Mathematics Homework Solutions
Problem
#46166

Wave Equation : Triangular Pulse

Let u(x; y) be the solution on 0 < x < 2 and 0 < y < 2 of uxx + uyy = 0
with u(0, y) = u(2, y) = u(x, 2) = 0 and u(x, 0) = f(x) the triangular pulse with f(0) = f(2) = 0 and f(1) = 2.
f(x) = {2x     0          {-2x+4  1Let uj,k = u(j/2; k/2) for j = 1; 2; 3 and k = 1; 2; 3 as obtained by the numerical
method.
(a) Write the linear equations for the 9 unknowns u11 ; : : : ; u33 in matrix form.
(b) Use symmetry to reduce these to six equations.
(c) Solve the equations to obtain the 9 unknowns u11 ; : : : ; u33.

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Homework Set 9 Problem


Let u(x, y) be the solution on 0 x 2 and 0 y 2 of uxx + uyy = 0
with u(0, y) = u(2, y) = u(x, 2) = 0 and u(x, 0) = f (x) the triangular pulse with
f (0) = f (2) = 0 and f (1) = 2.

2x, if 0 x 1;

f (x) =
-2x + 4, if 1 x 2.


Let uj,k = u(j/2, k/2) for j = 1, 2, 3 and k = 1, 2, 3 as obtained by the numerical
method.

(a) Write the linear equations for the 9 unknowns u11 , . . . , u33 in matrix form.

(b) Use symmetry to reduce these to six equations.

(c) Solve the equations to obtain the 9 unknowns u11 , . . . , u33 .

Solution Summary

A triangular pulse is investigated. The solution is detailed and well presented.

Solution
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