Close-contact melting within a spherical capsule its investigated both numerically and analytically. A complete mathematical model is solved numerically by utilizing the boundary fixing method. The approximate approach developed by Bareiss and Beer for the horizontal cylinder is applied to constructing an approximate mathematical model of contact melting in a spherical capsule with a non-isothermal wall. The main characteristic scales and dimensionless parameters which describe the principal features of the melting process are found. Due to the presence of the small parameter in governing equations the perturbation method is implemented. As a result, simple analytical solutions were found which describe close-contact melting inside the capsule with a non-isothermal wall and account for the streamwise convection in the molten layer. The extensive validation of the analytical solution, and its comparison with the numerical results, gives the proof of accuracy of the analytical solutions with estimated error of 10-15%. This conclusion is of crucial importance for evaluating the real latent heat thermal energy storage systems which contain thousands of capsules, since the simple closed-form solutions for a single capsule, used in the mathematical modeling of such kind of complex systems, significantly reduces the cost of numerical computations.