We will solve the heat equationut2 UI)0 <<<6, tN0with boundary/initial conditions:u(0, t) = 0,u(6, t) = 0,and u(r, 0) =S 2, 0Find Eigenfunctions for X(r).

Given that, ut=2uxx, 0<x<6, t≥0⋯⋯(1) BC: ux0,t=0, ux6,t=0 IC: ux,0=2, 0<x≤30,…

Answered: We will solve the heat equation u = 2…

We will solve the heat equation u = 2 uzz, 0 < < 6, t> 0 with boundary/initial conditions: u(0, t) = 0, u(6, t) = 0, 2, 0<<3 0, 3

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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

We will solve the heat equation
ut
2 UI)
0 <<<6, tN0
with boundary/initial conditions:
u(0, t) = 0,
u(6, t) = 0,
and u(r, 0) =
S 2, 0<a <3
0, 3<x < 6
6 with thermal diffusivity a =2 where the temperature at the ends is fixed at 0 and the initial temperature
This models temperature in a thin rod of length L = 6
distribution is u(x, 0).
For extra practice we will solve this problem from scratch.
Separate Variables.
Assume u(x, t) = X(x) T(t) and split the PDE into two differential equations, one with X and one with T.
%3D
%3D
(Notation: Write X" and T' for derivatives. Place all constants in the differential equation with T).
• DE for X(x):
Boundary conditions for X(x):
(Enter boundary equations: e.g. "X'(0) = 10")
= 0
• DE for T(t):
>Find Eigenfunctions for X(r).

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