New Runge–Kutta type symmetric two step finite difference pair with improved properties for second order initial and/or boundary value problems

A new three-stages symmetric two-step method with improved properties is developed in this paper and for the first time in the literature. The properties of the new proposed algorithm are:is a symmetric finite difference pair,is a scheme of two-step,is an algorithm of three-stages—i.e. hybrid or Runge–Kutta type,is of tenth-algebraic order,it has vanished the phase-lag and its first, second and third derivatives,it has improved stability properties for the general problems,it is a P-stable method since it has an interval of periodicity equal to 0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( 0, \infty \right) $$\end{document}. The new proposed scheme is constructed based on the following layers:An approximation denoted on the first layer on the point xn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n-1}$$\end{document},An approximation denoted on the second layer on the point xn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n}$$\end{document} and finally,An approximation denoted on the third (final) layer on the point xn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1}$$\end{document}, For the new proposed method we give a full theoretical analysis which consists of: (1) local truncation error analysis, (2) comparative local truncation error analysis, (3) stability analysis and (4) interval of periodicity analysis. The efficiency of the new proposed algorithm is tested on the approximate solution of systems of coupled differential equations arising from the Schrödinger equation.