:51
dkstackdkstack
204
$endgroup$
For $(t,x)in[0,T]times[0,X]$ let there be a partial differential equation (PDE) of the generic form
$$partial_ty=f(t,x,y,partial_xy)$$
along with some initial condition (i.c.)
$$y_i(x):=y(0,x)$$
but no boundary condition (b.c.).
The above initial-boundary value problem (IBVP) is profoundly underdetermined.
However one may come to think that for some infinitesimal $h$ there are two distinct approximations for the $x$-partial derivative:
$$partial_xyapproxfrac{y(t,x+h)-f(t,x)}{h}qquadqquadpartial_xyapproxfrac{y(t,x)-f(t,x-h)}{h}$$
Based on these approximations one can
- evaluate $partial_xy(0,x)$ for any $xin[0,X]$ and then
- evaluate $partial_ty(0,x)$ for any $xin[0,X]$ and then
- evaluate $y(0+dt,x)=y(0,x)+dtcdot partial_ty(0,x)$
In this way one is able to construct a unique solution $y(t,x)$ although the above IBVP is underdetermined. It is as if the missing boundary condition $y(t,X)$ is constructed in parallel with the evolution of initial data.
I would appreciate any recommendations of bibliography or ideas concerning this kind of integration.
ordinary-differential-equations pde numerical-methods partial-derivative boundary-value-problem
ordinary-differential-equations pde numerical-methods partial-derivative boundary-value-problem
asked Dec 29 '18 at 10:51
dkstackdkstack
204
asked Dec 29 '18 at 10:51
dkstackdkstack
204
edited Dec 29 '18 at 19:24
dkstack
asked Dec 29 '18 at 10:51
dkstackdkstack
204
asked Dec 29 '18 at 10:51
dkstackdkstack
204
asked Dec 29 '18 at 10:51
dkstackdkstack
204
204
add a comment |
add a comment |
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