NUMERICAL SOLUTIONS FOR PHASE-LAGGED VOLTERRA-FREDHOLM INTEGRAL EQUATIONS VIA SHIFTED LEGENDRE POLYNOMIALS

Mohamed E Nasr; Sahar M. Abusalim; Mohamed A. Abdel-Aty

doi:10.11948/20250197

2026 Volume 16 Issue 6

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Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(6): 3003-3028

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Mohamed E Nasr, Sahar M. Abusalim, Mohamed A. Abdel-Aty. NUMERICAL SOLUTIONS FOR PHASE-LAGGED VOLTERRA-FREDHOLM INTEGRAL EQUATIONS VIA SHIFTED LEGENDRE POLYNOMIALS[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3003-3028. doi: 10.11948/20250197

Citation:

Mohamed E Nasr, Sahar M. Abusalim, Mohamed A. Abdel-Aty. NUMERICAL SOLUTIONS FOR PHASE-LAGGED VOLTERRA-FREDHOLM INTEGRAL EQUATIONS VIA SHIFTED LEGENDRE POLYNOMIALS[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3003-3028. doi: 10.11948/20250197

NUMERICAL SOLUTIONS FOR PHASE-LAGGED VOLTERRA-FREDHOLM INTEGRAL EQUATIONS VIA SHIFTED LEGENDRE POLYNOMIALS

1.
Department of Mathematics, College of Science, Jouf University, Sakaka 72388, Saudi Arabia
2.
Department of Mathematics and Computer Science, Faculty of Science, Benha University, Benha 13518, Egypt

Author Bio: Email: sabosalem@ju.edu.sa(S. M. Abusalim); Email: mohammed.abdallah@fsc.bu.edu.eg(M. Abdel-Aty)
Corresponding author: Email: menasr@ju.edu.sa(M. E. Nasr)

Abstract

Abstract

In this paper, a numerical method based on shifted Legendre polynomials combined with a least-squares formulation is developed for solving a class of phase-lagged Volterra–Fredholm integral equations of the second kind. Phase-lag effects arise naturally in many physical and engineering models involving delayed responses, leading to increased analytical and computational complexity. By employing shifted Legendre polynomials as basis functions, the proposed approach transforms the original integral equation into a system of algebraic equations that can be solved efficiently. Sufficient conditions for the existence and uniqueness of the solution are established using the Banach fixed-point theorem, and a convergence analysis of the proposed method is provided. Several numerical examples are presented to demonstrate the accuracy, stability, and efficiency of the method, including comparisons with existing approaches. The results confirm that the proposed scheme offers an effective and reliable tool for the numerical solution of phase-lagged Volterra–Fredholm integral equations.
- Phase-lag /
- Volterra-Fredholm integral equations /
- Banach's ﬁxed point /
- shifted Legendre polynomials /
- least squares method
MSC: 41A10, 65D15, 65F05

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References

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