A localized collocation scheme with fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials. (December 2021) - Xi, Qiang; Fu, Zhuojia; Rabczuk, Timon; Yin, Deshun
Highlights: A localized collocation scheme is proposed for anomalous heat conduction analysis. Generalized reciprocity method is proposed to deal with the nonhomogeneous term. the effect of fractional derivative on anomalous heat conduction is investigated. Abstract: This paper makes the first attempt to develop a novel localized collocation scheme based on fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials. In the present numerical framework, the Laplace transformation and Fixed Talbot numerical inverse Laplace transformation are employed to deal with the time fractional derivative in anomalous time fractional heat conduction model. The localized collocation scheme based on fundamental solutions is applied to solve Laplace-transformed time-independent partial differential equations (PDEs), and the generalized reciprocity method (GRM) coupled with the derived problem-dependent generalized fundamental solution is proposed to deal with the inhomogeneous term in Laplace-transformed time-independent PDEs. Numerical tests demonstrate that the developed computational strategy can be considered as one of the competitive alternatives in the simulation of anomalous heat conduction behavior of functionally graded materials. Furthermore, the effect of time fractional derivative order on the behavior of anomalous heat conduction is numerically investigated.