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Consider the diffusion equation on (0, 1) with the Robin boundary condi- tions ux(0, t) - aou(0, t) = 0 and ux(l, t) + a₁u(l, t) = 0. If a > 0 and a > 0, use the energy method to show that the endpoints contribute to the decrease of fu²(x, t) dx. (This is interpreted to mean that part of the "energy" is lost at the boundary, so we call the boundary conditions "radiating" or "dissipative.")

Consider the diffusion equation on (0, 1) with the Robin boundary condi- tions ux(0, t) - aou(0, t) = 0 and ux(l, t) + a₁u(l, t) = 0. If a > 0 and a > 0, use the energy method to show that the endpoints contribute to the decrease of fu²(x, t) dx. (This is interpreted to mean that part of the "energy" is lost at the boundary, so we call the boundary conditions "radiating" or "dissipative.")

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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[Second Order Equations] How do you solve this?

8. Consider the diffusion equation on (0, 1) with the Robin boundary condi- tions ux(0, t) - aou(0, t) = 0 and ux(1, t) + a₁u(l, t) = 0. If ao > 0 and a₁ > 0, use the energy method to show that the endpoints contribute to the decrease of fu²(x, t) dx. (This is interpreted to mean that part of the "energy" is lost at the boundary, so we call the boundary conditions "radiating" or "dissipative.")
Transcribed Image Text:8. Consider the diffusion equation on (0, 1) with the Robin boundary condi- tions ux(0, t) - aou(0, t) = 0 and ux(1, t) + a₁u(l, t) = 0. If ao > 0 and a₁ > 0, use the energy method to show that the endpoints contribute to the decrease of fu²(x, t) dx. (This is interpreted to mean that part of the "energy" is lost at the boundary, so we call the boundary conditions "radiating" or "dissipative.")
Expert Solution
Step 1

Given That: Diffusion equation with a Robin Boundary conditions is  ux0,t-a0u0,t=0 and uxl,t+alul,t=0.

To Show: Endpoints contribute to the decrease of ∫0lu2x,tdx.

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