Approximate the solutions to the following elliptic partial differential equations, using Algorithm
12.1:
a. ∂2u / ∂x2 + ∂2u / ∂y2 = 0, 0 < x < 1, 0 < y < 1;
u(x, 0) = 0, u(x, 1) = x,...............0≤ x ≤ 1;
u(0, y) = 0, u(1, y) = y, ...............0≤ y ≤ 1.
Use h = k = 0.2, and compare the results to the actual solution u(x, y) = xy.
b. ∂2u / ∂x2 + ∂2u / ∂y2 = −(cos (x + y) + cos (x − y)), 0< x < π,0< y < π/2;
u(0, y) = cos y, u(π, y) = −cos y, ...............0≤ y ≤ π/2,
u(x, 0) = cos x, u (x, π/2)= 0, .....................0 ≤ x ≤ π.
Use h = π/5 and k = π/10, and compare the results to the actual solution u(x, y) = cos x cos y.
c. ∂2u / ∂x2 + ∂2u / ∂y2 = (x2 + y2)exy, 0< x < 2, 0 < y < 1;
u(0, y) = 1, u(2, y) = e2y, ...............0≤ y ≤ 1;
u(x, 0) = 1, u(x, 1) = ex, .................0≤ x ≤ 2.
Use h = 0.2 and k = 0.1, and compare the results to the actual solution u(x, y) = exy.
d. ∂2u / ∂x2 + ∂2u / ∂y2 = x/y + y/x, 1< x < 2, 1 < y < 2;
u(x, 1) = x ln x, u(x, 2) = x ln4x2, ................1≤ x ≤ 2;
u(1, y) = y ln y, u(2, y) = 2y ln(2y), ...............1≤ y ≤ 2.
Use h = k = 0.1, and compare the results to the actual solution u(x, y) = xy ln xy.