- #1

meatpuppet

- 5

- 0

I've tried and failed to search for this on the forum, so apologies if this has been answered many times before.

Given a variable [tex]u[/tex] which is a function of [tex]x[/tex] and [tex]y[/tex]:

[tex]u = u\left(x,y\right)\\[/tex]

is it possible to solve the pde:

[tex]Au_{xx} + 2Bu_{xy} + Cu_{yy} = D\\[/tex]

The knowns are:

The real coefficients:

[tex]A,~B,~C,~D[/tex]

the initial values

[tex]u(x_0,y_0)[/tex]

[tex]u_x(x_0,y_0)[/tex]

[tex]u_y(x_0,y_0)[/tex]

[tex]u_{xx}(x_0,y_0)[/tex]

and the values of the following along the line y=y

[tex]u(x,y_0)[/tex]

[tex]u_x(x,y_0)[/tex]

[tex]u_{xx}(x,y_0)[/tex]

The coefficents [tex]A,~B,~C[/tex] are such that the equation is parabolic ie [tex]B^2 - AC = 0[/tex]

The quantities I am trying to obtain are [tex]u_{xy},u_{yy}[/tex], but these can be back derived from the function [tex]u[/tex] if it can be obtained.

Given a variable [tex]u[/tex] which is a function of [tex]x[/tex] and [tex]y[/tex]:

[tex]u = u\left(x,y\right)\\[/tex]

is it possible to solve the pde:

[tex]Au_{xx} + 2Bu_{xy} + Cu_{yy} = D\\[/tex]

The knowns are:

The real coefficients:

[tex]A,~B,~C,~D[/tex]

the initial values

[tex]u(x_0,y_0)[/tex]

[tex]u_x(x_0,y_0)[/tex]

[tex]u_y(x_0,y_0)[/tex]

[tex]u_{xx}(x_0,y_0)[/tex]

and the values of the following along the line y=y

_{0}:[tex]u(x,y_0)[/tex]

[tex]u_x(x,y_0)[/tex]

[tex]u_{xx}(x,y_0)[/tex]

The coefficents [tex]A,~B,~C[/tex] are such that the equation is parabolic ie [tex]B^2 - AC = 0[/tex]

The quantities I am trying to obtain are [tex]u_{xy},u_{yy}[/tex], but these can be back derived from the function [tex]u[/tex] if it can be obtained.

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