Second‐Order Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach

Abstract

In this article, an efficient numerical approach for the solution of second-order delay differential equations to deal with the experimentation of the Internet of Industrial Things (IIoT) is presented. With the help of the Haar wavelet technique, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for various collocation points. The results show that the Haar wavelet method is an effective method for solving delay differential equations of second order. The convergence rate is also measured for various collocation points, which is almost equal to 2.

On the numerical solution of second order delay differential equations via a novel approach

“…A time-invariant system is one that runs independent of time. The ODEs with constant coefficients can defined such systems [34]. Recently, there are numerous research undertaken in engineering and other science subjects with a focus on the development of mathematical models using DDEs.…”

Application of Taylor Wavelets in Computation of Solution of Delay Differential Equations

“…Many authors have tried to find the solutions of delay differential equations using various techniques such as a collocation method based on Bernoulli operational matrix [10], Taylor polynomials [11,12,13,14], Euler bases together with operational matrices [15], perturbation-iteration algorithms [16], Laguerre series [17], Walsh stretch matrices and functional differential equation [18], Bernstein polynomials [19], Fourier operational matrices of differentiation and transmission [21] polynomial interpolation [20], Spline functions approximation [22], Adomian decomposition method [23,24], Hermite interpolation [25], collocation method [26], Chebyshev polynomials [27], Legendre polynomial approximation [28], differential transform method [29], block-pulse functions and Bernstein polynomials [30], Variational iteration method(VIM) [31,32], Jacobi rational-Gauss collocation (JRC) [33], successive interpolations [34], an efficient transferred Legendre pseudospectral method [35], Muntz-Legendre basis and operational matrices of fractional derivatives [36] etc.. Methods based on the wavelets are more attractive and considerable. Some of wavelets techniques are applied in order to solve the equation (1) namely, Chebyshev wavelets [37,38,39], Hermite wavelets [40,41], Legendre wavelets method [42,43], Haar wavelets method [44,46]] etc.…”