Bernoulli Polynomials Method for Solving Integral Equations with Singular Kernel

Authors

  • Muna M. Mustafa Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq. https://orcid.org/0000-0001-8620-4976
  • Heba A. Abd-Alrazak Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq.

DOI:

https://doi.org/10.21123/bsj.2024.9712

Keywords:

Abel’s integral equation, Bernoulli polynomials, Integral equation, Singular kernel, Weakly singular kernel

Abstract

There is always an interest in an effective technique to generate a numerical solution of integral equations with singular or weakly singular kernels more precisely because numerical methods have limitations. In this study, integral equations with singular or weakly singular kernels are solved using the Bernoulli polynomial approach. The primary goal of this study is to provide an approximate solution to such problems in the form of a polynomial in a series of straightforward steps. Also, assuming that the denominator of the kernel will never be zero or have an imaginary value due to the selected nodes of the unique two kernel variables. With the 4th and 8th-degree Bernoulli polynomials as an example, the current approach provides a solution very close to the exact solution in the test examples. While. The very modest magnitude of the errors in the test examples proves the effectiveness of the current strategy. Also, the ease with which a computer program can be implemented makes this technique very efficient.  Another objective is to determine the efficiency of the proposed method by comparing it with various approaches. The approximated solution for integral equations with singular or weakly singular kernels is demonstrated to significantly converge to the precise ones by using the Bernoulli polynomial and is superior to those found by other stated approaches. This guarantees the originality and high accuracy of the suggested method. Also, the convergent of the proposed method is discussed. The programs are implemented using the MATLAB program R2018a.

References

Saeed RK, Hassan JS. Solving Singular Integral Equations by Using Collocation Method. Math Sci Lett. 2014; 3(3): 185–187. http://dx.doi.org/10.12785/msl/030308

Behzadi R, Tohidi E, Toutounian F. Numerical Solution of Weakly Singular Fredholm Integral Equations via Generalization of the Euler–Maclaurin Summation Formula. J Taibah Univ Sci. 2014; 8(2): 199–205. http://dx.doi.org/10.1016/j.jtusci.2013.11.001

Paul S, Panja MM, Mandal BN. Approximate Solution of First Kind Singular Integral Equation with Generalized Kernel Using Legendre Multiwavelets. Comput Appl Math. 2019; 38(1). https://doi.org/10.1007/s40314-019-0770-3

Ali MR, Mousa MM, Ma W-X. Solution of Nonlinear Volterra Integral Equations with Weakly Singular Kernel by Using the HOBW Method. Adv Math Phys. 2019; 2019: 1–10. https://doi.org/10.1155/2019/1705651.

Wang T, Qin M, Lian H. The Asymptotic Approximations to Linear Weakly Singular Volterra Integral Equations via Laplace Transform. Numer Algor. 2020; 85: 683-711. https://doi.org/10.1007/s11075-019-00832-5

Abdullah JT. Approximate Numerical Solutions for Linear Volterra Integral Equations Using Touchard Polynomials. Baghdad Sci J. 2020; 17(4): 1241–9. http://dx.doi.org/10.21123/bsj.2020.17.4.1241

Qiu W, Xu D, Guo J. A Formally Second-Order Backward Differentiation Formula Sinc-Collocation Method for the Volterra Integro-Differential Equation with a Weakly Singular Kernel Based on the Double Exponential Transformation. Numer Methods Partial Differ Equ. 2020; 38(4): 830–47. https://doi.org/10.1002/num.22703.

Shoukralla ES, Ahmed BM, Sayed M, Saeed A. Interpolation Method for Solving Volterra Integral Equations with Weakly Singular Kernel Using an Advanced Barycentric Lagrange Formula. Ain Shams Eng J. 2022; 13(5): 1-7. https://doi.org/10.1016/j.asej.2022.101743

Singh UP. Applications of Orthonormal Bernoulli Polynomials for Approximate Solution of Some Volterra Integral Equations. Albanian J Math. 2016; 10(1): 47–80. https://doi.org/10.48550/arXiv.2007.10814

Mirzaee F, Samadyar N. Application of Bernoulli Wavelet Method for Estimating a Solution of Linear Stochastic Itô-Volterra Integral Equations. Multidiscip Model Mater Struct. 2019; 15(3): 575–598.

Shiralashetti SC, Kumbinarasaiah S, Mundewadi RA. Bernoulli Wavelet Based Numerical Method for the Solution of Abel’s Integral Equations. Int J Eng Sci Math. 2019; 6(8): 63–70.

Samadyar N, Mirzaee F. Numerical Scheme for Solving Singular Fractional Partial Integro-Differential Equation via Orthonormal Bernoulli Polynomials. Int J Numer Model El 2019; 32(6): 1–18. https://doi.org/10.1002/jnm.2652

Singh M, Singhal S, Handa N. Exact and Numerical Solution of Abel Integral Equations by Orthonormal Bernoulli Polynomials. Int J Appl Comput Math. 2019; 5(6). https://doi.org/10.1007/s40819-019-0734-8

Nemati S, Torres DFM. Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems. Axioms. 2020; 9(4): 1–18. https://doi.org/10.3390/axioms9040114

Mirzaee F, Samadyar N. Explicit Representation of Orthonormal Bernoulli Polynomials and its Application for Solving Volterra–Fredholm–Hammerstein Integral Equations. SeMA J. 2020; 77(1): 81–96. https://doi.org/10.1007/s40324-019-00203-z

Samadyar N, Mirzaee F. Orthonormal Bernoulli Polynomials Collocation Approach for Solving Stochastic Itô ‐ Volterra Integral Equations of Abel Type. Int J Numer Model. 2019;33(1):1–14. https://doi.org/ 10.1002/jnm.2688

Sahlan MN, Afshari H, Alzabut J, Alobaidi G. Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions. Fractal Fract. 2021; 5(4): 1–20. https://doi.org/10.3390/fractalfract5040212

Nemati S, Lima PM. Numerical Solution of Variable-Order Fractional Differential Equations Using Bernoulli Polynomials. Fractal Fract. 2021; 5(4): 1–15. https://doi.org/10.3390/fractalfract5040219

Abdullah JT, Ali HS, Ali WS. Numerical Solutions of Linear Abel Integral Equations Via Boubaker Polynomials Method. Baghdad Sci J. 2023; OnlineFirs. https://doi.org/10.21123/bsj.2023.8167

Kong D, Xiang S, Wu H. An Efficient Numerical Method for Volterra Integral Equation of the Second Kind with a Weakly Singular Kernel. J Comput Appl Math. 2023; 427: 115101. https://doi.org/10.1016/j.cam.2023.115101

Khan S, Nahid T. Finding Non-Linear Differential Equations and Certain Identities for the Bernoulli–Euler and Bernoulli–Genocchi Numbers. SN Appl Sci. 2019; 1(3): 1–9. https://doi.org/10.1007/s42452-019-0222-0

Komatsu T, De J. Pita Ruiz VC. Several Explicit Formulae for Bernoulli Polynomials. Math Commun. 2016; 21(1): 127–140.

Momani S, Abu Arqub O, Maayah B. Piecewise Optimal Fractional Reproducing Kernel Solution and Convergence Analysis for the Atangana-Baleanu-Caputo Model of the Lienard’s Equation. Fractals. 2020; 28(8): 1–13. https://doi.org/10.1142/S0218348X20400071

Diogo T, Ford NJ, Lima P, Valtchev S. Numerical Methods for a Volterra Integral Equation with Non-Smooth Solutions. J Comput Appl Math. 2006; 189(1–2): 412–423. https://doi.org/10.1016/j.cam.2005.10.019

Zarei E, Noeiaghdam S. Solving Generalized Abel’s Integral Equations of the First and Second Kinds via Taylor-Collocation Method. arXiv:180408571 [mathNA]. 2018;1–10. https://doi.org/10.48550/arXiv.1804.08571

Downloads

Issue

Section

article

How to Cite

1.
Bernoulli Polynomials Method for Solving Integral Equations with Singular Kernel. Baghdad Sci.J [Internet]. [cited 2024 Jun. 14];21(12). Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/9712