In borehole heat exchangers, a set of coupled partial differential equations describe heat transfer in the three components of the borehole such as pipe-in, pipe-out and grout. This paper presents an analytical model for understanding coupled conductive-convective heat transfer processes in a borehole heat exchanger subjected to defined initial and boundary conditions. This analytical model introduces a simulation of transient heat transfer in a single U-tube geothermal borehole heat exchanger. The main focus of this research is on the solution technique that can be useful for many other applications, including fluid flow in narrow pipes, high fluid velocities, high fluid viscosities, and pipes made of composite materials. Also, the method can be useful for solving other no homogeneous coupled partial differential equations. Results show that suggested technique is accurate and effective for solving this type of equations.