A Shifted Balanced Polynomial Collocation Framework for Variable-Coefficient Multi-Term Time-Fractional Convection–Diffusion–Reaction Problems

We develop a spectral collocation method based on shifted balanced polynomials for solving variable-coefficient multi-term time-fractional convection–diffusion–reaction equations. The proposed approach uses the Caputo fractional derivative and a global space–time approximation. First, explicit formulas are derived for expressing the shifted balanced polynomials in the power basis and for expressing monomials in terms of the shifted balanced polynomial basis. These formulas are then used to obtain a representation of the Caputo operator at the selected points. The approximate solution is expanded in a tensor-product basis, and the governing equation is enforced at selected interior collocation points together with the prescribed initial and boundary conditions. Theoretical results concerning the growth of the shifted balanced polynomials and the decay of the expansion coefficients are established. These results are used to prove the absolute convergence of the proposed expansion and to derive an explicit upper bound for the truncation error. Four numerical examples are presented to examine the accuracy and effectiveness of the proposed method for different polynomial degrees and fractional orders. In addition, an account of the extension of the proposed approach to problems involving nonlinear terms is presented. The numerical results show that accurate approximations can be obtained using relatively small polynomial degrees.