OICC Press International Journal of Mathematical Modelling & Computations 2228-6233 16 1 2026 03 31 An Effcient Numerical Method for Solving First OrderPantograph Equations via Shifted Müntz Orthogonal Functions 10.57647/ijm2c.2026.160106 EN Hakimeh Kasmaei Najaf Abadi Department of Mathematics, Isf. C., Islamic Azad University, Isfahan, Iran https://orcid.org/0009-0005-6494-3993 Majid Tavassoli Kajani Department of Mathematics, Isf. C., Islamic Azad University, Isfahan, Iran https://orcid.org/0000-0003-3592-1304 Masoud Allame Department of Mathematics, Isf. C., Islamic Azad University, Isfahan, Iran Journal Article 2026 03 31 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 Müntz (SM) orthogonal functions. Unlike classical polynomial bases, SM functions incorporate logarithmic terms and exhibit real, simple roots in the interval (0,1). Leveraging these roots as collocation points within a domain decomposition framework, we achieve high-precision solutionsparticularly 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® Core™ i5-10210U processor (1.60 GHz ) and 8 GB DDR4 RAM running Windows 10.