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24.9 Problem. Prove the following "comparison" principle for the heat equation. Suppose that fi, f2 C(R) are bounded with f₁(x) 0 u(x, 0) = f(x), ER and VtVxx, xЄR, t> 0 v(x, 0) = f(x), x ЄR. Prove that either u(x,t) = v(x,t) for all x R and t> 0 or that u(x, t) < v(x, t) for all xER and t> 0.

24.9 Problem. Prove the following "comparison" principle for the heat equation. Suppose that fi, f2 C(R) are bounded with f₁(x) 0 u(x, 0) = f(x), ER and VtVxx, xЄR, t> 0 v(x, 0) = f(x), x ЄR. Prove that either u(x,t) = v(x,t) for all x R and t> 0 or that u(x, t) < v(x, t) for all xER and t> 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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24.9 Problem. Prove the following "comparison" principle for the heat equation. Suppose that fi, f2 C(R) are bounded with f₁(x) <f2(x) for all x and u and v solve UtUrx, xER, t>0 u(x, 0) = f(x), ER and VtVxx, xЄR, t> 0 v(x, 0) = f(x), x ЄR. Prove that either u(x,t) = v(x,t) for all x R and t> 0 or that u(x, t) < v(x, t) for all xER and t> 0.
Transcribed Image Text:24.9 Problem. Prove the following "comparison" principle for the heat equation. Suppose that fi, f2 C(R) are bounded with f₁(x) <f2(x) for all x and u and v solve UtUrx, xER, t>0 u(x, 0) = f(x), ER and VtVxx, xЄR, t> 0 v(x, 0) = f(x), x ЄR. Prove that either u(x,t) = v(x,t) for all x R and t> 0 or that u(x, t) < v(x, t) for all xER and t> 0.
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