Abstract
A direct two‐point block one‐step method for solving general second‐order ordinary differential equations (ODEs) directly is presented in this paper. The one‐step block method will solve the second‐order ODEs without reducing to first‐order equations. The direct solutions of the general second‐order ODEs will be calculated at two points simultaneously using variable step size. The method is formulated using the linear multistep method, but the new method possesses the desirable feature of the one‐step method. The implementation is based on the predictor and corrector formulas in the PE(CE) m mode. The stability and precision of this method will also be analyzed and deliberated. Numerical results are given to show the efficiency of the proposed method and will be compared with the existing method.
Highlights
In this paper, we are considering solving directly the general second-order initial value problems IVPs for systems of ODEs in the form y f x, y, y, y a y0, y a y0, x ∈ a, b .Equation in 1.1 arises from many physical phenomena in a wide spectrum of applications especially in the science and engineering areas such as in the electric circuit, damped and Mathematical Problems in Engineering h h h h xn xn+1 xn+2 xn+3 xn+4 kth block (k + 1)th block undamped spring mass and some other areas of application
Block methods for numerical solutions of ODEs have been proposed by several researchers such as in 1–5
The common block methods used to solve the problems can be categorized as one-step block method and multistep block method
Summary
We are considering solving directly the general second-order initial value problems IVPs for systems of ODEs in the form y f x, y, y , y a y0, y a y0, x ∈ a, b. One-step block method such as the implicit Runge-Kutta method is being referred to as one previous point to obtain the solution. The works in 6 showed the proposed two-point four-step block method presented as in a simple form of Adams Moulton method for solving second-order ODEs directly. The block method of Runge-Kutta type has been explored in 1 , and it is suggested that a block of new approximation values is used simultaneously for solving first-order ODEs. The works in 3, 8, 9 have been considered in solving 1.1 using the block one-step method, while 3 has proposed a two-point implicit block one-step method for solving second-order ODEs directly and suggested that the method is suitable to be parallel. The approach in this research is to extend the idea in 9 for solving 1.1 directly without reducing system of first-order ODEs using two-point block one-step method
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