A new numerical method to solve pantograph delay differential equations with convergence analysis

Abstract

The main aim presented in this article is to provide an efficient transferred Legendre pseudospectral method for solving pantograph delay differential equations. At the first step, we transform the problem into a continuous-time optimization problem and then utilize a transferred Legendre pseudospectral method to discretize the problem. By solving this discrete problem, we can attain the pointwise and continuous estimated solutions for the major pantograph delay differential equation. The convergence of method has been considered. Also, numerical experiments are described to show the performance and precision of the presented technique. Moreover, the obtained results are compared with those from other techniques.

Some Properties of Numerical Solutions for Semilinear Stochastic Delay Differential Equations Driven by G-Brownian Motion

“…Recently, some results related to G-framework have been obtained; however, it is a pity that there are few numerical conclusions. In the future, we will further discover more efficient numerical methods, such as the transferred Legendre pseudospectral method in [40], to solve semilinear stochastic delay differential equations with time-variable delay driven by G-Brownian motion. Influenced by experience gained from solving stochastic fractional differential equations driven by fractional Brownian motion, we will further study stochastic fractional differential equations driven by G-Brownian motion with "dy (t)" as fractional order in future research [41].…”

An Analysis of the Theta-Method for Pantograph-Type Delay Differential Equations

“…A pseudospectral method based on the Legendre principle is examined in Ref. [12], and a modified procedure based on the residual power series method is explored in [13]. Other spectral methods and recent research about numerical treatments of pantograph equations exist in Refs.…”

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

“…In the present years, several numerical methods for approximating the solution of delay differential equations have been established. 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,…”