Numerical simulation of fractional evolution model arising in viscoelastic mechanics

This paper develops an efficient local meshless collocation algorithm for approximating the time fractional evolution model that is applied for the modeling of heat flow in materials with memory. The model is based on the Riemann-Liouville fractional integral. The proposed method discretizes the unknown solution using two main parts. First, the temporal direction is approximated through the second-order finite difference scheme. Second, the spatial direction of the governing problem is discretized via the local radial basis function partition of unity technique. The major drawback of global collocation techniques is the computational cost associated with arisen dense algebraic system. This localized method is based on partitioning the original domain to several subdomains by means of the kernel approximation on each local domain and allows one to significantly sparsify the algebraic system having small condition number in addition to lowering the computational cost. The stability and convergence of the time difference formulation are discussed in detail with respect to the H¹-norm. Numerical results and comparisons are illustrated in order to confirm theoretical analysis and the accuracy of the method.