1. Consider the following heat equation Ut 1 16 u(0, t) = u(1, t) = 0, u(x,0) = 2 sin 2πx, Use Ax = At = 0.05, and each of the following numerical method to approximate the solution till = ·Uxx 0 0 t> 0, 0 ≤ x ≤ 1. π² t = 0.1, also compare your results at t = 0.1 to the actual solution u(x, t) = 2e¯¯ sin 2x. (a) Use the Forward-Difference method to approximate the solution. (b) Use the Backward-Difference method to approximate the solution. (c) Use Crank-Nicolson method to approximate the solution.
1. Consider the following heat equation Ut 1 16 u(0, t) = u(1, t) = 0, u(x,0) = 2 sin 2πx, Use Ax = At = 0.05, and each of the following numerical method to approximate the solution till = ·Uxx 0 0 t> 0, 0 ≤ x ≤ 1. π² t = 0.1, also compare your results at t = 0.1 to the actual solution u(x, t) = 2e¯¯ sin 2x. (a) Use the Forward-Difference method to approximate the solution. (b) Use the Backward-Difference method to approximate the solution. (c) Use Crank-Nicolson method to approximate the solution.
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:1. Consider the following heat equation
Use Δχ
=
1
3'
At
=
1
Ut = 16 U₁₂
Uxx'
u(0,t) = u(1, t) = 0,
t> 0,
u(x,0) = 2 sin 2πx,
0 ≤ x ≤ 1.
0.05, and each of the following numerical method to approximate the solution till
0 < x < 1, t>0
t = 0.1, also compare your results at t = 0.1 to the actual solution u(x, t) = 2e¯
(a) Use the Forward-Difference method to approximate the solution.
(b) Use the Backward-Difference method to approximate the solution.
(c) Use Crank-Nicolson method to approximate the solution.
π²
t
4 sin 2πx.
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