Please try not to reject question? If it is rejected explain why? ## Heat Equation and Its ApplicationsThe heat equation describes the flow of thermal energy (heat) in materials. The temperature \( \Psi \) obeys:\[\frac{\partial \Psi}{\partial t} = \frac{k}{c_p \rho} \nabla^2 \Psi,\]where \( t \) is time, \( k \) is the thermal conductivity, \( c_p \) is the specific heat capacity, and \( \rho \) is the mass density.### Problem Breakdown**(a)** Assuming the problem is spherically symmetric, write the heat equation in spherical coordinates.**(b)** Apply the technique of separation of variables, assuming:\[\Psi = \frac{u(r) f(t)}{r}\]Find the resulting differential equations for \( u \) and \( f \), and provide their general solution.- **Note:** 1. By factoring \( r^{-1} \) out of \( u \), the spherically-symmetric Laplacian is simplified. The solution to the differential equation for \( u \) will be recognizable. 2. Choose the sign of the separation constant such that 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?**(d)** Application to Cooking a TurkeyLet's apply this to the cooking of a turkey. Assume the turkey is a sphere of radius \( R = 0.2 \) m, starting at room temperature \( T_0 = 25^\circ \)C, with an oven set at \( T_1 = 180^\circ \)C. Assume the turkey shares the thermal properties of water:- **Thermal Conductivity:** \( k = 0.90 \text{ W/(m K)} \)- **Heat Capacity:** \( \rho c_p = 4.18 \times 10^6 \text{ J/m}^3 \)The recipe implies an initial condition \( \Psi(r, 0) = T_0 \) and a spatial boundary condition \( \Psi(R, t) = T_1 \). The solution must be regular at \( r = 0 \).**(e)** Define \( \Phi = \Psi - T_1 \) so \( \Phi(R, t) =

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?