Abstract
Block methods that approximate the solution at several points in block form are commonly used to solve higher order differential equations. Inspired by the literature and ongoing research in this field, this paper intends to explore a new derivation of block backward differentiation formula that employs independent parameter to provide sufficient accuracy when solving second order ordinary differential equations directly. The use of three backward steps and five independent parameters are considered adequately in generating the variable coefficients of the formulas. To ascertain only one parameter exists in the derived formula, the order of the method is determined. Such independent parameter retains the favorable convergence properties although the values of parameter will affect the zero stability and truncation error. An ability of the method to compute the approximated solutions at two points concurrently is undeniable. Another advantage of the method is being able to solve the second order problems directly without recourse to the technique of reducing it to a system of first order equations. The essential of the error analysis is to observe the effect of independent parameter on the accuracy, in the sense that with certain appropriate values of parameter, the accuracy is improved. The performance of the method is tested with some initial value problems and the numerical results confirm that the maximum error and average error obtained by the proposed method are smaller at certain step size compared to the other conventional direct methods.
Highlights
The mathematical problems in the physical world such as chemical kinetics, vibrations and electrical circuits can be found in the second order ordinary differential equations (ODEs) of the following form:y f x, y, y ', (1)where y a y0, y ' a y '0 are initial conditions with the interval x a,b
Equations (2) can be solved numerically using any first order ODEs solver such as backward differentiation formula (BDF) as introduced by [1] and block backward differentiation formulas (BBDF), which was proposed by [2]
Discussion on a second order initial value problems (IVPs) of ODEs clearly shows its potential for solving second order ODEs directly
Summary
The mathematical problems in the physical world such as chemical kinetics, vibrations and electrical circuits can be found in the second order ordinary differential equations (ODEs) of the following form:. Equations (2) can be solved numerically using any first order ODEs solver such as backward differentiation formula (BDF) as introduced by [1] and block backward differentiation formulas (BBDF), which was proposed by [2] This approach is cumbersome, consume a lot of human effort and increase the computational time. Contrary to the conventional BBDF method, the proposed method is derived using three backward steps and five independent parameters by adopting the technique presented by [10]–[13] for solving (1) directly without reducing it to (2) This is the idea underlying the derivation of the new version of numerical method, where the value of parameter can be adjusted to improve the approximation of initial value problems (IVPs). The strategy to derive the method is presented
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