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Fractal Fract
Volume 9
Issue 12
10.3390/fractalfract9120775
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Article

Error Analysis and Numerical Investigation of an L1-2 Fourth-Order Difference Scheme for Solving the Time-Fractional Burgers Equation

by
Kanyuta Poochinapan
1,2,3 and
Ben Wongsaijai
1,2,3,*
1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2
Advanced Research Center for Computational Simulation, Chiang Mai University, Chiang Mai 50200, Thailand
3
Centre of Excellence in Mathematics, MHESI, Bangkok 10400, Thailand
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(12), 775; https://doi.org/10.3390/fractalfract9120775 (registering DOI)
Submission received: 20 October 2025 / Revised: 24 November 2025 / Accepted: 25 November 2025 / Published: 27 November 2025
(This article belongs to the Special Issue Recent Advances in the Spatial and Temporal Discretizations of Fractional PDEs, Second Edition)
Versions Notes

Abstract

This paper presents a finite difference approach for solving the time-fractional Burgers’ equation, which is a model for nonlinear flow with memory effects. The method leverages the L1-2 formula for the fractional derivative and provides a novel linearization strategy to efficiently transform the system into a stable linear problem. Rigorous analysis establishes the existence, uniqueness, and pointwise-in-time convergence of the numerical solution in the L2 norm. The proposed formulation achieves second-order time accuracy and fourth-order spatial accuracy under smooth initial conditions, with numerically verified temporal convergence rates of O(τ1+α+τ2tnα−2) for solutions with weak singularities. Critically, numerical findings demonstrate that the method is robust and highly efficient, offering high-resolution solutions at a substantially lower computational cost than equivalent graded-mesh formulations.
Keywords: fourth-order difference method; Caputo fractional derivative; L1-2 formula; linearized difference scheme; convergence analysis; time fractional Burgers’ equation fourth-order difference method; Caputo fractional derivative; L1-2 formula; linearized difference scheme; convergence analysis; time fractional Burgers’ equation

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MDPI and ACS Style

Poochinapan, K.; Wongsaijai, B. Error Analysis and Numerical Investigation of an L1-2 Fourth-Order Difference Scheme for Solving the Time-Fractional Burgers Equation. Fractal Fract. 2025, 9, 775. https://doi.org/10.3390/fractalfract9120775

AMA Style

Poochinapan K, Wongsaijai B. Error Analysis and Numerical Investigation of an L1-2 Fourth-Order Difference Scheme for Solving the Time-Fractional Burgers Equation. Fractal and Fractional. 2025; 9(12):775. https://doi.org/10.3390/fractalfract9120775

Chicago/Turabian Style

Poochinapan, Kanyuta, and Ben Wongsaijai. 2025. "Error Analysis and Numerical Investigation of an L1-2 Fourth-Order Difference Scheme for Solving the Time-Fractional Burgers Equation" Fractal and Fractional 9, no. 12: 775. https://doi.org/10.3390/fractalfract9120775

APA Style

Poochinapan, K., & Wongsaijai, B. (2025). Error Analysis and Numerical Investigation of an L1-2 Fourth-Order Difference Scheme for Solving the Time-Fractional Burgers Equation. Fractal and Fractional, 9(12), 775. https://doi.org/10.3390/fractalfract9120775

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