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We want to measure heat flow in an insulated sphere of radius R when there is spherical symmetry. We have u = u(t, p) with - 1 = x (Upp + 21/1Up) Ut up(t, R) = 0 u(0, p) = f(p). ρ Use 2 for the separation constant. Don't forget to check the zero eigen- value! a) You will get a transcendental equation for the non-zero eigenvalues, Znλn R. Use Mathematica to find the first 3 non-zero eigenvalues (1, 2, and 3) to at least 5 decimal places. b) Find the general solution to the problem and formulas for all coeffi- cients in terms of the initial data. c) The zero eigenvalue should give you a constant solution. What does the coefficient for this term represent?

We want to measure heat flow in an insulated sphere of radius R when there is spherical symmetry. We have u = u(t, p) with - 1 = x (Upp + 21/1Up) Ut up(t, R) = 0 u(0, p) = f(p). ρ Use 2 for the separation constant. Don't forget to check the zero eigen- value! a) You will get a transcendental equation for the non-zero eigenvalues, Znλn R. Use Mathematica to find the first 3 non-zero eigenvalues (1, 2, and 3) to at least 5 decimal places. b) Find the general solution to the problem and formulas for all coeffi- cients in terms of the initial data. c) The zero eigenvalue should give you a constant solution. What does the coefficient for this term represent?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 60E
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We want to measure heat flow in an insulated sphere of radius R when there is spherical symmetry. We have u = u(t, p) with - 1 = x (Upp + 21/1Up) Ut up(t, R) = 0 u(0, p) = f(p). ρ Use 2 for the separation constant. Don't forget to check the zero eigen- value! a) You will get a transcendental equation for the non-zero eigenvalues, Znλn R. Use Mathematica to find the first 3 non-zero eigenvalues (1, 2, and 3) to at least 5 decimal places. b) Find the general solution to the problem and formulas for all coeffi- cients in terms of the initial data. c) The zero eigenvalue should give you a constant solution. What does the coefficient for this term represent?
Transcribed Image Text:We want to measure heat flow in an insulated sphere of radius R when there is spherical symmetry. We have u = u(t, p) with - 1 = x (Upp + 21/1Up) Ut up(t, R) = 0 u(0, p) = f(p). ρ Use 2 for the separation constant. Don't forget to check the zero eigen- value! a) You will get a transcendental equation for the non-zero eigenvalues, Znλn R. Use Mathematica to find the first 3 non-zero eigenvalues (1, 2, and 3) to at least 5 decimal places. b) Find the general solution to the problem and formulas for all coeffi- cients in terms of the initial data. c) The zero eigenvalue should give you a constant solution. What does the coefficient for this term represent?
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