Research Article Local RBF-FD-Based Mesh-free Scheme for Singularly Perturbed Convection-Diffusion-Reaction Models with Variable Coefficients Ram Jiwari , 1 Sukhveer Singh, 2 and Paramjeet Singh 3 1 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India 2 Department of Mathematics, Graphic Era Hill University, Dehradun 248002, India 3 School of Mathematics, apar Institute of Engineering and Technology, Patiala 147001, India Correspondence should be addressed to Ram Jiwari; ram.jiwari@ma.iitr.ac.in Received 19 January 2022; Accepted 8 February 2022; Published 15 March 2022 Academic Editor: Antonio Di Crescenzo Copyright © 2022 Ram Jiwari et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is work analyze singularly perturbed convection-diffusion-reaction (CDR) models with two parameters and variable coef- ficients by developing a mesh-free scheme based on local radial basis function-finite difference (LRBF-FD) approximation. In the evolvement of the scheme, time derivative is discretized by forward finite difference. After that, LRBF-FD approximation is used for spatial discretization, and we obtained a system of linear equations. en, the obtained linear system is solved by LU de- composition method in MATLAB. For numerical simulation, four singularly perturbed models are pondered to check the efficiency and chastity of the proposed scheme. 1. Introduction Singularly perturbed models (SPMs) can be seen in different areas of science, medicine, and engineering. Herein, we consider the following class of parabolic SPMs with two small parameters and variable coefficients: L ε,μ w(x, t) ≡ εw xx (x, t)+ μa(x, t)w x (x, t)− b(x, t)w t (x, t)− c(x, t)w � f(x, t), (x, t) ∈Ω �(0, 1)×(0,T], (1) with initial and boundary conditions (BCs) w(x, 0)� w 0 (x), x ∈ (0, 1), w(0,t)� g 1 (t), w(1,t)� g 2 (t), t ∈ (0,T], (2) where ε and μ are small positive parameters and the variable coefficients a(x, t),b(x, t),c(x, t), and f(x, t) are suffi- ciently smooth functions on Ω and satisfies the following conditions: a(x, t) ≥ α > 0, b(x, t) ≥ β > 0, c(x, t) ≥ c > 0, ∀(x, t) ∈ Ω. (3) We also suppose that the initial and BCs have enough smoothness on zΩ and sufficient compatibility conditions on the edges of the domain Ω. ese conditions on the initial and BCs ensure that there exists a unique classical solution w(x, t) ∈ C 4,2 ( Ω) [1]. Hindawi Journal of Mathematics Volume 2022, Article ID 3119482, 11 pages https://doi.org/10.1155/2022/3119482