Mathematics and Statistics Vol. 11(3), pp. 464 - 489
DOI: 10.13189/ms.2023.110304
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Approximation Method Using DP Ball Curves for Solving Ordinary Differential Equations

Abdul Hadi Bhatti ^*, Sharmila Binti Karim
Department of Mathematics & Statistics, School of Quantitative Sciences, University Utara Malaysia, 06010 UUM Sintok, Kedah, Malaysia

ABSTRACT

Many researchers frequently developed numerical methods to explore the idea of solving ordinary differential equations (ODEs) approximately. Scholars started evolving approximation methods by developing algorithms to improve the accuracy in terms of error for the approximate solution. Polynomials, piece-wise polynomials in the form of Bézier curves, Bernstein polynomials, etc., are frequently used to represent the approximate solution of ODEs. To get the minimum error between the exact and approximate solutions of ODEs, the DP Ball curve (DPBC) using the least squares method (LSM) is proposed to improve the accuracy of the approximate solutions for the initial value problem IVPs. This paper explores the use of control points of the DPBC with error reduction by minimizing the residual function. The residual function is minimized by constructing the objective function by taking the sum of squares of the residue function for the least residual error. Then, by solving the constraint optimization problem, we obtained the best control points of DPBC. Two strategies are employed: investigating DPBC's control points through error reduction with LSM and computing the optimum control points through degree raising of DPBC for the best approximate solution of ODEs. Substituting the values of control points back into the DPBC allows for the best approximate solution to be obtained. Moreover, the convergence of the proposed method to the IVPs is successfully analyzed in this study. The error accuracy of the proposed method is also compared with the existing studies. Numerous numerical examples of first, second, and third orders are presented to illustrate the efficiency of the proposed method in terms of error. The results of the numerical examples are shown in which the error accuracy is considerably improved.

KEYWORDS
DP Ball Curve, Ordinary Differential Equations, Initial Value Problems, Residual Function, Least Squares Method

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Abdul Hadi Bhatti , Sharmila Binti Karim , "Approximation Method Using DP Ball Curves for Solving Ordinary Differential Equations," Mathematics and Statistics, Vol. 11, No. 3, pp. 464 - 489, 2023. DOI: 10.13189/ms.2023.110304.

(b). APA Format:
Abdul Hadi Bhatti , Sharmila Binti Karim (2023). Approximation Method Using DP Ball Curves for Solving Ordinary Differential Equations. Mathematics and Statistics, 11(3), 464 - 489. DOI: 10.13189/ms.2023.110304.