Let the vector field (x,y)→f(x,y) be l-one-sided Lipschitz in space, let h=0.1, and consider the implicit Euler map 8:RxRd Rd that maps a point to the solution z of the equation (x, y) ERX Rd z = y + hf(x + h, z). What is the smallest number L such that g is L-Lipschitz in space whenever f is (-5)-one-sided Lipschitz in space? Hint: You can solve this question directly, but you have also seen the answer in the lectures and in the lecture notes. Where?
Let the vector field (x,y)→f(x,y) be l-one-sided Lipschitz in space, let h=0.1, and consider the implicit Euler map 8:RxRd Rd that maps a point to the solution z of the equation (x, y) ERX Rd z = y + hf(x + h, z). What is the smallest number L such that g is L-Lipschitz in space whenever f is (-5)-one-sided Lipschitz in space? Hint: You can solve this question directly, but you have also seen the answer in the lectures and in the lecture notes. Where?
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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Transcribed Image Text:Let the vector field (x,y)→f(x,y) be l-one-sided Lipschitz in space, let h=0.1, and consider the implicit Euler map
g: RxRd.
Rd
that maps a point
to the solution z of the equation
z = y + hf(x + h, z).
What is the smallest number L such that g is L-Lipschitz in space whenever f is (-5)-one-sided Lipschitz in space?
Hint: You can solve this question directly, but you have also seen the answer in the lectures and in the lecture notes. Where?
a. We cannot conclude that g is Lipschitz when f is only one-sided Lipschitz. The one-sided Lipschitz property is weaker than the
Lipschitz property.
O b. L=0
(x, y) = R XRd
O c. L=1/3
O d. L-2/3
O e. L=1
O f. L-exp(0.1)
g. L-exp(-0.1)
Oh. L-exp(-0.5)
O i. L-exp(0.5)
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