# 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

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.

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.

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