Variable Step Hybrid Block Method for the Approximation of Kepler Problem

Abstract

In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the step size ratio r is left the same, halved, or doubled in order to optimize the total number of steps, minimize the number of formulae stored in the code, and ensure that the method is zero-stable. The method is formulated by integrating the Lagrange polynomial with limits of integration selected at special points. The article further analyzed the stability, order, consistency, and convergence properties of the VSHBM. The stability regions of the VSHBM at different values of the step size ratios were also plotted and plots showed that the method is fit for solving the Kepler problem. The results generated were then compared with some existing methods, including the MATLAB inbuilt stiff solver (ode 15 s), with respect to total number of failure steps, total number of steps, total function calls, maximum error, and computation time.

Second Derivative Block Hybrid Methods for the Numerical Integration of Differential Systems

“…The values r and s denote output and stage values, respectively. Applying Equati (22) to the linear test equation…”

“…Chan and Tsai [16] considered explicit two-derivative RKMs, which are cheaper to calculate with fewer function evaluations than the standard RKMs. Recently, many authors have worked on methods to obtain better approximate solutions to differential equations or on stability properties to improve the accuracy and efficiency of solution of differential equations (see, for example, [17][18][19][20][21][22][23][24]).…”

Generalized Class of Finite Population Variance in the Presence of Random Nonresponse Using Simulation Approach

“…1 241.0492 T (1) 181.4473 T (4) 1 267.6401 T (4) 2 206.5321 T (4) 3 181.4576 T (5) 1 267.6965 T (5) 2 206.5532 T (5) 3 181.4663 T (6) 1 267.1691 T (6) 2 206.3557 T (6) 3 181.3844 T (7) 1 267.6403 T (7) 2 206.5322 T (7) 3 181.4567 T (8) 1 267.4689 T (8) 2 206.468 T (8) 3 181.431 T (9) 1 267.7158 T (9) 2 206.5604 T (9) 3 181.4693 T (10) 1 266.8310 T (10) 2 206.2289 T (10) 3 181.3318 171.5264 T (5) 1 259.3661 T (5) 2 210.6753 T (5) 3 171.5099 T (6) 1…”

“…Many other researchers have also signifcantly contributed to estimating the fnite population variance. Sunday et al [6] adopted a variable step hybrid block method for the approximation of Kepler Problem by integrating the Lagrange polynomial with limits of integration selected at special points. Juraev et al [7,8] used regularization formula and matrix factorization for explicit form of the approximate solutions of the Cauchy Problem.…”

An Accuracy-preserving Block Hybrid Algorithm for the Integration of Second-order Physical Systems with Oscillatory Solutions

“…Furthermore, Figures 4, 5 and 6 respectively depict the trajectories in the phase plane flow using proposed BHA, BHA developed by [2] and the exact flow N = 160. For more details on Kepler equations, see [29][30][31][32][33][34][35].…”