A meshless collocation method for solving the inverse Cauchy problem associated with the variable-order fractional heat conduction model under functionally graded materials

Abstract

A localized meshless collocation method, namely the generalized finite difference method (GFDM), is introduced to cope with the inverse Cauchy problem associated with the fractional heat conduction model under functionally graded materials (FGMs). The variable-order time-fractional heat conduction equation under the Caputo definition is employed to describe anomalous heat conduction problems. In the present numerical framework, temporal-discretization is implemented by using the standard implicit finite difference method with the spatial-discretization through the GFDM. On the basis of moving least squares and Taylor series expansion, the GFDM is capable of avoiding the ill-posedness in inverse Cauchy problems and solving the time fractional heat conduction equations (TFHCE) under FGMs. To give evidence of the efficiency and accuracy of the proposed approach for solving inverse heat conduction of FGMs, three numerical experiments are considered in the results and discussions section.