Numerical study demonstrates high-order accuracy for fractional reaction–subdiffusion equations, highlighting rapid exponential convergence without expensive numerical quadratures.
Key Points
To develop an efficient spectral Galerkin numerical scheme for solving one-dimensional time-fractional reaction–subdiffusion equations with weak initial singularities.
Analytically transformed the governing fractional differential equation into a weakly singular Volterra integral equation to mitigate initial singularity challenges.
Constructed numerical solutions in a two-dimensional tensor-product space using orthogonal Legendre cardinal functions on Gauss and Gauss–Lobatto grids to bypass numerical quadratures.
Derived theoretical a priori error bounds alongside residual-based a posteriori error estimators.
Convergence rates scale algebraically with the Sobolev regularity of the exact solution and accelerate to exponential (spectral) rates for sufficiently smooth profiles.
Numerical tests confirm that the Legendre cardinal basis eliminates the need for costly quadrature evaluations while maintaining superior accuracy and computational efficiency compared to standard approaches.