A temporally local method for the numerical solution of transient diffusion problems in unbounded domains is proposed by combining the scaled boundary finite element method and a novel solution procedure for fractional differential equations. The scaled boundary finite element method is employed to model the unbounded domain. In the Fourier domain (x), an equation of the stiffness matrix for diffusion representing the flux-temperature relationship at the discretized near field/far field interface is established. A continued-fraction solution in terms of ffiffiffiffiffiffi ix p is obtained. By using the continued-fraction solution and introducing auxiliary variables, the flux-temperature relationship is formulated as a system of linear equations in ffiffiffiffiffiffi ix p . In the time-domain, it is interpreted as a system of fractional differential equations of degree a = 1/2. To eliminate the computationally expensive convolution integral, the fractional differential equation is transformed to a system of first-order differential equations. Numerical examples of two-and three-dimensional heat conductions demonstrate the accuracy of the proposed method. The computational cost of both the temporally global and local approach for transient analysis is examined.