Let us consider the following problem for the heat equation ôu d’u = 0 in (0,1) ×(0,+∞), ди -(0,t) ди “ (1,t) = 0,t > 0, (1) u(x,0) = sin x,0< x<1. Use the method of separation of variables to solve problem (1).

Let us consider the following problem for the heat equation ôu d’u = 0 in (0,1) ×(0,+∞), ди -(0,t) ди “ (1,t) = 0,t > 0, (1) u(x,0) = sin x,0< x<1. Use the method of separation of variables to solve problem (1).

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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Concept explainers

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

PROBLEM 1
Let us consider the following problem for the heat equation
ди ди
= 0 in (0,1)×(0,+o),
ди
(0,1):
ди
(1,t) = 0,t > 0,
%3D
(1)
|u(x,0) = sin x,0 < x <1.
Use the method of separation of variables to solve problem (1).
PROBLEM 2
Use the method of separation of variables to solve
V³u = 0 in R = (0,1)²,
ди
(x,1) = sin 7x on7,
ду
u = 0 on ôR\y,
(2)
with Y = {(x, y)|0<x<1, y=1}.
Hints: Draw a picture and use the functions cosh and sinh

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