Error of discretization of Caputo fractional derivative in weighted spaces

Abstract: We establish uniform error bounds of the L1 discretization of the Caputo fractional derivative of the function from the weighted Sobolev space with weight belonging to the Mucknenhoupt class. We present how our framework works for several examples of weight, which belong to the Muckenhoupt class. As and application, we show the convergence of the L1 scheme for the Fractional ODE. Finally, we verify the theoretical results with numerical illustrations.

• The paper establishes explicit error estimates for the L1 discretization of the Caputo fractional derivative in weighted Sobolev spaces, accurately capturing solution singularities through Muckenhoupt weights.
• The analysis leverages local Taylor expansions and weighted Hölder inequalities to derive uniform error bounds for various weight profiles, including power, Jacobi, and logarithmic weights.
• Numerical experiments validate the theoretical error rates and demonstrate the framework’s applicability to fractional differential equations with non-smooth initial data.

Error Estimates for Caputo Fractional Derivative Discretization in Weighted Sobolev Spaces

Introduction

The paper develops a comprehensive error analysis for the L1 scheme when discretizing the Caputo fractional derivative for functions in weighted Sobolev spaces $W^{2,p}_\omega(0,T)$, with the weight $\omega$ drawn from the Muckenhoupt class $A_p$. The Caputo fractional derivative, crucial in modeling subdiffusion and anomalous processes, inherently produces solutions with weak singularities near the initial time. This behavior makes classical smoothness assumptions ($C^2$ or standard Sobolev regularity) unsuitable for many practical fractional differential equations (FDEs). The paper argues that weighted Sobolev spaces, equipped with Muckenhoupt weights, provide the correct functional setting for capturing the true solution regularity; this approach harmonizes numerical analysis with the realistic singular behavior observed in fractional-order models.

Weighted Functional Framework and Well-posedness

Weighted spaces $L^p_\omega$ and $W^{2,p}_\omega$ are constructed to control singularity in derivatives via integrability, with the weight $\omega$ satisfying the $A_p$ condition. The paper establishes that the Caputo operator $D^\alpha$ acts boundedly from $W^{1,p}_\omega$ to $\omega$0. This result relies on harmonic analysis, specifically the boundedness of the Riemann-Liouville fractional integral in $\omega$1-weighted spaces, ensuring consistency between the functional setting and numerical methods for fractional calculus.

Uniform Error Bounds for L1 Discretization

The central theoretical result asserts that, for $\omega$2 and $\omega$3,

$\omega$4

where $\omega$5 encodes the influence of the weight's local behavior over discretization intervals. The proof draws on local Taylor expansion, the structure of the L1 finite difference scheme, and sophisticated usage of Hölder’s inequality adapted to weighted spaces.

Explicit Error Estimates for Common Singular Profiles

• Power Weights $\omega$6: The order is $\omega$7, matching the solution's singularity at the origin.
• Jacobi Weights $\omega$8: The order is again $\omega$9, with $A_p$0, reflecting both initial and terminal singularity.
• Logarithmic Weights $A_p$1: The order is $A_p$2, with an additional logarithmic factor $A_p$3.

These bounds are demonstrated to be sharp, as verified through numerical experiments.

Application to Fractional Differential Equations

The error estimate is extended to initial value problems for FDEs of the form $A_p$4, discretized via the L1 method. The global error between the true solution and the numerical approximation is shown to be governed directly by the uniform L1 truncation error, leading to the convergence rate $A_p$5 under unconditional stability. This result is significant as it applies to equations with solution singularities not covered under classical smoothness assumptions.

Numerical Validation

Numerical experiments strongly corroborate the analytical error bounds. For various choices of $A_p$6, $A_p$7, $A_p$8, and $A_p$9, the observed convergence rates closely align with theoretical predictions. Complications arising with logarithmic weights are resolved by refining the convergence order estimation to account for logarithmic scaling, yielding agreement between computation and theory across all tested scenarios.

Implications and Future Directions

The analytical framework eliminates reliance on restrictive smoothness assumptions and provides realistic, robust error estimates for L1 discretizations of fractional derivatives in practical applications. The Muckenhoupt-weighted Sobolev setting precisely characterizes solution behaviors in fractional diffusion and reaction models, opening avenues for adaptive weighted methods and rigorous analysis of more general fractional PDEs. Notably, the paper's approach can potentially be extended to fractional-order Sobolev spaces $C^2$0, $C^2$1, to accommodate intermediate regularity in actual solutions of time-fractional PDEs.

From a computational perspective, the results advance best-practice guidance for mesh refinement and stability analysis in the simulation of fractional dynamics, particularly for models with non-smooth or algebraically/logarithmically singular initial data. Theoretically, the identification of weight-dependent convergence rates motivates deeper exploration of harmonic analysis tools in fractional calculus and numerical methods, including optimal mesh grading and weighted residual approaches.

Conclusion

This study delivers a rigorous error characterization for the L1 scheme applied to Caputo fractional derivatives within weighted Sobolev spaces, with Muckenhoupt weights ensuring harmonic analytic compatibility and physical realism. The resulting bounds are explicit in both the discretization parameter and the weight's profile, and they correctly capture solution singularities. Extensions to intermediate regularity and more complex fractional PDEs, as well as fully adaptive schemes based on weight analytics, are natural future research directions.