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MATH 450: HOMEWORK 2

P2.1 (12 pts) Use a polynomial interpolant on the Chebyshev points {(x^, y )}务=° with n = 40 to approximate the solution to the Laplace's equation in 2D with the boundary conditions

V²u = 0 in (—1,1)2,

^ux (—^1,y) = ^ux^{(1, y)} = ^0, u(x, —1) = cos(2nx), u(x, 1) = sin (^x).

Numerically approximate the integral 带 u(0.5 cos 0,0.5sin。) d。by the sum

1 ^m 2n '

(1.2) Sm := 一 V* u(0.5 cos , 0.5sin ) where = ( ) j,

mJ \ m + 1 /

3=1

with m = 256, then compare this integral with the polynomial approximation at the point (0, 0).

P2.2 (12 pts) Use a polynomial interpolant on the Chebyshev points {(x^, yj)}另3=0 with n = 40 to approximate solution to the variable coefficient boundary value problem problem

V - KVu = 0 in (—1,1)²,

K(x, y) = exp[—(x — 0.1)² — 5(y — 0.2)2], u(—1, y) = y² + 1, u(x, —1) = x² + 1, u(1, y) = y² + 1, u(x, 1) = x² + 1.

P2.3 (6 pts) For the two polynomial approximations above, evaluate the solution u and its normal derivatives n - Vu on the Chebyshev points that lie on the boundary. Draw two scatter plots of these values, with values x^ := u(x^_fc, yj_k) on the x-axis and the value of y^ := n - Vu(x^_fc, yj_k) on the y-axis.