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Show that V(x, t) = π – S√4xt - e ds is a solution to the heat equation V₁ = xVxx, x = R.

Show that V(x, t) = π – S√4xt - e ds is a solution to the heat equation V₁ = xVxx, x = R.

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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Show that V(x, t) = - Sov √4xt e-s² ds is a solution to the heat equation V₁ = xV₁, x € R. with initial value =T- Suppose U solves the heat equation on the real line 4Uxx, x ER Ut= = U(x,0) = 4, x ≤0 2, x > 0. (i) Use the Fourier-Poisson formula to give an explicit expression for the solution U. (ii) Describe the qualitative behaviour of U in this case as t → ∞ and plot out the solution at several instants of time to explain your answer. What is the limit of U as t→∞?
Transcribed Image Text:Show that V(x, t) = - Sov √4xt e-s² ds is a solution to the heat equation V₁ = xV₁, x € R. with initial value =T- Suppose U solves the heat equation on the real line 4Uxx, x ER Ut= = U(x,0) = 4, x ≤0 2, x > 0. (i) Use the Fourier-Poisson formula to give an explicit expression for the solution U. (ii) Describe the qualitative behaviour of U in this case as t → ∞ and plot out the solution at several instants of time to explain your answer. What is the limit of U as t→∞?
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