Accurate computational study between spectral and wavelet methods for fractional differential equations

Abstract

This paper compares three numerical schemes for Caputo fractional differential equations: the Shifted Chebyshev Tau Method (SCTM), the Shifted Chebyshev Collocation Method (SCCM), and the Haar Wavelet Collocation Method (HWCM). In the two Chebyshev schemes, the unknown solution is approximated by shifted Chebyshev polynomials, whereas the Haar formulation uses localized piecewise-constant basis functions and fractional integration matrices. Each method reduces the governing equation to a finite algebraic system. The original contribution is a controlled like-for-like comparison in which the three formulations use the same numbers of unknowns, common independent test grids, and the same accuracy, conditioning, sparsity, and timing diagnostics. Six benchmark problems with known exact solutions are examined using the balanced approximation sizes N = 4, 8, 16, 32. The first five examples have smooth polynomial solutions. SCTM and SCCM recover the exact profiles to the adopted working precision whenever the exact polynomial lies in the selected approximation space. The sixth example has the nonpolynomial solution u(x) = x 5/2, whose third derivative is unbounded at the left endpoint, and therefore provides a genuine low-regularity convergence test. At N = 32, the maximum errors of SCTM and SCCM are 4.39523 × 10−6 and 4.31475 × 10−6 , respectively, whereas HWCM gives 4.33493 × 10−2 . The results show that the Chebyshev methods provide the highest accuracy for the problems studied, SCCM generally achieves this accuracy with a lower assembly cost than SCTM, and HWCM produces better-conditioned and less dense systems with generally first-order-type convergence.