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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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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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.
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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