Closed form solutions of the heat diffusion equation with a Gaussian source

In the context of the Large Hadron Collider, high energy particles are dumped onto blocks of material that are commonly called beam-dumps. The high energy deposition is almost instantaneous and can be represented by a very high power source for a short period of time. Temperatures can rise above failure limits at the locations of highest energy deposition. Particle beam accelerators are also widely used for medical applications where electron, proton or ion particle beams at energies of the order of megaelectronvolts are used to destroy cancer cells [1]. The motivation of this article is to be able to calculate the temperature fields without turning to numerical simulations that can be very time consuming. We present closed form solutions of the transient heat diffusion problem during the energy deposition. The transient heat diffusion equation assumes infinitely fast propagation of the information throughout the medium, but the diffusion process remains at finite speeds. Energy is deposited in such a small time interval so that the heat flux across the boundaries is assumed to be zero. It follows that the heat diffusion equation can be solved in an infinite medium, by the use of Green's functions, to provide closed form solutions for the transient temperature field within the pulse time. The range of applicability of these solutions is supported by a parametric relative error analysis. They demand very little computational power and time compared to commercial numerical software.