Thread: Laplace's equation in the square

Let

T(x, y) satisfy Laplace’s equation in the square 0 < x < ¼ and
0 < y < ¼. Let T = 1 on the side y = ¼, 0 < x < ¼, and let T = 0 on
the other three sides. Show that

T

(x, y) = big expression involving sinhs (I can derive this)

and, by considering three similar problems, that

T = 1/4at the centre of the square.

Which other 3 problems would I consider and then what would I do with these to get the solution at the centre of the square?

many thanks

Originally Posted by James0502

Let

T(x, y) satisfy Laplace’s equation in the square 0 < x < ¼ and
0 < y < ¼. Let T = 1 on the side y = ¼, 0 < x < ¼, and let T = 0 on
the other three sides. Show that

T

(x, y) = big expression involving sinhs (I can derive this)

and, by considering three similar problems, that

T = 1/4at the centre of the square.

Which other 3 problems would I consider and then what would I do with these to get the solution at the centre of the square?

many thanks

The other three problems are the same d.e. with (1) T= 0 on y=0, 0 on the other three sides, (2)T= 0 on x= 0, 0 on the other three sides, and (3)T= 0 on x= 1/4, 0 on the other three sides. Consider the average of the solutions of those four problems.