Theoretical and Computational Analysis of Delay Volterra Integro-Differential Equations via LaplaceTransform and Numerical Inversion

Authors

  • Kamran Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtoonkhwa, Pakistan
  • Nadeem Jan Department of Natural sciences and Humanities University of Engineering & Technology Mardan
  • Muhammad Ishfaq Khan College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, P.R, China
  • Ahmad Aloqaily Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
  • Nabil Mlaiki Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
  • Fady Hasan Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia

DOI:

https://doi.org/10.29020/nybg.ejpam.v18i3.6355

Keywords:

Delay integro-differential equation; Laplace transform; Existence; Uniqueness; Gauss-Hermite quadrature method; Weeks method

Abstract

Delay integro-differential equations (DIDEs) represent a significant class of integro-differential equations where state evolution depends on its past history. This paper presents a numerical approach for delay integro-differential equations (DIDEs), utilizing the Laplace transform (LT) and its inversion as the core methodology. The proposed technique begins by transforming the given equation into an algebraic equation in the Laplace domain using the LT. The resulting transformed equation is subsequently solved for the unknown function within the Laplace domain. Finally, two inversion methods the Gauss-Hermite quadrature method and the Weeks methods are used to invert the solution back to the time domain. Additionally, the existence and uniqueness of the solution are rigorously analyzed using the functional analysis. To demonstrate the effectiveness of the methods several examples from the literature are provided. The results obtained using the two techniques are compared and analyzed through tables and figures, highlighting their accuracy and computational efficiency.

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Published

2025-08-02

Issue

Vol. 18 No. 3: (July 2025)

Section

Mathematical Modeling and Numerical Analysis

License

Copyright (c) 2025 Kamran, Nadeem Jan, Muhammad Ishfaq Khan, Ahmad Aloqaily, Nabil Mlaiki, Fady Hasan

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How to Cite

Theoretical and Computational Analysis of Delay Volterra Integro-Differential Equations via LaplaceTransform and Numerical Inversion. (2025). European Journal of Pure and Applied Mathematics, 18(3), 6355. https://doi.org/10.29020/nybg.ejpam.v18i3.6355