(a) Use the finite difference method to approximate the solution to the elliptic PDE 0 < x < 1, 0
(a) Use the finite difference method to approximate the solution to the elliptic PDE 0 < x < 1, 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(a) Use the finite difference method to approximate the solution to the elliptic PDE
0 < x < 1, 0 <y<2
0≤x≤1
0 ≤ y ≤2
Uxx + Uyy = 4,
u(x, 0) = x²,
u(0, y) = y²,
u(x, 2) = (x - 2)²,
u(1, y) = (y − 1)²,
with Ax = and Ay
When you assemble the matrix form of the linear system, please keep all
elements in the matrix and the right hand side vector in the exact fraction expressions. After you solve the
system, write the solutions in decimal form with 4 decimal places.
(b) Compare the results of part (a) to the actual solution u(x, y) = (x − y)² by computing the error at each
mesh point. Then explain why the errors are all zeros.
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