Analytical and numerical study on a variable scalar equation

Volume 41, Issue 2, pp 183--194 https://dx.doi.org/10.22436/jmcs.041.02.03

Publication Date: October 03, 2025 Submission Date: May 17, 2025 Revision Date: June 12, 2025 Accteptance Date: July 24, 2025

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Authors

I. F. S. Alsahli - Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia. L. F. Seddek - Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia. A. Ebaid - Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia. E. R. El-Zahar - Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia.

Abstract

Basically, scalar equations have potential applications in various fields such as the transmission of nerve impulses between neurons through myelin substance and other disciplines. A particular model is well-known as the pantograph equation. The standard version of this scalar equation has been extensively investigated via different analytical and numerical techniques. This paper considers a variable version of the pantograph equation. Usually, constructing an exact or a closed form solution for a variable scalar equation is a challenge. However, this work proposes a developed hybrid approach to overcome such a difficulty. The solution of the current variable version is analytically obtained in different closed forms with addressing the convergence criteria. Under some conditions, such closed forms are successfully converted to different exact ones. Additionally, accurate approximations are provided and examined. Several comparisons with the available exact solutions are conducted as a validation of our approximations. Besides, the accuracy of our approximations is checked for some classes which have no exact solutions. Probably, the results demonstrate the elegance of the proposed approach to deal with a variable version of the Pantograph model.

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Keywords

• Pantograph
• delay
• scalar equation
• variable
• exact solution
• series solution

MSC

•  34K06
•  65L03

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