  Citation: Zaman, S.; Khan, L.U.; Hussain, I.; Mihet-Popa, L. Fast Computation of Highly Oscillatory ODE Problems: Applications in High-Frequency Communication Circuits. Symmetry 2022, 14, 115. https://doi.org/10.3390/sym14010115 Academic Editors: Lorentz JÄNTSCHI and Danny Arrigo Received: 17 October 2021 Accepted: 4 January 2022 Published: 9 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/). symmetry S S Article Fast Computation of Highly Oscillatory ODE Problems: Applications in High-Frequency Communication Circuits Sakhi Zaman 1 , Latif Ullah Khan 1 , Irshad Hussain 2, * and Lucian Mihet-Popa 3, * 1 Faculty of Architecture, Allied Sciences and Humanities, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan; sakhi.zaman@uetpeshawar.edu.pk (S.Z.); latif.shah@uetpeshawar.edu.pk (L.U.K.) 2 Faculty of Electrical and Computer Engineering, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan 3 Faculty of Information Technology, Engineering and Economics, Oestfold University College, 1757 Halden, Norway * Correspondence: ee.irshad@gmail.com (I.H.); lucian.mihet@hiof.no (L.M.-P.) Abstract: The paper demonstrates symmetric integral operator and interpolation based numerical approximations for linear and nonlinear ordinary differential equations (ODEs) with oscillatory factor x ′ = ψ(x)+ χ ω (t), where the function χ ω (t) is an oscillatory forcing term. These equations appear in a variety of computational problems occurring in Fourier analysis, computational harmonic analysis, fluid dynamics, electromagnetics, and quantum mechanics. Classical methods such as Runge–Kutta methods etc. fail to approximate the oscillatory ODEs due the existence of oscillatory term χ ω (t). Two types of methods are presented to approximate highly oscillatory ODEs. The first method uses radial basis function interpolation, and then quadrature rules are used to evaluate the integral part of the solution equation. The second approach is more generic and can approximate highly oscillatory and nonoscillatory initial value problems. Accordingly, the first-order initial value problem with oscillatory forcing term is transformed into highly oscillatory integral equation. The transformed symmetric oscillatory integral equation is then evaluated numerically by the Levin collocation method. Finally, the nonlinear form of the initial value problems with an oscillatory forcing term is converted into a linear form using Bernoulli’s transformation. The resulting linear oscillatory problem is then computed by the Levin method. Results of the proposed methods are more reliable and accurate than some state-of-the-art methods such as asymptotic method, etc. The improved results are shown in the numerical section. Keywords: symmetric integral operator; radial basis functions; Levin collocation quadrature; Bernoulli’s transformation; high frequencies; communication systems 1. Introduction Accurate computation of the initial value problems with oscillatory forcing term (IVPs) is one of the ambitious task in scientific computing. In this work, we focused on a particular case of the IVPs involving an ODE system x ′ (t)= Bx(t)+ h ω (t)f(x(t)), t ≥ 0, x(0)= x 0 ∈ R n , (1) where B is a square matrix of order n, f(x(t)) is a n-vector of functions, and h ω (t) is a rapidly oscillating scaler function with a frequency parameter ω ≫ 1. Particularly, we assume h ω (t)= e iωg(t) . Initial value problem (1) is the simplest model of the more complicated problems of electronic engineering. Generally, nonlinear ODEs and differential algebraic equations (DAEs) with oscillatory forcing term are frequently appeared in this field. The analytic solution of the ODEs (1) in terms of symmetric integral operator and is given as x = e At x 0 + L( f ), t ≥ 0, (2) Symmetry 2022, 14, 115. https://doi.org/10.3390/sym14010115 https://www.mdpi.com/journal/symmetry