A sparse spectral method for fractional differential equations in one-spatial dimension

• Published in: Advances in computational mathematics Vol. 50; no. 4
• Main Authors: Papadopoulos, Ioannis P. A., Olver, Sheehan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-08-2024; Springer Nature B.V
• Subjects: Affine transformations; Chebyshev approximation; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Differential equations; Fractional calculus; Hilbert transformation; Intervals; Linear systems; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Numerical methods; Operators (mathematics); Polynomials; Spectral methods; Visualization; Wave equations; 44A15; Sum spaces; 33C45; Sparse spectral methods; 26A33; Fractional; 65H05; Nonlocal operators
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10164-1

We develop a sparse spectral method for a class of fractional differential equations, posed on
R
, in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on
[
-
1
,
1
]
whereas the latter have global support. The global approximation space may contain different affine transformations of the basis, mapping
[
-
1
,
1
]
to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving
K
independent sparse linear systems of size
O
(
n
)
×
O
(
n
)
, with
O
(
n
)
nonzero entries, where
K
is the number of different intervals and
n
is the highest polynomial degree contained in the sum space. This results in an
O
(
n
)
complexity solve. Applications to fractional heat and wave equations are considered.