Please help to find the answer

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PA3040D

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Hi All Have a nice day. This is related to my second question based on my first question. Please see the following link. for the my first question

https://www.freemathhelp.com/forum/threads/please-advise-the-to-the-answer.138130/#post-590987

In the attached image 1, the boundary condition is given. However, in the image 2, the boundary condition is not given. Therefore, my question is: how do I identify boundary conditions? using image 2 Please advise.

Note In my previous question, ( Above Link ) the group members helped identify the boundary condition, and I were able to solve the problem. However, in this question, some information is missing, making it difficult for me to find the boundary condition

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This problem is almost similar to the last problem of the Laplace equation, except it is non-homogeneous. When the Laplace equation is non-homogeneous, it is called the Poisson equation.

Most of the time, when we are approximating the solutions of the Poisson equation, we are given the values of the boundary points directly without even mentioning the boundary conditions. If we are not given either the boundary conditions nor the values of the boundary points, we choose the values of the boundary points.

Usually, the values of the boundary points of the Poisson equation are zeros, ones, or mixed with zeros and ones. You can choose the same boundary points values that you found in the Laplace equation in the last problem. Or you can choose all zeros or all ones.

The central difference equation of the Poisson equation is a little different from that of the Laplace equation. Do you remember when we applied the difference equation to the point [imath]P_{11}[/imath] of the Laplace equation? We got the first equation as:

[imath]u_{21} + u_{12} + u_{01} + u_{10} - 4u_{11} = 0[/imath]

Now if we apply the difference equation to the point [imath]P_{11}[/imath] of the Poisson equation, we get the first equation as:

[imath]u_{21} + u_{12} + u_{01} + u_{10} - 4u_{11} = h^2f(x,y) = \left(\frac{1}{3}\right)^2*(-2) = -\frac{2}{9}[/imath]

[imath]h = \frac{1}{3}[/imath], is the mesh size.

And

[imath]f(x,y) = -2[/imath], has been taken from the original Poisson equation.

When you approximate the solutions of this Poisson equation just make sure that when you apply the difference equation on each interior point make it equal to [imath]h^2f(x,y) = -\frac{2}{9}[/imath].
 
PA3040D said:
View attachment 37731
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Look at image [imath]2[/imath] at line [imath]3[/imath], you will see [imath]u = xy[/imath]. This is the boundary condition. Since we were given the boundary condition, we have to use it to find the values of the boundary points.

We can write this boundary condition as:

[imath]u(x,y) = xy[/imath]

It is clear that the values of the boundary points, [imath]P_{10} = P_{20} = P_{01} = P_{02} = 0[/imath]. Can you see why?

Now all remain is to find the values of the boundary points, [imath]P_{13}, P_{23}, P_{31}[/imath], and [imath]P_{32}.[/imath]

Start finding the value of the boundary point [imath]P_{13}[/imath] by using the same method that we have used before.

[imath]P_{13} = P(h, 3h) = \ ?[/imath]

Hint: [imath]P(h, 3h) = u(h,3h)[/imath]
 
mario99 said:
Look at image [imath]2[/imath] at line [imath]3[/imath], you will see [imath]u = xy[/imath]. This is the boundary condition. Since we were given the boundary condition, we have to use it to find the values of the boundary points.

We can write this boundary condition as:

[imath]u(x,y) = xy[/imath]

It is clear that the values of the boundary points, [imath]P_{10} = P_{20} = P_{01} = P_{02} = 0[/imath]. Can you see why?

Now all remain is to find the values of the boundary points, [imath]P_{13}, P_{23}, P_{31}[/imath], and [imath]P_{32}.[/imath]

Start finding the value of the boundary point [imath]P_{13}[/imath] by using the same method that we have used before.

[imath]P_{13} = P(h, 3h) = \ ?[/imath]

Hint: [imath]P(h, 3h) = u(h,3h)[/imath]
Click to expand...
This is the superb answer. and the best answer I've seen to my questions. Great! Thanks for your kind help and detailed explanation.
 
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