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Let be the unit disc x² + y² ≤ 1. Suppose that the functions f(t, x, y), g(t, x, y) are smooth solutions of the system af Δg, Ət for tER and (x, y) =D, Əg = -Af, Ət 2² 8² where A = ?х2 and we have the boundary conditions " Əy2 f(t, x, y) = 0 = g(t, x, y) for teR and (x, y) = ID. Define M(t) = [] |ƒ(t, x,y)|² + [g(t, x, y)|² dA. Use integration by parts to show that dM = 0. dt =

Let be the unit disc x² + y² ≤ 1. Suppose that the functions f(t, x, y), g(t, x, y) are smooth solutions of the system af Δg, Ət for tER and (x, y) =D, Əg = -Af, Ət 2² 8² where A = ?х2 and we have the boundary conditions " Əy2 f(t, x, y) = 0 = g(t, x, y) for teR and (x, y) = ID. Define M(t) = [] |ƒ(t, x,y)|² + [g(t, x, y)|² dA. Use integration by parts to show that dM = 0. dt =

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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1. Let be the unit disc x² + y² ≤ 1. Suppose that the functions f(t, x, y), g(t, x, y) are smooth solutions of the system af = Ag, Ət for tER and (x, y) =D, Əg = -Af, Ət 22 where A = + მų2. , and we have the boundary conditions f(t, x, y) = 0 = g(t, x, y) for teR and (x, y) = 0D. Define M(t) = = [1₁₂1f² f(t, x, y)2 + g(t, x, y) |² dA. Use integration by parts to show that dM = 0. dt ?х2
Transcribed Image Text:1. Let be the unit disc x² + y² ≤ 1. Suppose that the functions f(t, x, y), g(t, x, y) are smooth solutions of the system af = Ag, Ət for tER and (x, y) =D, Əg = -Af, Ət 22 where A = + მų2. , and we have the boundary conditions f(t, x, y) = 0 = g(t, x, y) for teR and (x, y) = 0D. Define M(t) = = [1₁₂1f² f(t, x, y)2 + g(t, x, y) |² dA. Use integration by parts to show that dM = 0. dt ?х2
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