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(a) Show that V(x, t) = π - fx es² ds is a solution to the heat equation √4xt Vt=xVxx, xER.

(a) Show that V(x, t) = π - fx es² ds is a solution to the heat equation √4xt Vt=xVxx, xER.

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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T (a) Show that V(x, t) = π - SV₁xt √4xt e ds is a solution to the heat equation Vt = xVxx, x € R. (b) Suppose U solves the heat equation on the real line Ut = 4U, ER with initial value 4, x ≤ 0 2, x > 0. (i) Use the Fourier-Poisson formula to give an explicit expression for the solution (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→∞? U (x,0) =
Transcribed Image Text:T (a) Show that V(x, t) = π - SV₁xt √4xt e ds is a solution to the heat equation Vt = xVxx, x € R. (b) Suppose U solves the heat equation on the real line Ut = 4U, ER with initial value 4, x ≤ 0 2, x > 0. (i) Use the Fourier-Poisson formula to give an explicit expression for the solution (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→∞? U (x,0) =
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