Math Problem Solver

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)

Q2

(a) Given the mixed boundary value problem (BVP), 𝒚

′′ = 𝑦

3 − 𝑦𝑦

′

; 𝒚

′

(0) = −1,

𝑦(1) = 5. Using step size ℎ = 0.25. Construct the corresponding finite difference formula

to approximate 𝑦(𝑥𝑛

) at 𝑥 = (𝑥𝑛

) (b) Using step size ℎ = 0.2, construct the

corresponding finite difference formula to approximate 𝑦(𝑥𝑛

) .

Solve by giving all numerical solution in each steps