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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
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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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)
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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