Mathematical Problems in Engineering

Volume 2018, Article ID 3750274, 15 pages

https://doi.org/10.1155/2018/3750274

## Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN 37044, USA

Correspondence should be addressed to Samuel N. Jator; ude.uspa@srotaj

Received 5 March 2018; Revised 30 April 2018; Accepted 14 May 2018; Published 11 June 2018

Academic Editor: Zhangxin Chen

Copyright © 2018 Samuel N. Jator and Kindyl L. King. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In some cases, high-order methods are known to provide greater accuracy with larger step-sizes than lower order methods. Hence, in this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. The properties of the BHM are discussed and the performance of the method is demonstrated on some numerical examples. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.

#### 1. Introduction

General second-order differential equations frequently arise in several areas of science and engineering, such as celestial mechanics, quantum mechanics, control theory, circuit theory, astrophysics, and biological sciences. Several of these differential equations have oscillatory solutions, for instance, the initial value problem (IVP)where , is an integer, , is the dimension of the system, is a real nonsingular diagonalizable constant matrix having large eigenvalues, and has been studied (see [1–9]). Special cases of (1) have also been extensively discussed in the literature ([10–14], Hairer [15]).

The method proposed in this paper can solve (1), as well as the general second-order IVPwhere , is an integer, and is the dimension of the system. Equation (2) is conventionally solved by converting it into an equivalent first-order system of double dimension and then solved using standard methods that are available in the literature for solving systems of first-order IVPs (see Lambert [16], Hairer et al. [17], and Brugnano et al. [18]). In general, these methods are implemented in a step-by-step fashion in which, on the partition , an approximation is obtained at only after an approximation at has been computed, where for some constant step-size and integer ,

Several methods have been proposed for directly solving the special second-order IVP, where the function does not depend on . It has been shown that such methods have the advantages of requiring less storage space and fewer number of function evaluations (see Hairer [15], Hairer et al. [19], Lambert et al. [20], and Twizell et al. [21]). Nevertheless, these methods are implemented as predictor-corrector methods and require starting values as well. In which case, the cost of implementation is increased, especially, for higher order methods.

In this paper, we propose a high-order BHM of order 11 which provides greater accuracy even when larger step-sizes are used. We show that the BHM of order 11 is applied to directly solve Hamiltonian systems as well as the general form (2). The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. In this way, the method is self-starting and implemented without predictors (see Jator and Oladejo [22], Jator et al. [9], Jator [7], and Ngwane and Jator [6, 23, 24]). It is shown that the method can also be used to directly solve non-Hamiltonian systems with embedded first derivatives as well as partial differential equations via the method of lines. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.

The article is organized as follows. In Section 2, we derive our hybrid method and specify the coefficients as well. We discuss the properties and implementation of the method in Section 3. Numerical examples are given in Section 4 and concluding remarks are given in Section 5.

#### 2. Derivation

We propose a BHM for the IVP (2) which can be solved by advancing the from to by fixing two interpolation points and a set of distinct collocation points . We then choose the coefficients of the method, such that the method integrates the IVP exactly, whenever the solutions are members of the linear space . Thus, we initially seek a continuous local approximation on the interval , which is expressed in vector form aswhere are coefficients in to be uniquely determined. To determine these coefficients, we interpolate at and and then collocate at . In particular, we determine these coefficients by imposing the following conditions on (4):to obtain the following system of equations represented in matrix formThe system is solved with the aid of Mathematica to obtain the coefficients, , which are substituted into (4) to obtain the continuous formThe continuous scheme (7) is simplified and evaluated at , , to give the set of formulas (8), whose coefficients are specified in Tables 1 and 2. In the same vein, the first derivative of the continuous scheme (7) is simplified and evaluated at , , to give the set of formulas (9), whose coefficients are also specified in Tables 1 and 2.