Abstract
Numerical methods are regularly established for the better approximate solutions of the ordinary differential equations (ODEs). The best approximate solution of ODEs can be obtained by error reduction between the approximate solution and exact solution. To improve the error accuracy, the representations of Wang Ball curves are proposed through the investigation of their control points by using the Least Square Method (LSM). The control points of Wang Ball curves are calculated by minimizing the residual function using LSM. The residual function is minimized by reducing the residual error where it is measured by the sum of the square of the residual function of the Wang Ball curve's control points. The approximate solution of ODEs is obtained by exploring and determining the control points of Wang Ball curves. Two numerical examples of initial value problem (IVP) and boundary value problem (BVP) are illustrated to demonstrate the proposed method in terms of error. The results of the numerical examples by using the proposed method show that the error accuracy is improved compared to the existing study of Bézier curves. Successfully, the convergence analysis is conducted with a two-point boundary value problem for the proposed method.
Highlights
In real-world applications like various fields of engineering and computer science, many problems involve mathematical models which contain an ordinary differential equation (ODE)
The Least Square Method (LSM) is employed based on the control points of Bézier curves by developing the least square objective function for the discretization of integrals to improve the approximate solutions of higher order ODEs [29]
The algorithm to solve the ODEs by using the LSM through computing the control points of Wang Ball curves is shown in the following steps: Step 1: Suppose that a degree m and symbolically express A(x), the approximate solution of ODEs in the form of Wang Ball curves with degree m, respectively, is: A(x) = ∑mj=0 wj Ajm(x), 0 ≤ x ≤ 1 (4.4)
Summary
In real-world applications like various fields of engineering and computer science, many problems involve mathematical models which contain an ordinary differential equation (ODE). The LSM is employed based on the control points of Bézier curves by developing the least square objective function for the discretization of integrals to improve the approximate solutions of higher order ODEs [29]. The application of the LSM to find the approximate solution of higher order ODEs based on the Bézier curve’s control points, the result is only satisfied, but not on the required level in terms of error [30, 36, 39,40, 47]. The structural preservative properties between the Bézier curves and generalized Wang Ball curves are similar; the Wang Ball curves based on the control points have yet to be investigated to find the approximate solution of higher order ODEs. we propose the LSM to solve higher order ODEs by exploring the control points of Wang Ball curves to improve the accuracy in terms of error. The remainder of this paper is structured as follows: the Bézier curve’s representation in Section 2, the representations of Wang Ball curves along with their properties are briefly discussed in Section 3 while in Section 4, the new method is proposed for solving ODEs approximately by investigating the control points of Wang Ball curves; thereafter in Section 5, numerical examples are provided to demonstrate the newly proposed method; in Section 6, the convergence of the proposed method for the two-point value problem is analyzed; in Section 7 the conclusion is presented
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