Fractional order Chelyshkov wavelet for solving variable coefficients Caputo time-fractional convection-diffusion equation

Convection-diffusion equations are widely used to describe transport processes and arise in numerous scientific and engineering applications, including heat transfer, contaminant transport in porous media, groundwater flow, chemical reactions, and biological systems. However, many real-world transport phenomena exhibit memory properties that classical convection-diffusion equations cannot adequately represent. To address such nonlocal temporal effects, time-fractional convection-diffusion equations have been introduced, in which fractional derivatives provide a realistic description of anomalous diffusion. This work proposes a novel wavelet-based numerical method for solving a variable-coefficient Caputo time-fractional convection-diffusion equation using fractional-order Chelyshkov wavelets. The method constructs fractional Chelyshkov wavelets from fractional Chelyshkov polynomials and derives corresponding fractional operational matrices. The proposed wavelet basis functions, together with an operational matrix formulation, transform the time-fractional convection-diffusion equation into a system of algebraic equations. The stability, convergence, and error estimate are established. To demonstrate the effectiveness and precision of the proposed method, several numerical examples are provided. CPU time and memory usage are reported for different numbers of basis functions. The numerical results indicate that the fractional Chelyshkov wavelets provide an efficient representation of smooth solutions, achieving high accuracy with a relatively small number of basis functions.

کلیدواژه ها:

Caputo time fractional convection-diffusion equation ، Fractional Chelyshkov wavelet ، Operational matrix ، Stability ، Convergence and error analysis