- #1
Jhenrique
- 685
- 4
Hellow everybody!
A form how various ODE are intercorrelated can be sinterized like this:
##t = t##
##y = y(t)##
##y' = y'(t,\;y)##
##y'' = y''(t,\;y,\;y')##
##y''' = y'''(t,\;y,\;y',\;y'')##
Until here, no problems!
But, how is such relationship wrt the PDE?
Would be this:
##x = x##
##y = y##
##u = u(x,\;y)##
##u_x = u_x(x,\;y,\;u)##
##u_y = u_y(x,\;y,\;u)##
##u_{xx} = u_{xx}(x,\;y,\;u,\;u_x,\;u_y)##
##u_{yy} = u_{yy}(x,\;y,\;u,\;u_x,\;u_y)##
##u_{xy} = u_{xy}(x,\;y,\;u,\;u_x,\;u_y)##
##u_{yx} = u_{yx}(x,\;y,\;u,\;u_x,\;u_y)##
Ie, the derivative of order n wrt x no depends of the derivative of order n wrt y (and vice versa), or depends? If depends, so the relationship would be this:
##x = x##
##y = y##
##u = u(x,\;y)##
##u_x = u_x(x,\;y,\;u,\;u_y)##
##u_y = u_y(x,\;y,\;u,\;u_x)##
##u_{xx} = u_{xx}(x,\;y,\;u,\;u_x,\;u_y,\;u_{yy},\;u_{xy},\;u_{yx})##
##u_{yy} = u_{yy}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{xy},\;u_{yx})##
##u_{xy} = u_{xy}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{yy},\;u_{yx})##
##u_{yx} = u_{yx}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{yy},\;u_{xy})##
Wich 2 last relation is correct?
A form how various ODE are intercorrelated can be sinterized like this:
##t = t##
##y = y(t)##
##y' = y'(t,\;y)##
##y'' = y''(t,\;y,\;y')##
##y''' = y'''(t,\;y,\;y',\;y'')##
Until here, no problems!
But, how is such relationship wrt the PDE?
Would be this:
##x = x##
##y = y##
##u = u(x,\;y)##
##u_x = u_x(x,\;y,\;u)##
##u_y = u_y(x,\;y,\;u)##
##u_{xx} = u_{xx}(x,\;y,\;u,\;u_x,\;u_y)##
##u_{yy} = u_{yy}(x,\;y,\;u,\;u_x,\;u_y)##
##u_{xy} = u_{xy}(x,\;y,\;u,\;u_x,\;u_y)##
##u_{yx} = u_{yx}(x,\;y,\;u,\;u_x,\;u_y)##
Ie, the derivative of order n wrt x no depends of the derivative of order n wrt y (and vice versa), or depends? If depends, so the relationship would be this:
##x = x##
##y = y##
##u = u(x,\;y)##
##u_x = u_x(x,\;y,\;u,\;u_y)##
##u_y = u_y(x,\;y,\;u,\;u_x)##
##u_{xx} = u_{xx}(x,\;y,\;u,\;u_x,\;u_y,\;u_{yy},\;u_{xy},\;u_{yx})##
##u_{yy} = u_{yy}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{xy},\;u_{yx})##
##u_{xy} = u_{xy}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{yy},\;u_{yx})##
##u_{yx} = u_{yx}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{yy},\;u_{xy})##
Wich 2 last relation is correct?