1. **Problem Statement:** We are given the partial differential equation (PDE) in spherical coordinates: $$\frac{\partial u}{\partial t} = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial u}{\partial r}\right) + \frac{1}{r^2 \sin \phi} \frac{\partial}{\partial \phi} \left(\sin \phi \frac{\partial u}{\partial \phi}\right) + \frac{1}{r^2 \sin^2 \phi} \frac{\partial^2 u}{\partial \theta^2} + k$$ with boundary conditions (BC): $$u(R, \theta, \phi, t) = 0$$ $$\frac{\partial u}{\partial \theta}(r, 0, \phi, t) = \frac{\partial u}{\partial \theta}(r, \frac{\pi}{2}, \phi, t) = 0$$ $$\frac{\partial u}{\partial \phi}(r, \theta, 0, t) = \frac{\partial u}{\partial \phi}(r, \theta, \pi, t) = 0$$ and initial condition (IC): $$u(r, \theta, \phi, 0) = \beta(r, \theta, \phi)$$ 2. **Method: Separation of Variables** We assume a solution of the form: $$u(r, \theta, \phi, t) = R(r) \Theta(\theta) \Phi(\phi) T(t)$$ 3. **Substitute into PDE:** Substitute $u$ into the PDE and divide both sides by $R \Theta \Phi T$ to separate variables. 4. **Separate time variable:** We get an equation of the form: $$\frac{1}{T} \frac{dT}{dt} = \text{spatial terms} + k$$ Set the time-dependent part equal to a separation constant $-\lambda$: $$\frac{1}{T} \frac{dT}{dt} = -\lambda$$ which solves to: $$T(t) = T_0 e^{-\lambda t}$$ 5. **Spatial part:** The spatial part satisfies: $$\frac{1}{r^2 R} \frac{d}{dr} \left(r^2 \frac{dR}{dr}\right) + \frac{1}{r^2 \sin \phi \Theta \Phi} \frac{\partial}{\partial \phi} \left(\sin \phi \frac{\partial (\Theta \Phi)}{\partial \phi}\right) + \frac{1}{r^2 \sin^2 \phi \Theta \Phi} \frac{\partial^2 (\Theta \Phi)}{\partial \theta^2} + k = -\lambda$$ 6. **Angular separation:** Separate $\theta$ and $\phi$ variables by assuming: $$\Theta(\theta)$$ and $$\Phi(\phi)$$ satisfy eigenvalue problems with boundary conditions: - $\frac{d\Theta}{d\theta}(0) = \frac{d\Theta}{d\theta}(\frac{\pi}{2}) = 0$ - $\frac{d\Phi}{d\phi}(0) = \frac{d\Phi}{d\phi}(\pi) = 0$ 7. **Radial equation:** The radial part satisfies: $$\frac{1}{r^2} \frac{d}{dr} \left(r^2 \frac{dR}{dr}\right) + \left(\lambda - k - \frac{\mu}{r^2}\right) R = 0$$ where $\mu$ is the separation constant from angular parts. 8. **Boundary condition for $R$:** $$R(R) = 0$$ 9. **Summary:** - Solve angular eigenvalue problems for $\Theta$ and $\Phi$ with given Neumann BCs. - Use eigenvalues to solve radial ODE with Dirichlet BC. - Time dependence is exponential decay with rate $\lambda$. - Initial condition $u(r, \theta, \phi, 0) = \beta(r, \theta, \phi)$ determines coefficients in series expansion. This method fully separates variables and reduces the PDE to ODEs solvable by standard techniques. **Final answer:** The solution is expressed as a series: $$u(r, \theta, \phi, t) = \sum_{n,m,l} A_{nml} R_n(r) \Theta_m(\theta) \Phi_l(\phi) e^{-\lambda_{nml} t}$$ where $A_{nml}$ are coefficients from initial condition expansion.