Given an elliptic equation
𝜕
2𝑢
𝜕𝑥
2
+
𝜕
2𝑢
𝜕𝑦2
= 0. For 0 < 𝑥 < 1, 0 < 𝑢 < ∞, ℎ =
∆𝑥, 𝑘 = ∆𝑦 and 𝑢 = 𝑢(𝑥, 𝑦), subjected to the boundary condition (B.C)
𝑢(𝑥, 𝑦) = 𝑔1
(𝑥, 𝑦) for 𝑗 = 1,2,3,4.
(b) Find its matrix and show that 𝑀 is invertible, Show that 𝑀 is diagonally dominant
and decide whether the solution 𝑈 is unique at
i) 𝑈0,𝑗 = 𝑓 ii) 𝑈4,𝑗 = 𝑓
ii) 𝑈𝑖,4 = 𝑥𝑖(𝑥𝑖−1)
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