Answer to Question #91730 in Classical Mechanics for ss

Question #91730
Solve the heat conduction equation:
for the following boundary and initial conditions:

u(0,t) = u(5,t) = 0,
u(x,0) =2sin(πx)− 4sin(2πx)
1
Expert's answer
2019-07-17T09:51:13-0400
"\\frac{\\partial u}{\\partial t} = -\\kappa \\Delta u""u(0, t) = u(5, t) = 0""u(x, 0) = 2\\sin(\\pi x) -4\\sin(2\\pi x)"

Let's find the solution in such a way:

"u = T(t) X(x)""\\frac{T'}{T}(t) = -\\kappa \\frac{X''}{X}(x) = -\\kappa\\lambda"

where lambda is a constant.

"X'' + \\lambda X = 0""X(0) = X(5) = 0"

hence

"X = \\sin \\frac{n\\pi}{5}x"

while n is integer and

"\\lambda = \\bigg(\\frac{n\\pi}{5}\\bigg)^2"

Thus

"T' = - \\kappa\\lambda T""T = C e^{-\\kappa\\lambda t}"

The heat conduction equation is linear, therefore a sum of solutions is also a solution.

"u(x, 0) = 2\\sin(\\pi x) -4\\sin(2\\pi x)""\\lambda_1 = \\bigg(\\frac{5\\pi}{5}\\bigg)^2, \\qquad \\lambda_2 = \\bigg( \\frac{5\\cdot 2\\pi}{5}\\bigg)^2""u(x, t) = 2e^{-\\kappa \\pi^2 t} \\sin(\\pi x) -4e^{-4\\kappa\\pi^2 t}\\sin(2\\pi x)"


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