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Problem 5.2. Assume that u(0) = u(1) = 0, and that u satisfies 1 [u'v u'v' dx = f fv dx, 0 0 for all v € V = {v : ||v|| + ||v′|| < ∞, v(0) = v(1)=0}. a. Show that u minimizes the functional F (v) = 1/2 [ " (v²)² dx · - for fv dx. 0 0 Hint: F(v) = F(u+w) = F(u) +... ≥ F(u). b. Prove that the above minimization problem is equivalent to -u" = f, 0

Problem 5.2. Assume that u(0) = u(1) = 0, and that u satisfies 1 [u'v u'v' dx = f fv dx, 0 0 for all v € V = {v : ||v|| + ||v′|| < ∞, v(0) = v(1)=0}. a. Show that u minimizes the functional F (v) = 1/2 [ " (v²)² dx · - for fv dx. 0 0 Hint: F(v) = F(u+w) = F(u) +... ≥ F(u). b. Prove that the above minimization problem is equivalent to -u" = f, 0

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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Problem 5.2. Assume that u(0) = u(1) = 0, and that u satisfies 1 So [Wide-L u'v' dx = fv dx, for all v € V = {v : ||v|| + ||v′|| < ∞, v(0) = v(1) = 0}. a. Show that u minimizes the functional F(x)=L(1) dz-f fode, dx da fv dx. Hint: F(v) = F(u+w) = F(u)+... ≥ F(u). b. Prove that the above minimization problem is equivalent to -u"=f, 0< x < 1; u(0) = u(1) = 0. (5.6.2)
Transcribed Image Text:Problem 5.2. Assume that u(0) = u(1) = 0, and that u satisfies 1 So [Wide-L u'v' dx = fv dx, for all v € V = {v : ||v|| + ||v′|| < ∞, v(0) = v(1) = 0}. a. Show that u minimizes the functional F(x)=L(1) dz-f fode, dx da fv dx. Hint: F(v) = F(u+w) = F(u)+... ≥ F(u). b. Prove that the above minimization problem is equivalent to -u"=f, 0< x < 1; u(0) = u(1) = 0. (5.6.2)
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