Now we have an existence and uniqueness result for the heat equation (pending some housekeeping with unproven results). 24.5 Theorem. Let f = C(R) be bounded in the sense that there exists M> 0 such that |f(x) ≤ M for all x E R. Then the only solution to the heat IVP Su₁ = Uxx, -∞0 < x < ∞, t>0 | u(x, 0) = f(x), −xx<∞ (24.2) is u(x,t) = √ √ H(x − y, t)f(y) dy, x = R, t>0 f(x), x = R, t = 0, e-82/4t H(s, t) = So what else is this solution doing? First, no matter what the initial temperature distri- bution is, eventually everything "cools all the way down." 24.6 Problem. Let f = C(R) be bounded and let |f| be integrable. Let u solve (24.2). Prove that lim u(x,t) = 0 847 for each R. Go further and explain how this limit is "uniform" in x by finding a bound |u(x, t) M(t) valid for all x R and t>0 with lim₁→∞ M(t) = 0.

Now we have an existence and uniqueness result for the heat equation (pending some housekeeping with unproven results). 24.5 Theorem. Let f = C(R) be bounded in the sense that there exists M> 0 such that |f(x) ≤ M for all x E R. Then the only solution to the heat IVP Su₁ = Uxx, -∞0 < x < ∞, t>0 | u(x, 0) = f(x), −xx<∞ (24.2) is u(x,t) = √ √ H(x − y, t)f(y) dy, x = R, t>0 f(x), x = R, t = 0, e-82/4t H(s, t) = So what else is this solution doing? First, no matter what the initial temperature distri- bution is, eventually everything "cools all the way down." 24.6 Problem. Let f = C(R) be bounded and let |f| be integrable. Let u solve (24.2). Prove that lim u(x,t) = 0 847 for each R. Go further and explain how this limit is "uniform" in x by finding a bound |u(x, t) M(t) valid for all x R and t>0 with lim₁→∞ M(t) = 0.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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