Symmetric integrators based on continuous-stage Runge–Kutta–Nyström methods for reversible systems

Abstract

In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nyström methods. The construction of such methods heavily relies on the Legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions. New families of symmetric integrators as illustrative examples are presented. For comparing the numerical behaviors of the presented methods, some numerical experiments are also reported.

Delay dynamic double integral inequalities on time scales with applications

“…Due to its vast contributions in different branches of mathematics, it attracts the researchers and mathematicians to work on it. The role of inequalities cannot be forgot because they have huge contributions in the theory of differential equations , bivariate means [23][24][25][26][27][28][29][30][31], calculation and optimization [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], special functions [50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69], probability and statistics [70][71]…”

On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

“…Khataybeh et al [3] introduced operational matrices of Bernstein polynomials method for solving a class of third-order ODEs directly. Tang and Zhang [4] constructed continuous-stage Runge-Kutta-Nyström methods with Legendre expansion technique in conjunction with the symmetric conditions for solving second-order ordinary differential equations. On the other hand, certain two derivative Runge-Kutta-Nyström methods with inclusion of third derivative have been presented to solve second order ODEs (see Fang et al [5], Chen et al [6], Ehigie et al [7], and Mohamed et al [8]).…”

“…Problem 3 Consider the nonlinear homogeneous problem u = −6u 4 , u(0) = 1, u (0) = −1, u (0) = 2, x ∈ [0, 5] whose analytic solution is u (x) = 1 1+x .…”

Using of PQWs for solving NFID in the complex plane

“…where ω(x) is an unknown complex function to be found, μ(x) : [0, T] → C and R(x, t, ω(t)) : [0, T] 2 × C → C are continuous and Lipschitzian periodic functions such that as Maxwell's equations, biological, radiative energy, engineering problems, potential theory, and transfers problems of oscillations that can be formulated by this equation and fractional integro-differential equations; see [5][6][7]. Some numerical algorithms that discuss the approximation of the solution of IDE can be listed such as the nonsmooth initial data arising method [8], Haar and RH methods [9][10][11], cubic B-spline finite element method [12], Runge-Kutta-Nystrom methods [13,14], and high-rank constant terms [15]. Furthermore, in [16,17], by using a system of Cauchy type and numerical method with graded meshes, singular integral equations were solved.…”