  Citation: González-Rodelas, P.; Pasadas, M.; Kouibia, A.; Mustafa, B. Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions. Mathematics 2022, 10, 223. https://doi.org/10.3390/ math10020223 Academic Editors: Sara Remogna, Domingo Barrera, María José Ibáñez and Simeon Reich Received: 30 November 2021 Accepted: 4 January 2022 Published: 12 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions Pedro González-Rodelas † , Miguel Pasadas * ,† , Abdelouahed Kouibia † and Basim Mustafa † Department of Applied Mathematics, Granada University, 18071 Granada, Spain; prodelas@ugr.es (P.G.-R.); kouibia@ugr.es (A.K.); bmustafa@correo.ugr.es (B.M.) * Correspondence: mpasadas@ugr.es † These authors contributed equally to this work. Abstract: In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost. Keywords: Volterra integral equations system; radial basis functions; variational methods 1. Introduction A considerable large amount of research literature and books on the theory and applica- tions of Volterra’s integral equations have emerged over many decades since the apparition of Volterra’s book “Leçons sur les équations intégrales et intégro-différentielles” [1] in 1913. The applications include elasticity, plasticity, semi-conductors, scattering theory, seis- mology, heat and mass conduction or transfer, metallurgy, fluid flow dynamics, chemical reactions, population dynamics, and oscillation theory, among many others (see for exam- ple [2]). Other important references more related with the numerics of this type of equation are [3,4]. In fact, Volterra integral equations (VIEs) appear naturally when we try to transform an initial value problem into integral form, so that the solution of this integral equation is usually much easier to obtain than the original initial value problem. In the same way, some nonlinear Volterra integral equations are equivalent to an initial-value problem for a system of ordinary differential equations (ODEs). So, some authors (like for example [5]) have sought to exploit this connection for the numerical solution of the integral equations as well, since very effective ODE codes are widely available. Volterra integral equations arise in many usual applications of technology, engineering and science in general: as in population dynamics, the spread of epidemics, some Dirichlet problems in potential theory, electrostatics, mathematical modeling of radioactive equilib- rium, the particle transport problems of astrophysics and reactor theory, radiative energy and/or heat transfer problems, other general heat transfer problems, oscillation of strings and membranes, the problem of momentum representation in quantum mechanics, etc. However, many other complex problems of mathematics, chemistry, biology, astrophysics and mechanics, can be expressed in the terms of Volterra integral equations. Moreover, some practical problems, where impulses arise naturally (like in population dynamics or many biological applications) or are caused by some control system (like electric circuit problems and simulations of semiconductor devices) can be modeled by a differential equa- tion, an integral equation, an integro-differential equation, or a system of these equations all combined. Mathematics 2022, 10, 223. https://doi.org/10.3390/math10020223 https://www.mdpi.com/journal/mathematics