In this work, a forward-backward time fractional heat equation is introduced for the first time. A non-overlapping domain decomposition procedure with a finite difference scheme is used to solve the underlying problem in one-dimension. According to the nature of forward and backward property, the physical domain is divided into two separate subdomians. Time discretization of the fractional time derivative is performed by a defferential operator namely V1 method and the second order central difference scheme is employed to deal with the spatial derivative. In addition, a matrix form for the discretized equation is obtained. Furthermore, the convergence and stability of the proposed method is investigated. Finally, some numerical examples are presented to demonstrate the effectiveness of the proposed method.