Numerical Solutions of Third Kind Volterra Integral Equations Using Simpson’s Rule
DOI:
https://doi.org/10.14419/gq7zz405Published
23-11-2025Keywords:
Volterra Integral equation; Simpson's 1/3 rd Rule; Simpson's 3/8 th RuleAbstract
Volterra integral equations appear when we convert an initial value problem to an integral equation. The solution of the Volterra integral equation is much easier than the original initial value problem. Many problems of thermodynamics, biology, chemistry, medical sciences, physics, heat flow, neutron diffusion problem, electric circuit problem, transform problem mechanics, and engineering can be represented mathematically in terms of Volterra integral equations of the first and second kind. Numerical Method for solving Third Kind Volterra Integral Equations (VIEs) using Simpson's Rule. By approximating the unknown function using Lagrange Polynomial interpolation and applying Simpson's rule for numerical integration, we obtain a system of linear algebraic equations. The solution of this system provides an approximation to the original integral equation.
References
Abdul Majid Wazwaz, “Linear and Non-linear integral equations Method and Applications,” Higher Education Press, Beijing, and Springer – Ver-lag Berlin Heidelberg 2011.
Adem Kılıc¸man, L. Kargaran Dehkordi, M. Tavassoli Kajani “Numerical Solution of Nonlinear Volterra Integral Equations System Using Simp-son’s 3/8 Rule” Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2012, Article ID 603463, 16 pages https://doi.org/10.1155/2012/603463.
D.C. Sharma and M.C. Goyal, “Integral Equations” PHI Learning Private Limited, Delhi 110092, 2017.
Danna Wu, Weishan Zheng, and Yanfeng Chen. “Error Estimates of a Symmetric Spectral Method for a Linear Volterra Integral Equation” Sym-metry 2023, 15, 60. https://doi.org/10.3390/sym15010060.
Dr. Subhamay Dutta, “The Method of Successive Approximation (Neumann’s Series) of Volterra Integral Equation of the second kind, “Pure and Applied Mathematical Journal Volume 5 Issue 6 Pages 211-219. https://doi.org/10.11648/j.pamj.20160506.16.
G.A. Afrouzi, D. D. Ganji, H. Hosseinzadeh, R.A. Talarposhti “Fourth order Volterra integro-differential equations using modified Homotopy-perturbation method” The Journal of Mathematics and Computer Science Volume 3 No 2 (2011) 179-191. https://doi.org/10.22436/jmcs.03.02.10.
H. Tajadodi “Spectral Method for Third-Kind Volterra Integral Equation” Journal of Mathematical Extension Vol. 15, SI-NTFCA, (2021) (29)1-17.
Imtiyaz Ahmad Bhat and Lakshmi Narayan Mishra “Numerical Solutions of Volterra Integral Equations of Third Kind and Its Convergence Analy-sis” Symmetry 2022, 14, 2600. https://doi.org/10.3390/sym14122600.
I.G. Burova, A.G. Doronina, D.E. Zhilin “Splines of the Fourth Order Approximation and the Volterra Integral Equations” Wseas Transactions on Mathematics, Volume 20, 2021 https://doi.org/10.37394/23206.2021.20.2.
M. A. Elsayed “Some Different Methods via the Solution of Volterra Integral Equation” Applied Mathematics & Information Sciences. 16, No. 6, 973-981 (2022) https://doi.org/10.18576/amis/160614.
M. A. Abdou, A. A. Abbas “New technique of two numerical methods for solving integral equation of the second kind” IOSR Journal of Engineer-ing (IOSRJEN) Vol. 08, Issue 9 (September 2018), Pages 34-40.
M.Rahman, “Integral equations and their applications,” WIT Press.
Mustapha Yahaya_1 And Sirajo Lawan Bichi “Application of Simpson’s method for solving singular Volterra integral equation” Malaya Journal of Matematik 9(04) (2021), 206–215. https://doi.org/10.26637/mjm904/004.
P. Jami1, E. Hashemizadeh “A Method for Numerical Solution of Third-Kind Volterra Integral Equations Using Krall-Laguerre Polynomials” De-partment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
Poonam Jagtap, Avinash Khambayat “Coparision of Volterra Integral Equation by Successive Approximation and Adomain Decomposition Meth-ods” Journal of Emerging Technologies and Innovative Research (JETIR) Volume 11, Issue 8 August 2024.
Sonia Seyed Allaei, Zhan-Wen Yang and Hermann Brunner “Existence, Uniqueness and Regularity of Solutions to A Class of Third-Kind Volterra Integral Equations” Journal of Integral Equations and Applications Volume 27, Number 3, Fall 2015. https://doi.org/10.1216/JIE-2015-27-3-325.
Sudhanshu Aggarwal, Aakansha Vyas “Laplace Transform for the solution of Non-Linear Volterra Integral equation of second kind” Journal of Ad-vanced Research in Applied Mathematics and Statistics Volume 8 Issue 3& 4 Page no 18-25 https://doi.org/10.24321/2455.7021.202304.
Sahar Muhar Jaabar, Ahmed Hadi Hussain “A Move Recent Review of the Integral Equations and their applications” Journal of Physics: conference seric 1818 (2021) 012170. https://doi.org/10.1088/1742-6596/1818/1/012170.
Y. Talaeia, P. M. Limab “An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions” July 18, 2022. https://doi.org/10.1007/s40314-023-02333-7.
Zana Mohammed Hassan, Sadeq Taha Abdulazeez, Sarbast Kamal Rasheed “Investigating Solutions of Volterra Integral Equations Using the Suc-cessive Approximations” Mathematical Statistician and Engineering Applications Volume 72 Issue 2 2023 Pages 75-82.