A Fast Operator-Interpolation Mapped Spectral Method for Variable-Order Fractional PDEs

We propose a fast mapped spectral method for a class of variable-order fractional partial differential equations posed on the real line. The main difficulty arises from the spatially dependent fractional order of the operator, which prevents the direct diagonalisation techniques available for constant-order fractional Laplacians and leads, in general, to expensive nonlocal discretisations. To overcome this limitation, we introduce an operator-interpolation strategy in which the variable-order fractional Laplacian is approximated by a finite combination of constant-order fractional operators evaluated at suitably chosen interpolation nodes in the fractional-order variable. Each constant-order contribution is treated through a Fourier-like mapped Chebyshev representation on R, allowing the corresponding nonlocal operator to be evaluated efficiently in spectral space. Chebyshev interpolation with respect to the fractional order is employed to obtain an accurate approximation over a prescribed interval s(x) ∈ [s_min, s_max]. This construction separates the difficulties associated with spatial unboundedness and variable nonlocality, and leads to an implementation whose cost is governed by a small number of fast constant-order operator evaluations. The spatial discretisation is combined with a pseudospectral treatment of nonlinear terms, avoiding the explicit construction of dense high-order interaction tensors. Approximation properties are analysed by separating the error due to interpolation in the fractional order from the mapped spectral discretisation error in space. The proposed approach is first validated on variable-order fractional problems with prescribed or manufactured solutions. It is then applied to stationary variable-order fractional Allen–Cahn equations in heterogeneous media, with particular attention to the influence of spatial variations of the fractional order on interface profiles, asymmetry, and far-field decay. Numerical experiments are designed to assess accuracy, convergence, computational complexity, and robustness with respect to both the spectral resolution and the variation of the fractional order.