A pseudospectral method for time-fractional PDEs with shifted Chebyshev and Lagrange interpolating polynomials on overlapping decomposed domains

Abstract

Time-fractional partial differential equations (TFPDEs) are powerful mathematical tools for modeling a wide range of physical and biological phenomena that exhibit memory effects, anomalous diffusion and non-local behavior. These classes of equations are crucial in capturing dynamics where the influence of past states affects future evolution, making them essential in many areas of applied science, such as heat transfer, viscoelasticity and anomalous diffusion. This study proposes a pseudospectral method that combines the weighted sums of the Chebyshev and Lagrange polynomials to numerically approximate the solutions of TFPDEs. The spatial domain is partitioned into uniform, overlapping subdomains, where the solution in each subdomain is represented as a weighted sum of the Lagrange interpolating polynomials. On the other hand, the time domain is treated as a whole without decomposition, and the solution in the temporal dimension is expanded using the first-kind shifted Chebyshev polynomials. We validate the accuracy and performance of the method through a series of test cases, covering both linear and nonlinear TFPDEs in one and multiple spatial dimensions. These examples showcase the method’s capability to handle the computational challenges associated with TFPDEs and underline its potential for broader applications in problems involving fractional dynamics. Specifically, the proposed technique is applied to resolve TFPDE, which models heat transfer on a disk, a problem relevant to modeling heat conduction in circular plates and semiconductor wafers. A time-dependent Gaussian heat source concentrated in a specific region of the disk is introduced to accurately simulate practical thermal diffusion dynamics. The gradual increase of the source term over time offers a more realistic representation of the evolving thermal diffusion process.