NUMERICAL SOLUTION OF A MULTI-TERM TIME-FRACTIONAL VISCOELASTIC NON-NEWTONIAN FLUID MODEL VIA A HYBRID SPECTRAL METHOD

The application of a hybrid spectral collocation method (HSCM) to a class of new multi-term time-fractional viscoelastic non-Newtonian fluid models is studied in this work. The noteworthy addition of this work is that the new model in this study includes a novel time-fractional operator on the spatial derivative, also the considered model includes multi-term time-fractional derivatives with fractional orders ranging from 0 to 2. The proposed HSCM combines the Laplace transform (LT) for temporal discretization and the spatial operators are discretized via the Chebyshev collocation method (CM). It has been proven that the LT is beneficial for solving diffusion-type problems and presents a substitute for the finite difference method (FDM). In FDM, small time step is required to achieve high accuracy and stability, due to such limitations the computational cost is also enhanced. On the other hand, CM utilizes Langrange’s interpolation polynomial, based on Chebyshev collocation points. For smooth problems, the implementation of CM is straightforward and has spectral convergence. The HSCM has three basic steps. Initially, the LT is utilized to reduce the proposed model into a time-independent model; second, the CM is used for spatial discretization; and finally, to get the original solution of the model, inverse LT is used. The Ulam–Hyers stability of the considered model is also discussed. To ensure the stability, accuracy, and computational efficiency of the proposed method, four numerical examples have been examined.