Laplace PDE that will Blow your Mind: partial^2-u/partial-x^2 + partial^2-u/partial-y^2 = 0,...

[mario99]

[math]\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0[/math]
[math]0 < x < 1, \ \ \ \ \ 0 < y < 2[/math]
[math]u(x,0) = x, \ \ \ \ \ u(x,2) = x^2[/math]
[math]u(0,y) = y, \ \ \ \ \ u(1,y) = y^2[/math]
This is a Laplace partial differential equation.

I don't know how to solve this problem. I looked for similar problems in the book, but did not find. I also looked for similar problems in the web, but did not find.

A very important note: If you think that this is an easy problem, don't underestimate my skills. I have solved the Airy Equation before from scratch.

 

[khansaheb]

[math]\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0[/math]
[math]0 < x < 1, \ \ \ \ \ 0 < y < 2[/math]
[math]u(x,0) = x, \ \ \ \ \ u(x,2) = x^2[/math]
[math]u(0,y) = y, \ \ \ \ \ u(1,y) = y^2[/math]
This is a Laplace partial differential equation.

I don't know how to solve this problem. I looked for similar problems in the book, but did not find. I also looked for similar problems in the web, but did not find.

A very important note: If you think that this is an easy problem, don't underestimate my skills. I have solved the Airy Equation before from scratch.

Have you read:

Elliptic Partial Differential Equation -- from Wolfram MathWorld

A second-order partial differential equation, i.e., one of the form Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, (1) is called elliptic if the matrix Z=[A B; B C] (2) is positive definite. Elliptic partial differential equations have applications in almost all areas of mathematics, from...

mathworld.wolfram.com

If you did,

How far did you go following the procedure outlined?

Exactly where are you stuck?

 

[khansaheb]

Did you investigate:

https://tutorial.math.lamar.edu/classes/de/laplaceseqn.aspx

 

[mario99]

Have you read:

Elliptic Partial Differential Equation -- from Wolfram MathWorld

A second-order partial differential equation, i.e., one of the form Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, (1) is called elliptic if the matrix Z=[A B; B C] (2) is positive definite. Elliptic partial differential equations have applications in almost all areas of mathematics, from...

mathworld.wolfram.com

If you did,

How far did you go following the procedure outlined?

Exactly where are you stuck?

This link does not have a procedure to follow. It just shows when to call a second-order PDE elliptic. Then it confirms that a Laplace equation is elliptic and this equation is called Poisson equation when it is nonhomogeneous. At end it talks about non-linear PDE.

And if you mean the procedure outlined by the Fourier transform, it is true that the coefficients are constant, but unfortunately, the domain is finite. The Fourier transform works effectively in infinite domains, or at least half infinite.

Did you investigate:

https://tutorial.math.lamar.edu/classes/de/laplaceseqn.aspx

This link is helpful. I will read it carefully but I think that I will need a hand in some points. Thank you professeur Khan.