1. **Problem Statement:** We need to solve the partial differential equation (PDE) for $v(r, \theta, \varphi, t)$ given by the heat/diffusion equation in spherical coordinates: $$\frac{\partial v}{\partial t} = \nabla^2 v = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial v}{\partial r}\right) + \frac{1}{r^2 \sin \varphi} \frac{\partial}{\partial \varphi}\left(\sin \varphi \frac{\partial v}{\partial \varphi}\right) + \frac{1}{r^2 \sin^2 \varphi} \frac{\partial^2 v}{\partial \theta^2}$$ with boundary conditions: $$\begin{cases} v(R, \theta, \varphi, t) = 0 \\ \frac{\partial v}{\partial \theta}(r, 0, \varphi, t) = \frac{\partial v}{\partial \theta}\left(r, \frac{\pi}{2}, \varphi, t\right) = 0 \\ \frac{\partial v}{\partial \varphi}(r, \theta, 0, t) = \frac{\partial v}{\partial \varphi}(r, \theta, \pi, t) = 0 \end{cases}$$ 2. **Method and Formula:** We use separation of variables and spherical harmonics expansion to solve this PDE. The Laplacian in spherical coordinates suggests eigenfunctions of the form: $$v(r, \theta, \varphi, t) = R(r) \Theta(\theta) \Phi(\varphi) T(t)$$ where each function satisfies an ODE derived from the PDE. 3. **Boundary Conditions Explanation:** - $v(R, \theta, \varphi, t) = 0$ means the solution vanishes on the sphere of radius $R$. - Neumann boundary conditions on $\theta$ and $\varphi$ imply symmetry or insulated boundaries in angular directions. 4. **Angular Solutions:** - The $\theta$-dependence with Neumann BCs at $0$ and $\pi/2$ suggests cosine Fourier modes: $$\frac{\partial v}{\partial \theta} = 0 \Rightarrow \Theta(\theta) = \cos(n \theta), \quad n=0,1,2,...$$ - The $\varphi$-dependence with Neumann BCs at $0$ and $\pi$ corresponds to Legendre polynomials $P_l(\cos \varphi)$, which satisfy: $$\frac{d}{d\varphi} P_l(\cos \varphi) = 0 \text{ at } \varphi=0, \pi$$ 5. **Radial Equation:** The radial part satisfies: $$\frac{1}{r^2} \frac{d}{dr}\left(r^2 \frac{dR}{dr}\right) - \frac{\lambda}{r^2} R = \frac{1}{\alpha} \frac{dT}{dt} \frac{1}{T}$$ where $\lambda$ is the separation constant from angular parts. 6. **Time Dependence:** The time part satisfies: $$\frac{dT}{dt} = -\alpha \mu T$$ with $\mu$ related to eigenvalues from spatial parts. 7. **Eigenvalue Problem and Solution:** - Solve angular eigenvalue problems to find $\lambda$. - Solve radial Sturm-Liouville problem with boundary condition $R(R) = 0$. - Construct full solution as series: $$v(r, \theta, \varphi, t) = \sum_{n,l} A_{n,l} e^{-\alpha \mu_{n,l} t} R_{n,l}(r) \cos(n \theta) P_l(\cos \varphi)$$ 8. **Summary:** The problem reduces to finding eigenvalues and eigenfunctions of the angular and radial parts with given boundary conditions, then combining them with exponential decay in time. This approach is standard for heat/diffusion equations in spherical coordinates with mixed boundary conditions.