Exercise 4. At each point x = (0, 1) an elastic string subjected to a force (x (-(-0.5)²), f(x; a) == 1 V2πα exp where a > 0 is a force parameter that can be adjusted. The string is fixed at the endpoints and the displacement from its stationary state 0 at the point x € (0,1) is given by u(x). For a given a > 0, the string displacement is described by the differential equation x = (0, 1) -u" (x) = f(x; a) u(0) = u(1) = 0 (3) a) Show that u(x) ≤0 for all x € (0, 1). Hint: Use the Green's function solution representation and properties of the Green's function. b) Show that for any a > 0, the maximial displacement of the solution of (3) satisfies the inequality: sup u(x)| ≤ 1/4 x=[0,1]
Exercise 4. At each point x = (0, 1) an elastic string subjected to a force (x (-(-0.5)²), f(x; a) == 1 V2πα exp where a > 0 is a force parameter that can be adjusted. The string is fixed at the endpoints and the displacement from its stationary state 0 at the point x € (0,1) is given by u(x). For a given a > 0, the string displacement is described by the differential equation x = (0, 1) -u" (x) = f(x; a) u(0) = u(1) = 0 (3) a) Show that u(x) ≤0 for all x € (0, 1). Hint: Use the Green's function solution representation and properties of the Green's function. b) Show that for any a > 0, the maximial displacement of the solution of (3) satisfies the inequality: sup u(x)| ≤ 1/4 x=[0,1]
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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![Exercise 4.
At each point x € (0, 1) an elastic string subjected to a force
(x -0.5)2\
2a²
f(x; a):
==
1
√2πa
exp
·(-
where a > 0 is a force parameter that can be adjusted. The string is fixed at the
endpoints and the displacement from its stationary state 0 at the point x € (0,1) is
given by u(x).
For a given a > 0, the string displacement is described by the differential equation
x = (0, 1)
-u" (x) = f(x; a)
u(0) = u(1) = 0
a) Show that u(x) ≤0 for all x € (0, 1).
(3)
Hint: Use the Green's function solution representation and properties of the
Green's function.
b) Show that for any a > 0, the maximial displacement of the solution of (3) satisfies
the inequality:
sup u(x)| ≤ 1/4
x€[0,1]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F702b72a5-f07f-4c22-a3f5-6956511ed203%2Fd3d6a792-4db5-465e-a0bb-d90daf7058fc%2Fk6ap59n_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 4.
At each point x € (0, 1) an elastic string subjected to a force
(x -0.5)2\
2a²
f(x; a):
==
1
√2πa
exp
·(-
where a > 0 is a force parameter that can be adjusted. The string is fixed at the
endpoints and the displacement from its stationary state 0 at the point x € (0,1) is
given by u(x).
For a given a > 0, the string displacement is described by the differential equation
x = (0, 1)
-u" (x) = f(x; a)
u(0) = u(1) = 0
a) Show that u(x) ≤0 for all x € (0, 1).
(3)
Hint: Use the Green's function solution representation and properties of the
Green's function.
b) Show that for any a > 0, the maximial displacement of the solution of (3) satisfies
the inequality:
sup u(x)| ≤ 1/4
x€[0,1]
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