A Hybrid Method Based on Block-Pulse Functions and Bernoulli Polynomials for the Efficient Numerical Solution of Two-Dimensional Fractional Differential Equations

Hadi, Mohammed Saleh

Jurnal MIPA dan Pembelajarannya
Vol. 6 No. 3 (2026): March

The current paper is a new hybrid numerical approach to the solution of two-dimensional fractional differential equations (FDEs) based on Block-Pulse Functions (BPFs) and compact support as well as Bernoulli polynomials (BPs) with fast global convergence. The fractional differential equation is converted into a system of linear algebraic equations through the proposed method based on collocation methods at Gauss-Lobotto points, using operational matrices of fractional integral and derivative operators.  The algorithm can be used with small bases (M≤5) to achieve high accuracy. The effectiveness of the method is confirmed by four numerical examples (including linear and nonlinear fractional differential equations) and a comparison with classical numerical methods. The approach is also used to a practical model to simulate the dispersion of the pollutants in the multi-layered soil systems. Findings indicate that hybrid approach provides a good compromise between computational cost and computational accuracy even with non-smooth solutions or complicated domains.