Vieta–Lucas wavelets method for fractional linear and nonlinear delay differential equations

Abstract

PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

Improving the accuracy for amplitude and frequency of analytical equation of single-ended ring oscillators based on circuit and transistor parameters

“…and the function f(•) depends on the operating region of the transistor and t is delay between consecutive inverting blocks. Equation ( 2) is a nonlinear delay differential equation that generally does not have an explicit and closed-form answer and is often analyzed by numerical methods (Ghaneai et al, 2012;Fridman, 2014;Idrees and Saeed, 2022). Therefore, obtaining a closed-form relationship for the ring oscillator parameters such as amplitude and frequency based on the circuit parameters has not been possible accurately and has always been a problem and challenge in the design of ring oscillators.…”