1. The heat equation describes the flow of thermal energy (heat) in materials. The temperature V obeys: ᏭᏪ k Ət Cpp = where t is time, k is the thermal conductivity, c, is the specific heat capacity, and p is the mass density. 2 (a) Assuming the problem is spherically symmetric, write the heat equation in spherical coordinates. (b) Apply the technique of separation of variables, assuming. V = u(r) f(t) T Find the the resulting differential equations for u and f, and provide their general solution. Note: (1) By factoring r- out of u as we have, the spherically-symmetric Laplacian is simplified and the solution to the differential equation for u should be easy to recognize. (2) Here I would like you to choose the sign of the separation constant sthat the solutions for f(t) are exponentials that decay with increasing time. (c) What is the form of the solution when the separation constant is zero?
1. The heat equation describes the flow of thermal energy (heat) in materials. The temperature V obeys: ᏭᏪ k Ət Cpp = where t is time, k is the thermal conductivity, c, is the specific heat capacity, and p is the mass density. 2 (a) Assuming the problem is spherically symmetric, write the heat equation in spherical coordinates. (b) Apply the technique of separation of variables, assuming. V = u(r) f(t) T Find the the resulting differential equations for u and f, and provide their general solution. Note: (1) By factoring r- out of u as we have, the spherically-symmetric Laplacian is simplified and the solution to the differential equation for u should be easy to recognize. (2) Here I would like you to choose the sign of the separation constant sthat the solutions for f(t) are exponentials that decay with increasing time. (c) What is the form of the solution when the separation constant is zero?
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