A continued-fraction approach for transient diffusion in unbounded medium

Abstract

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 ( ω), 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 i ω 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 i ω . In the time-domain, it is interpreted as a system of fractional differential equations of degree α = 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.