A two-dimensional hybrid Legendre-improved block-pulse function method for Caputo-Fabrizio time-fractional Cattaneo equations

A two-dimensional hybrid Legendre--improved block-pulse function method is developed for the Caputo--Fabrizio time-fractional Cattaneo equation. The two-dimensional LIBPF basis combines normalized shifted Legendre polynomials and improved block-pulse functions in a tensor-product form. The highest mixed derivative is approximated in this basis, and the numerical solution is recovered through a boundary-conforming repeated-integration reconstruction. Closed-form expressions are derived for the required repeated integrals and Caputo--Fabrizio kernel terms. Under suitable regularity assumptions, projection and reconstruction error estimates are established, showing a two-order gain in the mesh-dependent error after reconstruction. A weighted oversampled least-squares discretization based on mapped Gauss--Legendre nodes is used to improve discrete stability, and the resulting system is solved by QR factorization. Numerical results confirm the predicted convergence orders, demonstrate the stabilizing effect of oversampling, and show consistent accuracy for different fractional orders and for both separable and nonseparable space--time solutions.