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HOMEWORK #1 due Wednesday Jan 28
Text references are to
Introduction to partial differential equations with
MATLAB
Jeffrey Cooper (1998) (Birkhauser)
NOTE: In the following, ^ means superscript, _ (underscore) means subscript, and Sum(i=1,9) means the sum for i=1 to 9.
1. (a) Section 2.1 (page 21) Exercise 1
(b) Section 2.1 (page 21) Exercise 3
2. (Like Section 2.2, page 27, Exercise 2) Assume
for t > 0. Find a formula for u(x,t) in terms of f, x, t, and c.
3. (Like Section 2.2, page 27, Exercise 7) Do parts (a) and (b).
4. (Like Section 2.2, page 27, Exercise 3) Show that the equation
(where f(x) is positive and continuously differentiable) has a unique solution u(x,t) for all x for 0 <= t < C_x where 0 < C_x <infinity. Find a formula for the solution.
5. (Like Section 2.3, page 37, Exercise 2) Show that the equation
(where f(x) is continuously differentiable) cannot have a continuously differentiable solution u(x,t) for all x and t > 0 if f(x_1) > f(x_2) for some pair of points x_1,x_2 with x_1 < x_2.
6. (Like Section 2.3, page 37, Exercise 3) Show that the equation
(where f(x) is continuously differentiable and f'(x) > 0 for all x) has a continuously differentiable solution u(x,t) for all x and t > 0. Show how to find u(x,t) in terms of f, x, and t.