Sinc-Collocation Method: Exponential Accuracy

Updated 13 April 2026

Sinc-collocation method is a spectral algorithm that uses Sinc basis functions and conformal mapping to achieve high-accuracy solutions for differential and integral equations.
It employs explicit interpolation, differentiation, and integration formulations to effectively resolve boundary singularities and unbounded domains.
The method attains almost-exponential convergence rates, with double-exponential variants offering superior performance for a range of applications.

The Sinc-collocation method is a class of exponentially convergent spectral algorithms for the numerical solution of differential, integral, and eigenvalue problems using Sinc basis functions. The approach leverages the analytic properties of the Sinc function on infinite and semi-infinite intervals and, through conformal mappings, generalizes this framework to finite and semi-infinite domains. It is applicable to ordinary and partial differential equations, Volterra and Fredholm integral equations, weakly singular equations, and eigenvalue problems with singular or infinite domains. The Sinc-collocation method is characterized by spectrally accurate representations, explicit convergence theory, and broad applicability to problems with boundary singularities and unbounded geometry.

1. Sinc Basis Construction, Cardinal Properties, and Mapping Techniques

The normalized Sinc function on $\mathbb{R}$ is defined by

$\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$

The Sinc-collocation method builds the solution space using Sinc basis functions localized at mesh points. To handle domains of practical interest, a conformal map $\phi$ maps the physical domain (for example, $[0,\infty)$ or $(a,b)$ ) to the entire real line, and the mesh points are defined as $t_k = k h$ ( $h>0$ ), pulling back to the physical domain via $x_k = \phi(t_k)$ .

For $[0,\infty)$ , a typical mapping is $\phi(t) = e^t$ , so $\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$0. On finite intervals, maps such as the single-exponential (SE) $\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$1 or the double-exponential (DE) $\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$2 are used (Bayatbabolghani et al., 2017, Gaudreau et al., 2014). This mapping enables the Sinc basis to effectively resolve boundary layers and singularities.

The Sinc-cardinal basis $\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$3 on the physical domain is given by

$\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$4

satisfying the interpolation (cardinal) property $\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$5. For periodic problems, the basis is extended through periodic repetition, and for domains with endpoint singularities, basis modifications may include explicit singular factors (Marconcini et al., 2013, Nedaiasl, 2019).

2. Collocation System Formulation for Differential and Integral Equations

The Sinc-collocation method approximates the solution $\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$6 by a Sinc expansion

$\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$7

and imposes the governing equations at the collocation nodes $\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$8: $\sinc\,z = \frac{\sin(\pi z)}{\pi z}, \qquad \sinc(0) = 1.$9 where $\phi$ 0 is a differential or integral operator. For ODEs, collocation requires the computation of differentiation matrices, which, via chain rule and Sinc derivative formulas, can be explicitly evaluated at the mesh points (Bayatbabolghani et al., 2017). For m-th order derivatives: $\phi$ 1

For Volterra/Fredholm integral equations, Sinc quadrature and indefinite-integration approximations are combined by replacing integrals with Sinc-discretized sums, e.g.: $\phi$ 2 where $\phi$ 3 is a precomputed indefinite-integration weight function (Okayama, 14 Mar 2025).

For boundary value problems, endpoint conditions are enforced either by replacing equations at boundary nodes or by modifying the expansion to include linear "hat" functions, e.g.

$\phi$ 4

where $\phi$ 5 and $\phi$ 6 (Okayama, 14 Mar 2025).

3. Error Analysis and Convergence Rates

The analytic structure of the Sinc basis and conformal mapping enables root-exponential or almost-exponential convergence for functions analytic in a strip about the real axis and satisfying algebraic boundary regularity. Specifically, with $\phi$ 7 a boundary regularity index and $\phi$ 8 the half-width of the analyticity strip, choosing mesh size $\phi$ 9 (SE methods) yields (Bayatbabolghani et al., 2017, Okayama, 2013): $[0,\infty)$ 0 Double-exponential (DE) methods achieve the faster rate for $[0,\infty)$ 1: $[0,\infty)$ 2 For eigenvalue problems with double-exponential basis decay, the error in eigenvalues satisfies

$[0,\infty)$ 3

where $[0,\infty)$ 4 quantify the double-exponential decay of the transformed solution (Gaudreau et al., 2014, Gaudreau et al., 2014, Cassidy et al., 2015).

4. Applications: ODEs, PDEs, Integral and Eigenvalue Problems

The Sinc-collocation method is used in a broad range of problems:

Linear and nonlinear ODEs/IVPs/BVPs: By collocation with Sinc-Lagrange or Poly-Sinc bases, including adaptive piecewise variants for stiff or layer problems; supports exponential convergence under analytic data (Khalil et al., 2022).
Volterra and Fredholm equations: Discretization via Sinc-collocation produces algebraic systems that attain root-exponential or almost-exponential accuracy even in the presence of endpoint singularities (Okayama, 27 Feb 2025, Okayama, 2023, Okayama, 14 Mar 2025).
Weakly singular equations: The method incorporates explicit singular factors to resolve algebraic kernel singularities, maintaining convergence rates (Nedaiasl, 2019).
Eigenvalue and Sturm-Liouville problems: DE Sinc-collocation yields symmetric/centrosymmetric generalized eigenproblems for quantum, Sturm-Liouville, and Schrödinger operators with singular or infinite domains (Gaudreau et al., 2014, Gaudreau et al., 2014, Gaudreau et al., 2015, Sharaf et al., 2019).
Fractional PDEs: Tensor-product Sinc bases combined with collocation enable discretization of fractional Laplacians and other nonlocal operators in higher dimensions (Antil et al., 2022).
Stochastic PDEs: Sinc-collocation at spatial nodes is combined with spectral polynomial-chaos expansions for elliptic SPDEs (Youssef et al., 2019).

5. Implementation and Computational Properties

The discrete algebraic systems arising from Sinc-collocation are dense and of size $[0,\infty)$ 5 to $[0,\infty)$ 6 for direct solvers; FFT-based strategies are available for convolution structures (Antil et al., 2022). For eigenvalue problems, symmetry and centrosymmetry under parity can be exploited to reduce the problem size and computational resources (Gaudreau et al., 2015).

Sinc-collocation is spectrally accurate for analytic functions, and, for classical analytic ODEs/integrals, $[0,\infty)$ 7– $[0,\infty)$ 8 typically suffices for $[0,\infty)$ 9– $(a,b)$ 0 accuracy (Bayatbabolghani et al., 2017). For strong endpoint singularities, polynomial or fractional factors may be included in the basis (Nedaiasl, 2019).

Special care is needed in the evaluation of high-order Sinc derivatives at nearly coincident nodes; stable recurrence formulas are documented in the literature (Bayatbabolghani et al., 2017). The DE mapping is preferred for functions with severe singularities or infinite/zero-boundary layers.

For nonlinear collocation systems (from nonlinear ODEs or integral equations), Newton-type or Gauss-Seidel fixed-point iterations are employed, with explicit contraction bounds available for the DE Sinc setting (Yamamoto, 3 Jan 2026). The convergence factor for Gauss-Seidel iteration decays as $(a,b)$ 1 for $(a,b)$ 2, enabling rapid solution of large nonlinear systems.

6. Comparison to Other Spectral and Collocation Methods

Sinc-collocation is equivalent, in the sense of spectral accuracy and conditioning, to Fourier pseudospectral collocation when the conformal map reduces to the identity (periodic/whole-line problems). For periodic or band-limited functions, it reproduces Fourier interpolation and derivative properties precisely (Marconcini et al., 2013). Relative to finite-element, polynomial collocation, or harmonic oscillator bases, Sinc-collocation yields substantially faster convergence for problems with analytic data or boundary singularities (Sharaf et al., 2019), with fewer basis functions needed for high accuracy.

Compared to Sinc-Nyström methods, Sinc-collocation avoids the repeated use of special functions (e.g., sine integrals) in the reconstruction phase, offering both computational efficiency and comparable or better error control (Okayama, 2013, Okayama et al., 2023). Adaptive and piecewise Poly-Sinc variants further enhance efficiency for problems with steep gradients or localized layers (Khalil et al., 2022).

7. Significance and Current Research Directions

The Sinc-collocation method, especially in its double-exponential (DE) variant, is established as a theoretically rigorous and practically efficient method for singular problems, problems on infinite and semi-infinite domains, and for equations with endpoint singularities or non-smooth solutions. Recent research focuses on:

Establishing rigorous convergence theorems for new Sinc-collocation variants and for equations involving more complex and variable transformations (Okayama, 14 Mar 2025).
Improving node consistency and computational efficiency, especially for large-scale and integral equation problems (Okayama, 2023).
Analyzing nonlinear system solvers for collocation equations, demonstrating super-rapid anomaly correction with Gauss-Seidel (Yamamoto, 3 Jan 2026).
Extending applications to stochastic PDEs, fractional operators, and domain-decomposition via adaptive collocation (Antil et al., 2022, Khalil et al., 2022, Youssef et al., 2019).
Exploiting symmetries (e.g., centrosymmetry) to dramatically reduce computational cost (Gaudreau et al., 2015).