University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 16 3 2025 09 01 Fractional-Order‎ ‎Pell Hybrid‎ ‎Functions for Solving Variable-Order Fractional Differential Equations and their Application in Chemistry 223 244 115048 10.22052/ijmc.2024.255544.1911 EN Naghmeh Shiri Department of Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎Alzahra University‎, ‎Tehran‎, ‎Iran Yadollah Ordokhani Department of Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎Alzahra University‎, ‎Tehran‎, ‎Iran Sedigheh Sabermahani Department of Mathematics Education, Farhangian University, Tehran, P.O. Box 14665-889, Iran Journal Article 2024 09 29 ‎Herein‎, ‎we introduce a novel computational approach to find a solution of variable-order fractional differential equations (VO-FDE)‎. ‎The desired method is proposed by combining fractional-order Pell functions‎, ‎the Ritz and collocation methods‎. ‎First‎, ‎we define a new set of hybrid functions named fractional-order Pell hybrid functions‎. ‎Next‎, ‎to approximate the solution of VO-FDEs‎, ‎we obtain an extra pseudo-operational matrix of the Caputo variable-order derivative‎. ‎Then‎, ‎by using the Ritz method‎, ‎the operational matrix approach‎, ‎and the collocation method‎, ‎the problem is transformed to a system of algebraic equations‎, ‎which is solved by Newton's iterative method‎. ‎The error estimation of the proposed method is also examined‎. ‎Finally‎, ‎some examples (especially in chemistry) are presented to illustrate the effectiveness of this method‎. ‎ Hybrid functions Variable-order fractional differential equation Ritz method https://ijmc.kashanu.ac.ir/article_115048_665885af5d7a886ec3149eb852ea9160.pdf