Question

Solve Laplace’s equation inside a rectangle 0 <= x <= L, 0 <= y <= H, with the following boundary conditions

Answer 1

u(0,y) = g(y), u(L,y)=0,
\[\frac{\partial u}{\partial x} (x,0)=0\]
, u(x,H) = 0

Answer 2

I think it helps to draw a picture, although it's unnecessary it can really help to organize your thoughts.

Make sure to use separation of variables to save yourself. You can also use the picture you draw to help you flop the boundary conditions over to the ODEs you get from the separation. \[u(x,y)=X(x)Y(y)\]

Answer 3

\[\nabla^2u(x,y)=0\\[1ex]
0<x<L,\qquad\ \ u(0,y) = g(y), \qquad u(L,y)=0\\
0<y<H,\qquad u_x(x,0)=0,\qquad \ \ \ \ u(x,H) =0\]

\begin{align}
u_{xx}+u_{yy}&=0\\
&& u(x,y) &= X(x)Y(y)\\
&& u_{xx}&=X''Y\\
&& u_{yy}&=XY''\\
X''Y+XY''&=0\\[1ex]
X''/X=-Y''/Y&=\text{const.}=-\lambda
\end{align}

Answer 4

Solve Laplace’s equation inside a rectangle 0

Answer 5

Please help, I am asked to solve the given differential equation subject to the indicated boundary conditions.

Answer 6

Solve the wave equation for u(x, y, t) in the rectangle 0 < x < L, 0 < y < H with
initial conditions u(x, y, 0) = φ(x, y), ∂u/∂t(x, y, 0) = ψ(x, y) a

Answer 7

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Answer 8

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Answer 9

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Answer 10

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A rectangle of length 10 units and width 5 units is shown.

Answer 11

The figure below shows a shaded rectangle inside a large rectangle:

A rectangle of length 10 units and width 5 units is shown.

Answer 12

Quick question on boundary conditions:

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Answer 13

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Answer 14

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Answer 16

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Answer 17

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Answer 18

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Answer 19

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Answer 20

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Answer 21

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Answer 22

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Answer 23

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Answer 24

help