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Accurate Solution for the Pantograph Delay Differential Equation via Laplace Transform

Author

Listed:
  • Reem Alrebdi

    (Department of Mathematical Sciences, Umm Al-Qura University, Makkah 24382, Saudi Arabia
    Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia)

  • Hind K. Al-Jeaid

    (Department of Mathematical Sciences, Umm Al-Qura University, Makkah 24382, Saudi Arabia)

Abstract

The Pantograph equation is a fundamental mathematical model in the field of delay differential equations. A special case of the Pantograph equation is well known as the Ambartsumian delay equation which has a particular application in Astrophysics. In this paper, the Laplace transform is successfully applied to solve the Pantograph delay equation. The solution is obtained in a closed series form in terms of exponential functions. This closed form reduces to the corresponding solution in the relevant literature for the Ambartsumian delay equation as a special case. In addition, the convergence of the obtained series is proved theoretically and validated graphically. Furthermore, the accuracy of the numerical results are estimated through several computations of the residual errors. It is shown that such residuals tend to zero, even in a huge domain. The obtained results reveal that the Laplace transform is a powerful approach to solve linear delay differential equations, including the Pantograph model.

Suggested Citation

  • Reem Alrebdi & Hind K. Al-Jeaid, 2023. "Accurate Solution for the Pantograph Delay Differential Equation via Laplace Transform," Mathematics, MDPI, vol. 11(9), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2031-:d:1132217
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    References listed on IDEAS

    as
    1. S. M. Khaled & Abdelhalim Ebaid & Fahd Al Mutairi, 2014. "The Exact Endoscopic Effect on the Peristaltic Flow of a Nanofluid," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-11, November.
    2. Huda O. Bakodah & Abdelhalim Ebaid, 2018. "Exact Solution of Ambartsumian Delay Differential Equation and Comparison with Daftardar-Gejji and Jafari Approximate Method," Mathematics, MDPI, vol. 6(12), pages 1-10, December.
    3. Aneefah H. S. Alenazy & Abdelhalim Ebaid & Ebrahem A. Algehyne & Hind K. Al-Jeaid, 2022. "Advanced Study on the Delay Differential Equation y ′( t ) = ay ( t ) + by ( ct )," Mathematics, MDPI, vol. 10(22), pages 1-13, November.
    4. Hoda S. Ali & Elham Alali & Abdelhalim Ebaid & Fahad M. Alharbi, 2019. "Analytic Solution of a Class of Singular Second-Order Boundary Value Problems with Applications," Mathematics, MDPI, vol. 7(2), pages 1-10, February.
    5. Abdulrahman B. Albidah & Nourah E. Kanaan & Abdelhalim Ebaid & Hind K. Al-Jeaid, 2023. "Exact and Numerical Analysis of the Pantograph Delay Differential Equation via the Homotopy Perturbation Method," Mathematics, MDPI, vol. 11(4), pages 1-14, February.
    6. Rajan Arora & Hariom Sharma, 2018. "Application of HAM to seventh order KdV equations," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 9(1), pages 131-138, February.
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