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Let u be the solution to the initial boundary value problem for the Heat Equation, d,u(t, x) = 5 du(t, x), te (0, 00), x € (0,5); with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial condition 0, € [0, € [$., 15). u(0, x) = f(x) = {2, 0, XE XE x € [15,5]. XE

Let u be the solution to the initial boundary value problem for the Heat Equation, d,u(t, x) = 5 du(t, x), te (0, 00), x € (0,5); with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial condition 0, € [0, € [$., 15). u(0, x) = f(x) = {2, 0, XE XE x € [15,5]. XE

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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Let u be the solution to the initial boundary value problem for the Heat Equation, du(t, x) = 5 du(t, x), t € (0, ∞0), x € (0,5); with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial condition 0, 15 u(0, x) = f(x) = 2, = [0, 2), = [5, ¹5). € [15.5]. 0, XE The solution u of the problem above, with the conventions given in class, has the form u(t, x) = Ï Cn Un(t) wn(x), n=1 with the normalization conditions U,(0) = 1 and wn (5) = 1. Find the functions Un, Wn, and the constants Cn. Un(t) = wn(x) = Cn = x E XE M M
Transcribed Image Text:Let u be the solution to the initial boundary value problem for the Heat Equation, du(t, x) = 5 du(t, x), t € (0, ∞0), x € (0,5); with Dirichlet boundary conditions u(t,0) = 0 and u(t, 5) = 0, and with initial condition 0, 15 u(0, x) = f(x) = 2, = [0, 2), = [5, ¹5). € [15.5]. 0, XE The solution u of the problem above, with the conventions given in class, has the form u(t, x) = Ï Cn Un(t) wn(x), n=1 with the normalization conditions U,(0) = 1 and wn (5) = 1. Find the functions Un, Wn, and the constants Cn. Un(t) = wn(x) = Cn = x E XE M M
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