Consider a two-dimensional circular metal plate of radius a, as represented by the shaded region below, and suppose that the boundary is held at 0°C around the circumference. Let u(r, 0, t) be the temperature distribution of the metal plate. At time t = 0, the metal plate is suddenly intensely heated up in such a way that its initial temperature distribution can be idealised as 10 u(r,0,0) = f(r) = 8 (r - 2). a X State clearly any symmetry, and its reasoning, that could simplify the problem. [Hint: Observe the dependence on 0.] Then, using the method of separation of variable, determine u(r, 0, t) for t > 0. You may assume that the relevant separation constant is negative and you may leave your final answer in terms of the Bessel functions Jo and J₁.
Consider a two-dimensional circular metal plate of radius a, as represented by the shaded region below, and suppose that the boundary is held at 0°C around the circumference. Let u(r, 0, t) be the temperature distribution of the metal plate. At time t = 0, the metal plate is suddenly intensely heated up in such a way that its initial temperature distribution can be idealised as 10 u(r,0,0) = f(r) = 8 (r - 2). a X State clearly any symmetry, and its reasoning, that could simplify the problem. [Hint: Observe the dependence on 0.] Then, using the method of separation of variable, determine u(r, 0, t) for t > 0. You may assume that the relevant separation constant is negative and you may leave your final answer in terms of the Bessel functions Jo and J₁.
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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![Consider a two-dimensional circular metal plate of radius a, as represented by the shaded region
below, and suppose that the boundary is held at 0°C around the circumference.
Let u(r, 0, t) be the temperature distribution of the metal
plate. At time t = 0, the metal plate is suddenly intensely
heated up in such a way that its initial temperature distribution
can be idealised as
u(r, 0,0) = f(r) : 108 (r-2).
=
a
State clearly any symmetry, and its reasoning, that could
simplify the problem. [Hint: Observe the dependence on 0.]
Then, using the method of separation of variable, determine
u(r, 0, t) for t > 0. You may assume that the relevant separation constant is negative and you
may leave your final answer in terms of the Bessel functions Jo and J₁.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2e48a5e-4a4f-4f1f-9f12-eb70816ab8e6%2Ff847fec7-5a33-4bae-8358-45c5864d3e50%2Fpdtrla_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a two-dimensional circular metal plate of radius a, as represented by the shaded region
below, and suppose that the boundary is held at 0°C around the circumference.
Let u(r, 0, t) be the temperature distribution of the metal
plate. At time t = 0, the metal plate is suddenly intensely
heated up in such a way that its initial temperature distribution
can be idealised as
u(r, 0,0) = f(r) : 108 (r-2).
=
a
State clearly any symmetry, and its reasoning, that could
simplify the problem. [Hint: Observe the dependence on 0.]
Then, using the method of separation of variable, determine
u(r, 0, t) for t > 0. You may assume that the relevant separation constant is negative and you
may leave your final answer in terms of the Bessel functions Jo and J₁.
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