Analytical solution of the heat equation for an instantaneous point source in a hollow sphere

In this technical note, we add to the literature the analytical solution of the heat equation with radiation boundary conditions for an instantaneous point source in a hollow sphere. Instead of using the conventional method of the Laplace transfer and inverse Laplace transfer that requires intricate integral in the complex plane, we present a new approach to derive the analytical solution in the time domain. First, we express the instantaneous point source as the initial condition in terms of Dirac delta, e.g., Eq. (2), which is further expressed in the form of the general solution, i.e., Eq. (18). Subsequently, we determine the solution coefficients and obtain the analytical solution. Two special cases with prescribed surface temperature and no-flux boundaries are presented to demonstrate the solution. This solution is of practical importance in many scientific and engineering applications, such as pore pressure prediction in reservoir engineering, thermal analysis of casing in drilling engineering, and determining the relaxation process in nuclear magnetic resonance (NMR).