Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

Abstract

There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2k multi-step formulas (although we will see that this number can be reduced to k+1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

A Novel Optimized Exponentially Fitted Algorithm with Time-Efficiency

“…whose solutions in closed form given by v 1 (x) = cos (10x) + sin (x) and v 2 (x) = sin (5x) − cos (x) characterize a periodic motion with two central frequencies and a small perturbation of small frequency. The fitting frequency is selected as ω = 10 for equation (26), while h = 1 2 i , i = 1, 2, 3, 4 are selected as the integration steps. It can be seen in Table 4 and from Figure 5 that the newly proposed method shows a more efficient behavior than the methods it compared.…”

“…The methods with constant coefficients, like the majority of classical methods, are derived primarily using polynomials as basis functions, such as power series and orthogonal polynomials ( [6,16,17,26,28,31]). Many of these approaches do not perform well in the case of oscillating outcomes due to the nature of the solutions.…”

A Chebyshev polynomial based block integrator for the direct numerical solution of fourth order ordinary differential equations

“…Block linear multistep methods (BLMMs) have proven effective for solving IVPs associated with ordinary differential equations, offering stability and computational advantages. Various polynomials have been employed by different authors in BLMMs, such as Power series by Awoyemi [1], Familua & Omole [2], Ramos et al [5], Atabo & Adee [6] and Modebei et al [7], Taylor series by Adoghe & Omole [8], Lucas Polynomials by Adeniran & Longe [9], Legendre Polynomials by Nazreen & Zanariah [10], Chebyshev Polynomials by Olabode & Momoh [11] and Alabi et al [12] and Hermite polynomials by Ogunlaran & Kehinde [3]. The effectiveness of Chebyshev polynomials is widely recognized across scientific and engineering fields, including weather forecasting, surface and interface stress effects in thin films, solving integral equations and two-point boundary value problems, and so on, Khater et al [13].…”

A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

“…Speed, accuracy, and precision are the most common evaluation metrics used to evaluate these numerical solutions. Due to significant advances in the processing capabilities and speed of the processors and computers in the late 20th century, numerical solutions became more prevalent and continue to expand [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. The main topic presented in this paper is about the approximate solution of the following equation: y = f (x, y), y(x 0 ) = y 0 , y (x 0 ) = y 0 , (1) where f is a sufficiently continuous and derivative function of any order.…”