[4] Let X = C(0, 1; R)={@:[0,1]–→ R:0(;) is continuous} , with inner product (Aø().w() = fu(s)ø(s)ds . Consider the differential operator A: D(A)c X → X defined by A[@()](x)= g'(x), with domain D(A)= {@() e C' (0,1; R) : P(1) = 0} . (4a) Show that A is dissipative: (i.e., show that (A@(:),@())< 0 for all ø(-) e D(A)). (A9(), pO) = (4b) Could y =1 be an eigenvalue of A ? Give a reason for your answer. (4c) Let ø(x)= (x – 1) and w(x) = x² – x. Note both functions are in D(A) = {@() e C'(0,1; R): 9(1) = 0}: Compute: (Aø(), y(). (Aø(),w(O) =
[4] Let X = C(0, 1; R)={@:[0,1]–→ R:0(;) is continuous} , with inner product (Aø().w() = fu(s)ø(s)ds . Consider the differential operator A: D(A)c X → X defined by A[@()](x)= g'(x), with domain D(A)= {@() e C' (0,1; R) : P(1) = 0} . (4a) Show that A is dissipative: (i.e., show that (A@(:),@())< 0 for all ø(-) e D(A)). (A9(), pO) = (4b) Could y =1 be an eigenvalue of A ? Give a reason for your answer. (4c) Let ø(x)= (x – 1) and w(x) = x² – x. Note both functions are in D(A) = {@() e C'(0,1; R): 9(1) = 0}: Compute: (Aø(), y(). (Aø(),w(O) =
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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