Consider a solid spherical metal ball of radius a and let u = u(r, t) be the spherically symmetric temperature distribution inside the ball. %3D a. Assuming the separable solution u(r, t) = R(r)G (t), derive the differential equations governing R and T. You may assume that the separation constant is non- positive. b. By letting R(r) = p(r)/r, derive the differential equation governing p for r>0, and hence, find the general solutions for R(r). c. Supposing that the surface of the metal ball is held at 0°C, derive the general solution for u(r, t). d. Given that the initial temperature inside the metal ball is 200 u(r, 0): sin COS 2a 2a Determine the temperature distribution u (r, t) at any time t > 0.

Consider a solid spherical metal ball of radius a and let u = u(r, t) be the spherically symmetric temperature distribution inside the ball. %3D a. Assuming the separable solution u(r, t) = R(r)G (t), derive the differential equations governing R and T. You may assume that the separation constant is non- positive. b. By letting R(r) = p(r)/r, derive the differential equation governing p for r>0, and hence, find the general solutions for R(r). c. Supposing that the surface of the metal ball is held at 0°C, derive the general solution for u(r, t). d. Given that the initial temperature inside the metal ball is 200 u(r, 0): sin COS 2a 2a Determine the temperature distribution u (r, t) at any time t > 0.

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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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

Consider a solid spherical metal ball of radius a and let u = u(r, t) be the spherically
symmetric temperature distribution inside the ballI.
a. Assuming the separable solution u(r, t) = R(r)G (t), derive the differential
equations governing R and T. You may assume that the separation constant is non-
positive.
b. By letting R(r) = p(r)/r, derive the differential equation governing p for r>0,
and hence, find the general solutions for R(r).
c. Supposing that the surface of the metal ball is held at 0°C, derive the general
solution for u(r, t).
d. Given that the initial temperature inside the metal ball is
200
u(r, 0)
COS
sin
2a
Determine the temperature distribution u(r, t) at any time t > 0.

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