A Chebyshev Spectral Method for Time–Fractional Heat Equations on Irregular Domains

Abstract

In this paper, we use Chebyshev polynomials to create a spectral collocation method that can be used to numerically solve the two-dimensional time-fractional heat equation with the Caputo derivative. Our method is different from traditional ones because it uses the high accuracy of global spectral discretization in space along with the shifted Grünwald–Letnikov approximation in time. The suggested scheme works on irregular computational domains by using coordinate transformations to map the geometry onto a standard domain. We look at how the method converges, how stable it is, and how fast it works. We test the scheme on a number of benchmark problems in both standard and irregular domains, such as a right trapezoid and a pipe-like shape. Numerical tests show that the spectral method achieves exponential convergence for smooth solutions, and it is more accurate than other methods of discretization. The numerical experiments were conducted using MATLAB R2023a on a Windows 11 system equipped with an Intel Core i5 processor and 16 GB of RAM.