Question 1 Consider the homogeneous du Pu Ət ər² heat problem with boundary condition ди (0₁ t) = 0 = u(1. t) ; u (x,0) = f(x) ər where t > 0, 0 ≤ x ≤ 1 and f is a piecewise smooth function on [0,1]. (a) Find the eigenvalues An and the eigenfunctions X₁ (x) for this problem. Write the formal solution of the problem (a), and express the constant coefficients as integrals involving f(x). = (b) Find a series solution in the case that f(x) = uo, uo a constant. Find an approximate solution good for large times.
Question 1 Consider the homogeneous du Pu Ət ər² heat problem with boundary condition ди (0₁ t) = 0 = u(1. t) ; u (x,0) = f(x) ər where t > 0, 0 ≤ x ≤ 1 and f is a piecewise smooth function on [0,1]. (a) Find the eigenvalues An and the eigenfunctions X₁ (x) for this problem. Write the formal solution of the problem (a), and express the constant coefficients as integrals involving f(x). = (b) Find a series solution in the case that f(x) = uo, uo a constant. Find an approximate solution good for large times.
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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![Question 1
Consider the homogeneous
ди
J²u
Ət Ər²¹
heat problem with boundary condition
ди
(0, t) = 0 = u(1, t) ;
u (x,0) = f(x)
ər
where t > 0, 0 ≤ x ≤ 1 and f is a piecewise smooth function on [0,1].
(a) Find the eigenvalues An and the eigenfunctions X₁ (x) for this problem.
Write the formal solution of the problem (a), and express the constant
coefficients as integrals involving f(x).
=
(b) Find a series solution in the case that f(x) = uo, uo a constant. Find an
approximate solution good for large times.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf99633b-9e51-4e49-86a8-33175d1f06ff%2F87d95bfc-3198-4b2c-aea2-8738bc1aa2e9%2Fylf6gbc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 1
Consider the homogeneous
ди
J²u
Ət Ər²¹
heat problem with boundary condition
ди
(0, t) = 0 = u(1, t) ;
u (x,0) = f(x)
ər
where t > 0, 0 ≤ x ≤ 1 and f is a piecewise smooth function on [0,1].
(a) Find the eigenvalues An and the eigenfunctions X₁ (x) for this problem.
Write the formal solution of the problem (a), and express the constant
coefficients as integrals involving f(x).
=
(b) Find a series solution in the case that f(x) = uo, uo a constant. Find an
approximate solution good for large times.
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