. Consider the partial differential equation u₁(x, t) = KUxx(x, t) + au(x, t), where a is a constant. (a) Suppose we introduce a new dependent variable w(x, t) by defining u(x, t) = est w(x, t), where is a constant. Show that if d is chosen properly, then w(x, t) is a solution of w₁(x, t) = KW Kwar(x, t). What is the value of d? (b) Show that w(x, t) =e-47²t cos 2x is a solution of the initial-boundary value problem w₁(x, t) = wxx(x, t), 0
. Consider the partial differential equation u₁(x, t) = KUxx(x, t) + au(x, t), where a is a constant. (a) Suppose we introduce a new dependent variable w(x, t) by defining u(x, t) = est w(x, t), where is a constant. Show that if d is chosen properly, then w(x, t) is a solution of w₁(x, t) = KW Kwar(x, t). What is the value of d? (b) Show that w(x, t) =e-47²t cos 2x is a solution of the initial-boundary value problem w₁(x, t) = wxx(x, t), 0
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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Transcribed Image Text:39. Consider the partial differential equation ut(x, t) = kuxx(x, t) + au(x, t), where a is a constant.
(a) Suppose we introduce a new dependent variable w(x, t) by defining u(x, t) est w(x, t),
where is a constant. Show that if d is chosen properly, then w(x, t) is a solution of
w₁(x, t) = kwxx(x, t). What is the value of 8?
₂-4π²t
cos 2x is a solution of the initial-boundary value problem
(b) Show that w(x, t)
= e
w₁(x, t) = wxx (x, t), 0<x< 1, 0 < t < ∞
wx (0, t) = wx (1, t) = 0, 0≤ t <∞
Ox
w(x, 0) = cos 2πx, 0≤ x ≤ 1.
=
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