A High-Accuracy Hybrid Method for Linear Fredholm Integral Systems Using Bernoulli Polynomials Coupled with Enhanced Block-Pulse Functions

Alqarni, Mohammed Z.; Ramadan, Mohamed A.; Elaaser, Esraa G.; Osheba, Heba S.

doi:10.3390/math14173240

This is an early access version, the complete PDF, HTML, and XML versions will be available soon.

Open AccessArticle

A High-Accuracy Hybrid Method for Linear Fredholm Integral Systems Using Bernoulli Polynomials Coupled with Enhanced Block-Pulse Functions

¹

Mathematics Department, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia

²

Mathematics & Computer Science Department, Faculty of Science, Menoufia University, Shebin El-Kom 6131567, Egypt

³

Faculty of Computers and Information Technology, Innovation University, 10th of Ramadan City 7055501, Egypt

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(17), 3240; https://doi.org/10.3390/math14173240

Submission received: 30 July 2026 / Revised: 23 August 2026 / Accepted: 31 August 2026 / Published: 7 September 2026

(This article belongs to the Section C: Mathematical Analysis)

Download Versions Notes

Abstract

This paper proposes a novel mixed numerical scheme for approximating linear Fredholm integral equation systems (LFISs) using a combination of Bernoulli polynomials (BPs) and enhanced block-pulse functions

(E B P F s)

. This suggested representation makes use of both the local support nature and computation efficiency of the

(E B P F s)

as well as the high-order approximating nature of BPs. Using the operational matrices, the system of coupled integrals can be transformed into a finite-dimensional algebraic system (AS) of expansion coefficients. A theoretical analysis is established to investigate the solvability, convergence, stability, and approximation error of the resulting scheme. In addition, the effects of polynomial degree and partition refinement on the numerical accuracy are examined. Several test problems are considered, and the obtained results demonstrate that the proposed

B E B P F

approach provides highly accurate approximations while requiring relatively small basis dimensions. Comparisons with previously reported numerical techniques further illustrate their computational effectiveness and accuracy.

Keywords: Bernoulli polynomials; enhanced block-pulse functions; Fredholm integral systems; hybrid approximation; operational matrices; convergence analysis; numerical solution

Share and Cite

MDPI and ACS Style

Alqarni, M.Z.; Ramadan, M.A.; Elaaser, E.G.; Osheba, H.S. A High-Accuracy Hybrid Method for Linear Fredholm Integral Systems Using Bernoulli Polynomials Coupled with Enhanced Block-Pulse Functions. Mathematics 2026, 14, 3240. https://doi.org/10.3390/math14173240

AMA Style

Alqarni MZ, Ramadan MA, Elaaser EG, Osheba HS. A High-Accuracy Hybrid Method for Linear Fredholm Integral Systems Using Bernoulli Polynomials Coupled with Enhanced Block-Pulse Functions. Mathematics. 2026; 14(17):3240. https://doi.org/10.3390/math14173240

Chicago/Turabian Style

Alqarni, Mohammed Z., Mohamed A. Ramadan, Esraa G. Elaaser, and Heba S. Osheba. 2026. "A High-Accuracy Hybrid Method for Linear Fredholm Integral Systems Using Bernoulli Polynomials Coupled with Enhanced Block-Pulse Functions" Mathematics 14, no. 17: 3240. https://doi.org/10.3390/math14173240

APA Style

Alqarni, M. Z., Ramadan, M. A., Elaaser, E. G., & Osheba, H. S. (2026). A High-Accuracy Hybrid Method for Linear Fredholm Integral Systems Using Bernoulli Polynomials Coupled with Enhanced Block-Pulse Functions. Mathematics, 14(17), 3240. https://doi.org/10.3390/math14173240

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

A High-Accuracy Hybrid Method for Linear Fredholm Integral Systems Using Bernoulli Polynomials Coupled with Enhanced Block-Pulse Functions

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI