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A nonstandard fractional Jacobi pseudospectral method for weakly regular right-sided Caputo terminal problems

Ibrahem G. Ameen, Magda Hammad, Mohsen Razzaghi, Mahmoud A. Zaky

arXiv:2610.20829v1math.NAcs.NA

TL;DR

Weak terminal singularities in right-sided fractional systems can reduce the effectiveness of standard numerical methods. The paper proposes a fractional Jacobi pseudospectral method for coupled nonlinear terminal systems, using a basis designed for that singular structure. Its analysis and numerical experiments report high-order accuracy for weakly regular solutions, while fast evaluation for the right-sided framework remains an open direction.

  • Problem

    Weak algebraic singularities near the terminal point challenge standard numerical methods for right-sided fractional equations.

    Source: §1 · pp. 1–2

  • Method

    The method uses an FJFs2–Gauss collocation scheme that approximates the system in a fractional Jacobi basis.

    Source: §1 · p. 3, §3 · p. 7

  • Results

    The proposed scheme retains high-order accuracy for weakly regular solutions, with numerical results showing rapid weighted-error decrease for increasingly many basis functions, including more singular cases.

    Source: This problem admits the exact solution · pp. 19–20, §7 · p. 20

  • Takeaways & Limitations

    The singularity-preserving basis supports accurate approximation of low-regularity right-sided fractional systems, including nonlinear coupled systems.

    Source: §7 · p. 20

    Whether fast evaluation can be designed for the global right-sided pseudospectral scheme, and what efficiency gains it could offer, remains open.

    Source: §7 · p. 20

Abstract

from arXiv · show

Right-sided fractional differential equations arising in terminal value problems present substantial analytical and numerical challenges due to their intrinsic nonlocal structure and the presence of weak algebraic-type singularities near the terminal point. These singular behaviors significantly degrade the performance of classical finite-difference, finite-element, and standard spectral methods, which typically rely on smoothness assumptions that break down in the fractional setting. In this work, we develop a comprehensive fractional pseudospectral framework specifically designed for nonlinear right-sided Caputo terminal value systems. The proposed method employs a versatile family of nonstandard fractional Jacobi basis functions whose endpoint-weighted structure naturally reflects the asymptotic behavior of right-sided fractional operators. This construction enables the accurate approximation of solutions with limited regularity near the terminal point, without requiring smoothing transformations or mesh grading. We establish the existence and uniqueness of the discrete numerical solution and derive rigorous convergence estimates that faithfully capture the inherent singularity structure of the problem. A series of numerical experiments validates the theoretical developments and demonstrates that the proposed approach delivers high-order accuracy and superior resolution of terminal-layer behavior compared with traditional spectral techniques. The resulting framework provides an efficient and reliable computational tool for challenging nonlinear fractional terminal value systems characterized by weak solution regularity.

1. Introduction

Right-sided fractional equations often produce weak algebraic singularities near the terminal point, reducing the accuracy or convergence of standard numerical methods. This work extends fractional collocation to fully coupled nonlinear terminal systems using a nonstandard Jacobi basis designed to reflect those singularities.

1. Introduction

  • Right-sided fractional kernels become weakly singular near the terminal point, where solutions often develop algebraic endpoint singularities and lose classical differentiability.The resulting sharp gradients or reduced smoothness can cause loss of accuracy and nonuniform convergence in standard methods.Source: §1 · pp. 1–2
  • Weak terminal singularities can cause standard spectral and finite-difference methods to lose order or converge inefficiently unless approximation strategies reflect the endpoint structure.Source: §1 · p. 2
  • Coupled nonlinear terminal systems are especially challenging because singular behavior can propagate through all solution components, motivating approximation spaces tailored to terminal singularities.Source: §1 · p. 2
  • The paper assumes homogeneous terminal conditions for notational simplicity; a nonhomogeneous condition is shifted to an equivalent homogeneous one.The shifted right-hand side retains the smoothness and Lipschitz properties, so the stated theorems apply unchanged.Source: §1 · p. 3
  • The nonstandard fractional Jacobi basis is designed to resolve terminal singularities in fully coupled nonlinear terminal systems while retaining high-order accuracy for nonsmooth solutions.The paper also develops a rigorous convergence analysis for this setting.Source: §1 · p. 3

2. Properties of FJFs2

The FJFs2 approximation space is built from fractional powers of the distance to the right endpoint and uses a weight adapted to that fractional scaling. The section also connects its fractional Sobolev framework to the classical Jacobi-weighted space.

2. Properties of FJFs2

  • The FJFs2 space is generated by fractional powers of (1 − ς), with fractional scaling parameter µ in (0, 1].Source: §2 · p. 3
  • For µ = 1, the FJFs2 family reduces to the classical shifted Jacobi polynomials.Source: §2 · p. 4
  • The FJFs2 form an orthogonal system on χ with respect to the stated weight, which defines the associated weighted L2 norm.Source: §2 · p. 4
  • The fractional Sobolev-type space is related to its classical counterpart through a fractional change of variables, yielding equivalent norms.Source: §2 · p. 5

Then the interpolation error estimate

The section gives an interpolation error estimate with a constant independent of the approximation order and function. It also sets out weighted norms and componentwise notation for vector-valued functions.

Then the interpolation error estimate

3. The FJFs2–Gauss collocation method

The collocation scheme rewrites the right-sided fractional system in integral form and approximates each component in the FJFs2 basis. Interpolation and quadrature then yield a nonlinear algebraic system for the unknown coefficients.

3. The FJFs2–Gauss collocation method

  • The system is rewritten through the right-sided fractional integral representation, then approximated using FJFs2 interpolation and Gauss quadrature.Source: §3 · pp. 7–8
  • The scheme seeks each component of the approximate solution as an expansion in the FJFs2 basis.Source: §3 · p. 7
  • The resulting nonlinear algebraic system determines the expansion coefficients and can be solved with standard iterative methods such as Newton-type methods.The numerical experiments use Mathematica FindRoot with zero initial guesses.Source: §3 · p. 8

4. Convergence analysis

The convergence analysis bounds the error between the exact and discrete solutions under regularity and Lipschitz assumptions. It derives the estimate by separating interpolation, quadrature/interpolation, and nonlinear contributions.

4. Convergence analysis

  • The error analysis separates contributions from interpolation in z, quadrature/interpolation in η, and the nonlinearity of bF.The decomposition is introduced before estimating the terms individually.Source: §4 · p. 10
  • Theorem 4.1 gives an error estimate for the discrete solution under a regularity assumption on U and a Lipschitz condition on bF.The stated hypotheses include L < Γ(β + 1).Source: §4 · p. 10
  • Under condition L < Γ(β + 1), the proof absorbs the last term into the left-hand side, from which estimate (4.8) follows.Source: §4 · p. 15

5. Existence and uniqueness of the discrete collocation scheme

The discrete collocation system is recast as a fixed-point problem. Under a contraction condition, the iteration converges in the weighted norm and the discrete solution is unique.

5. Existence and uniqueness of the discrete collocation scheme

  • The discrete collocation system admits a unique solution when the stated Lipschitz condition yields L Γ(β + 1) < 1.The theorem also states convergence of the generated sequence in the weighted norm.Source: §5 · pp. 15, 17
  • The fixed-point iteration produces a Cauchy sequence in the weighted norm, whose limit solves the discrete collocation system.Source: §5 · pp. 15, 17
  • Uniqueness follows from the contraction property, equivalently by the Banach fixed-point theorem.Source: §5 · p. 17

6. Numerical results

The numerical section tests the method on linear and nonlinear systems, including rational and irrational fractional orders. Errors are assessed in a weighted L2 norm, with a linear benchmark set up for comparison to a prior method.

6. Numerical results

  • Three examples assess accuracy, robustness, and computational efficiency for linear and nonlinear systems with rational and irrational fractional orders.Source: §6 · p. 17
  • The examples progress from a linear benchmark with a literature comparison to nonlinear coupled systems, including irrational orders and nonlinear interactions.Source: §6 · pp. 17–18
  • Errors are measured in the weighted L2 norm, and Example 1 reports error norms for β = 0.7, 0.5, and 0.3.The benchmark problem is linear and has terminal condition U(1) = 0.Source: §6 · p. 18

This problem admits the exact solution

Across the reported examples, the proposed method improves approximation of weakly singular right-sided problems and retains rapid convergence for linear and nonlinear systems. The results include more singular cases and irrational fractional orders.

Linear benchmark

  • The proposed scheme outperforms the method in on the linear benchmark with very few modes, which the authors attribute to its singularity-preserving basis.The comparison is described as showing substantially reduced approximation error across tested β values.Source: This problem admits the exact solution · p. 18

Example 2

  • For all fractional order values of β, computed weighted errors rapidly decrease as M increases; the method preserves spectral convergence even for more singular solutions with pronounced endpoint singularities near t = 1.Source: This problem admits the exact solution · p. 19

Example 3

  • In the nonlinear coupled system, weighted errors for U1 and U2 decrease rapidly as basis functions increase, despite irrational fractional orders.The authors identify the ability to handle both rational and irrational orders as an advantage of the framework.Source: This problem admits the exact solution · p. 20

7. Conclusion

The conclusion presents a fractional Jacobi pseudospectral method designed to preserve the terminal singular structure of low-regularity right-sided fractional systems. Its analysis establishes discrete well-posedness and error estimates, while experiments support its accuracy; extensions and acceleration remain future directions.

7. Conclusion

  • The method preserves terminal singular structure and achieves high-order accuracy for weakly regular solutions without smoothing transformations, singularity corrections, or auxiliary regularization.Experiments report highly accurate approximations with relatively few collocation points.Source: §7 · p. 20
  • The interpolation and collocation analysis yields explicit error estimates and proves existence and uniqueness of the discrete numerical approximation.Source: §7 · p. 20
  • Future work includes extending the framework to multidimensional systems, nonlinear and tempered operators, distributed-order models, and inverse problems.Acceleration for this global, non-marching right-sided formulation remains an open direction.Source: §7 · p. 20
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