In this chapter, a heat conduction equation in terms of the Caputo-Fabrizio derivative of order 0< α≤ 1 is considered. Caputo-Fabrizio derivative is defined by a non-singular exponential decay kernel function. Thanks to this feature, it eliminates the computational difficulties arising from the singular power kernel functions of the traditional fractional derivatives such as Riemann-Liouville and Caputo. In the present study, the fundamental solutions to the Cauchy and Dirichlet problems based upon a heat conduction equation equipped with the Caputo-Fabrizio derivative are investigated on a line segment. To obtain the fundamental