10.57647/ijm2c.2026.160106

An efficient numerical method for solving first order pantograph equations via shifted Muntz orthogonal functions

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  • Hakimeh Kasmaei Najaf Abadi1,
  • Majid Tavassoli Kajani*1,
  • Masoud Allame1,
  1. Department of Mathematics, Isf. C., Islamic Azad University, Isfahan, Iran

Received: 12-07-2025

Revised: 27-09-2025

Accepted: 01-10-2025

Published in Issue 31-03-2026

Published Online: 02-10-2025

Copyright (c) 2025 Hakimeh Kasmaei Najaf Abadi, Majid Tavassoli Kajani, Masoud Allame (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Kasmaei Najaf Abadi, H., Tavassoli Kajani, M., & Allame, M. (2026). An efficient numerical method for solving first order pantograph equations via shifted Muntz orthogonal functions. International Journal of Mathematical Modelling & Computations. https://doi.org/10.57647/ijm2c.2026.160106
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Abstract

Pantograph equations are considered as a special type of delay differential equations with proportional delay and have numerous applications. This paper introduces a collocation method for solving first-order pantograph equations using shifted Muntz (SM) orthogonal functions. Unlike classical polynomials or Muntz-Legendre bases (e.g., [2]), SM functions incorporate logarithmic terms and exhibit real, simple roots in (0,1). Leveraging these roots as collocation points within a domain decomposition framework, we achieve high-precision solutions-particularly advantageous for inherently non-polynomial pantograph solutions. We derive rigorous error estimates, establish method stability, and demonstrate significant accuracy gains over existing techniques. Numerical experiments confirm the method’s efficacy, underscoring the superior approximation capability of SM functions for pantograph-type problems. All computations in this study were performed using Maple 2021 software. The codes were executed on a PC equipped with an Intel® CoreTM i5-10210U processor (1.60 GHz ) and 8 GB DDR4 RAM running Windows 10. 

Keywords

  • Pantograph equation,
  • Shifted Muntz orthogonal functions,
  • Collocation,
  • Stability
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