Research Article
Least Square Methods Based on Control Points of Said Ball Curves for Solving
Ordinary Differential Equations
Abdul Hadi Bhatti1*,Sharmila Binti Karim2
1Department of Mathematics & Statistics, School of Quantitative Sciences, Universiti Utara Malaysia, 06010 UUM Sintok, Kedah, Malaysia.
2Department of Mathematics & Statistics, School of Quantitative Sciences, Universiti Utara Malaysia, 06010 UUM Sintok, Kedah, Malaysia.
abdulhadi.math@gmail.com1*, mila@uum.edu.my2
Article History: Received: 10 November 2020; Revised: 12 January 2021; Accepted: 27January 2021; Published online: 05April 2021
Abstract:This paper presents the use of Said Ball curve’s control points to approximate the solutions of linear ordinary
differential equations (ODEs). Least squares methods (LSM) is proposed to find the control points of Said Ball curves by minimizing the error of residual function.The residual error is measured by taking the sum of squares of the Said Ball curve’s control points of the residual function. Then the approximate solution of ODEs is obtained by minimizing residual error.Two numerical examples are given in term of error and compared with the exact solution to demonstrate the efficiency of the proposed method.
Keywords:Said Ball curves, Control points, Ordinary differential equations, Least Square Methods, Residual function.
1. Introduction
In general, there are three ways to find the solutions of ordinary differential equations (ODEs), analytically, approximately, and numerically. In the various fields of engineering and computer sciences, mathematical models involving ODEs, did not have exact or analytical solutions.Thus,it required to consider the approximate solutions which are computed by the different numerical methods.
In order to obtain the approximate solution of ODEs numerical solutions of the differential equations, the piecewise polynomial, and polynomial functions are regularly employed (Lai, M. J, et al.. 2000).Hence these approximations provide a process of solutions which represented by a function value on mutually linked control points through interpolation of the function.For the approximate solutions of differential equations, the control points of Bézier curve are found suitable and computationally beneficial technique (Ghomanjani, F, et al..,2017) As well as Bézier curves involved to approximate the functions or polynomial functions and data (Nürnberger, G, et all.,2000)Bézier curves was adopted as a solution in solving the dynamical system (Gachpazan, M, et all.,2011)Furthermore,Bézier curves are used to solve the partial differential equations (Beltran, J, et al.,2004)
In the Bezier curves, if the approximate solution is not acceptable then the degree of Bezier curve can be increased in order to obtain the best approximate solution and vice versa the degree can be reduced to overcome the burden of higher degrees which are not necessary (Farin, G, 1997)
Since 1970’s, some research scholars have introduced thebasis functions of generalized Ball curves which are more efficient and effectivecompared to the Bezier curves(Ball, A. A, 1974;1975;1977; Said, H. B,1989; Wang, G, 1987)The generalized Ball curves have the similar properties of the curve or shape preservative as the Bézier curves. Evaluation of any point of generalized Ball curves in recursive algorithms is more efficient as compare to the Bézier curve in the de Casteljau algorithmSaid, H. B, et al..,1989; Hongyi W, 2000)Generalized Ball curve is the best suited for the degree elevation and reduction compared to the Bézier curve(Goodman, T. N, et al.,1991)Since the control point base is found as animportant geometric feature of the shape of Ball curves, the computations based on the control points will be conducted.
A cubic polynomial curve described mathematically during the eminent aircraft design system for the conic lofting surface program CONSURF (Ball, A. A. 197; Ball, A. A. 1977;Delgado, J et all.,2003).It is extended to three further distinct generalizations called Said Ball curves, DP Ball curves, and Wang Ball curves for higher degree𝑛polynomials
The degree of the curve of the Said Ball curves, DP Ball curves, and Wang Ball curves can be found by overlying their control points(Aphirukmatakun, C et all.,2007).Many research scholars have comeout with the Ball curves based on the theoretical calculation, elevation and reduction of their degree to obtain the more
accurate shape of curveBeltran. J. V., et al..,2004; Said. H. B,1989;Goodman. T. N., et al..,1991;Delgado. J., et al..,2003; Delgado. J,et all.,2003; Aphirukmatakun.C, et all.,2007; Hu. S, et al.,1996; Pihien, H. N, et al 2000)
Mathematical form and graphical representation of the basis function of cubic polynomial curve given as follows:
Figure 1.1.Basis function of Cubic Ball
Two advantages of Ball function are identified, first cubic Ball function can be reduced to quadratic Bézier function when the interior control point of Ball functions was combined with the Ball basis function. The second was the Ball function is more efficient in term of computation whengeneralized representations of Ball curves is used(Said, H. B. 1989). Meanwhile the Said Ball curve is more competent in term of computation compared to the Bézier curve and as well as the shape preservative construction properties are similar between Bernstein Bézier basis and Said Ball basis (Said, H. B. 1989; Jaafar, W, et al.,2018)
Anyhow, our study aim is to develop efficient numerical method in order to improve the level of accuracy in term of error and computational performance for the approximate solution of ODEs.In order to find the maximum accuracy of approximate solutions for higher order ODEs, a lot of algorithmswere developed using different numerical methods such as reduction method (Lambert. J. D., et all.,1976; Hull,T. E., et all.,1972; Sarafyan, D.1990; Bun, R. A, 1992).However, the reduction method has many hindrances such as complications in writing and developing computer program as well as the load of computation affects the error accuracy which consume more of humanoid effort. Due to that the direct method for solving the higher ODEs and its correspondent system of first order ODEs was introduced in 1960 by (RutishauserAnake, T. A. 2011). A lot of research scholars suggested that the direct method is more efficient and suitable method for solving the higher order and its correspondent system of first order ODEswithout reduction to the first order ODEs. According to Bjorck(1996), the Least Squares Methods (LSM) was first discovered from Gauss in 1975 and found well recognize as the direct methods to develop the best approximate solutions of ODEs by employing piecewise polynomial functions. Least squares methods also involved for integrals’ discretization (Ascher, U. 1978).
As a discretization alternative of integrals,the least squares objective function in LSM was developed to find the approximate solutions of ODEs based on the control points of Bézier curve(Zheng, J, et al,2004)Moreover, in literature review, some different type of differential equations solved by the same technique LSM(Mehrkanoon.S.,et al..,2012;Amaal A., M. 2012; Chistyakov, V. F, et all., 2013;Ghomanjani, F.,et all.,2013;Alavizadeh, S. R, et al..,2014; Yilmaz, B., et al., 2017; Monterde, J.2004; Husin, S. F., 2019).The advantage of the least square’s method is that it gives practically simple technique for finding the better approximate solution to ODEs (Husin, S. F., 2019).
For the best of our current knowledge, the Said Ball curves representations have yet to be investigated by using LSM to approximate the ODEs solutions.This paper proposes the least squares methods (LSM) will be applied investigation the control points of Said Ball curves representationsin order to approximate solutions of ODEs.
This research paper is organized as follows: Section 2 briefly overviewson Said Ball curves with its important properties. In section 3, a new method is proposed for solving ODEs based on control points paradigm of the Said Ball curves representation. Meanwhile in section 4, the numerical examples are given and solved through the proposed method. Lastly, the conclusion is given.
2. Said Ball Curves Representations
The Said Ball curves with degree 𝑚 and (𝑚 + 1) control points {𝒗𝒋}𝒋=𝟎𝒎 is as follows[10, 17, 20-21].
𝑆 𝑥 = 𝑆𝑗𝑚 𝑥 𝑣𝑗 𝑚
𝑗 =0
(2.1)
where the Said Ball Polynomials 𝑆𝑗𝑚 𝑥 are defined as follows
𝑆𝑗𝑚 𝑥 = 𝑚 − 12 + 𝑗 𝑗 𝑥𝑗(1 − 𝑥)𝑚 −12 +1 , 𝑓𝑜𝑟 0 ≤ 𝑗 ≤𝑚 − 1 2 𝑚 − 1 2 + 𝑚 − 𝑗 𝑚 − 𝑗 𝑥𝑚 −12 +1(1 − 𝑥)𝑚 −𝑗, 𝑓𝑜𝑟𝑚 + 1 2 ≤ 𝑗 ≤ 𝑚 (2.2) when 𝑚 is odd and 𝑆𝑗𝑚 𝑥 = 𝑚2 + 𝑗 𝑗 𝑥𝑗(1 − 𝑥)𝑚2+1 , 𝑓𝑜𝑟 0 ≤ 𝑗 ≤𝑚 2 − 1 𝑚 𝑚 2 𝑥𝑚2 1 − 𝑥 𝑚 2 , 𝑓𝑜𝑟𝑗 =𝑚 2 𝑚 2 + 𝑚 − 𝑗 𝑚 − 𝑗 𝑥 𝑚 2 +1 1 − 𝑥 𝑚 −𝑗 , 𝑓𝑜𝑟𝑚 2 ≤ 𝑗 ≤ 𝑚 (2.3) when 𝑚 is even.
The following figure (1.6), (1.7), (1.8) and (1.9) each shows the graphical representations of the basis function of Said Ball of degree 2, 3, 4 and 5 respectively.
Figure 1.2.Basis function of Said Ball with degree 2
Figure 1.3.Basis function of Said Ball with degree 3
Figure 1.5.Basis function of Said Ball with degree 5 Properties of Said Ball basis functions:
i. Said Ball basis function is non-negative.
𝑆𝑗𝑚 𝑥 ≥ 0 , 𝑓𝑜𝑟𝑎𝑙𝑙𝑗 = 0, 1, 2, … , 𝑚 (2.4)
ii. Partition of Said Ball basis function is unity.
𝑆𝑗𝑚 𝑥 𝑚
𝑗 =0
= 1 (2.5)
The satisfaction of the two properties of the Said Ball basis function in Equation (2.4), and (2.5) indicated the convex combination of its control points. Therefore, the Said Ball curve is in the convex hull of its control polygon with control points(Hu, S. M., et al..,1996)
3. Least squares method for solving ODE’s
The technique based on the control points of Said Ball curves
Control point-based technique is proposed to represent the approximate solution in the form of Said Ball curve,due to the Said Ball curve has more computational competency, compared to the Bézier curve. Moreover, the construction properties in term of shape preservative are similar between Said Ball basis and Bernstein Bézier basis. We take the sum of squares of the Said Ball curve’s control points of the residual function 𝑅 𝑥 = 𝐿𝑔 𝑥 − 𝑓 𝑥 to compute the residual error. If this residual error quantity equal to zero then obviously the residual function also will be equal to zero, which implies that the approximate solution of ODEs is equal to the exact solution.
Thestrategies in this workare as follows:
(i) Finding control points of Said Ball curves by using LSM.
(ii) Investigating Said Ball curve’s control points by minimizing the residual function.
The following IVPs and BVPs are considered to find their approximate solution:
𝐿𝑔 𝑥 = ∅𝑗 𝑥 𝑔𝑗 𝑥 = 𝑓 𝑥, 𝑦, 𝑦′, 𝑦′′, … , 𝑦(𝑗 −1), 𝑦(𝑗 ) 2𝑚
𝑗 =0
, 0 ≤ 𝑥 ≤ 1 (3.1)
with initial value problem as follows
𝑔 0 = 𝑔0, 𝑔𝑗 0 = 𝑔𝑗, 𝑗 = 1, 2, 3, … , 𝑚 − 1 (3.2)
and with boundary value problem as follows
𝑔𝑗 0 = 𝑔
Where 𝐿 = ∅𝑗 𝑥 𝑑𝑗 𝑑𝑥𝑗 2𝑚 𝑗 =0
is(2𝑚)𝑡 order differential operator with the polynomial coefficients ∅𝑗 𝑥 in term of 𝑥, and as well as also a
polynomial function 𝑓(𝑥) in term of 𝑥. The main goal of thispaper is to develop the generalized solution of ODEs by reducing the error of the residual function 𝑅 𝑢 = 𝐿𝑔 𝑥 − 𝑓 𝑥 , then an algorithm is developed for the approximate solution as a function 𝑔 𝑥 of ODEs such that the function 𝑔 𝑥 satisfies the conditions of IVPs
(3.2) and BVPs (3.3).
The details of algorithm using LSMmethodfor computing the approximate solutions given in the following steps:
Step 1.
Setting a degree 𝑚 representatively and shows as solutions 𝑆 𝑥 in the form of Said Ball curves with degree 𝑚 respectively.
𝑆 = 𝑆 𝑥 = 𝑆𝑗𝑚 𝑥 𝑣𝑗 𝑚
𝑗 =0
, 0 ≤ 𝑥 ≤ 1 (3.4)
where (𝑚 + 1) control points {𝑣𝑗}𝑗 =0𝑚 = 𝑣0, 𝑣1, 𝑣2, … … … , 𝑣𝑚 are to be determined.
Step 2.
Finding the residual functions for the Ball curves 𝑅𝑆 𝑥 = 𝐿𝑔 𝑥 − 𝑓 𝑥 , by replacing the solutions 𝑆 𝑥
into the IVPs and BVPs of ODEs in the form of Said Ball curves. The polynomial
𝑅𝑆 𝑥 = 𝑐𝑆𝑗𝑆𝑗𝑚 𝑥 𝑘
𝑗 =0
can be expressed with degree 𝑘 ≥ 𝑚 in the forms of Said Ball curves.
We note that 𝑘 = 𝑚𝑎𝑥𝑗{𝑚 − 𝑗 + deg ∅𝑗 , deg 𝑓 𝑥 } and where the linear functions𝑐𝑆𝑗are control points in
term of unknowns 𝑣𝑗and 𝑗 = 0, 1, 2, 3, … … … , 𝑘. These functions will be obtained by adopting the procedures
of multiplication, degree elevation and differentiation for Said Ball curves.
Step 3.
Developing the objective functions for Said Ball curves.
𝐹𝑆= 𝑐𝑆𝑗2 𝑘
𝑗 =0
𝑣𝑗
Step 4.
Solving the constrained optimization problems for 𝑣𝑗
min𝐹𝑆= 𝑐𝑆𝑗2 𝑘 𝑗 =0 𝑣𝑗 , s.t 𝑔𝑗 0 = 𝑔 𝑗 , 𝑗 = 0, 1, 2, 3, … … . . . , 𝑚 − 1 (IVPs) or 𝑔𝑗 0 = 𝑔 𝑗𝑎𝑛𝑑𝑔𝑗 1 = 𝑧𝑗 , 𝑗 = 0, 1, 2, 3, … . . . … , 𝑚 − 1 (BVPs),
The Lagrange Multiplier method or any other suitable method will be employed to solve the constrained optimization problems.
Step 5.
By replacing these minimum solutions which obtain in previous Step 4 back into representatively solutions 𝑆 𝑥 in the Step 1 into the form of Said Ball curves, the approximate solutions of the ODEs will be obtained.
4. Numerical Examples
In this section, we demonstrate the two numerical examples for validation of error accuracy and as well as the computational performance of proposed method based on the control points of Said Ball curves.
4.1. Example 1:2nd order ODE with IVP
We consider a simple example to demonstrate the method.
𝑦′′ 𝑥 − 6𝑥 = 0, with 𝑦 0 = 0, 𝑦 1 = 1.
The exact solution is 𝑦𝑒(𝑥) = 𝑆 𝑥 = 𝑥3. Next, we try to find the approximate solution using LSM based on
the control points of Said Ball curves.
Suppose that the approximate solution in the form of Said Ball curves with degree 2is as follows:
𝑦 = 𝑆 𝑥 = 𝑆𝑗𝑚 𝑥 𝑣𝑗 2
𝑗 =0
= 𝑣0𝑆02 𝑥 + 𝑣1𝑆12 𝑥 + 𝑣2𝑆22 𝑥
𝑦𝑎(𝑥) = 𝑆𝑎(𝑥) = 𝑣0(1 − 𝑥)2+ 𝑣1 2𝑥 1 − 𝑥 + 𝑣2𝑥2(3.5)
By replacing the solutions 𝑦 = 𝑆 𝑥 into ODEs in the form of Said Ball curves, the residual functions for the Ball curves is obtained as:
𝑅𝑆 𝑥 = 2𝑣0− 4𝑣1+ 2𝑣2− 6𝑥 = 2𝑣0− 4𝑣1+ 2𝑣2 𝑆02 𝑥 + 2𝑣0− 4𝑣1+ 2𝑣2− 6𝑥 𝑆12 𝑥 .
Then, the next step is developing the objective functions for Said Ball curves:
𝐹𝑆(𝑣0, 𝑣1, 𝑣2) = 2𝑣0− 4𝑣1+ 2𝑣2 2+ 2𝑣0− 4𝑣1+ 2𝑣2− 6 2.
Then, solving the constrained optimization problems for 𝑣0, 𝑣1, 𝑣2 by minimizing the objective function with
given condition 𝑦 0 = 0, 𝑦 1 = 1, we found 𝑣0, 𝑣1, and 𝑣2.
Table 1. Values of Said Ball curve’s Control points for the example 1
Control Points Values
𝑣0 0
𝑣1 0
𝑣2 1
By substituting thesecontrol points’ values of Said Ball curves back into Equation (3.5), hence weobtain the approximate solution as follows:
𝑦𝑎(𝑥) = 𝑆𝑎(𝑥) = 𝑥2
The maximum error-0.002375 at value of 𝑥 = 0.05which is liesin the interval [0, 1] between the approximate solution and exact solution.
4.2. Example 2: A fourth order boundary value problem involved in transversal bending clamped beam mathematical problem.
( 𝑥2+ 1 𝑦′′ 𝑥 )′′= 𝑥 + 1, 0 < 𝑥 < 1,
with 𝑦 0 = 𝑦 1 = 0, 𝑦′ 0 = 𝑦′ 1 = 0.
𝑦𝑎 𝑥 = 𝑆𝑎 𝑥 = 𝑆𝑗𝑚 𝑥 𝑣𝑗 8 𝑗 =0 = 𝑣0𝑆08 𝑥 + 𝑣1𝑆18 𝑥 + 𝑣2𝑆28 𝑥 + 𝑣3𝑆38 𝑥 + 𝑣4𝑆48 𝑥 + 𝑣5𝑆58 𝑥 + 𝑣6𝑆68 𝑥 + 𝑣7𝑆78 𝑥 + 𝑣8𝑆88 𝑥 𝑦𝑎 𝑥 = 𝑆𝑎 𝑥 = 𝑣0 1 − 𝑥 5+ 5 𝑣1𝑥 1 − 𝑥 5+ 15 𝑣2𝑥2 1 − 𝑥 5+ 35 𝑣3𝑥3 1 − 𝑥 5 +70 𝑣4𝑥4 1 − 𝑥 4+ 35 𝑣5𝑥5 1 − 𝑥 3+ 15 𝑣6𝑥5 1 − 𝑥 2+ 5 𝑣7𝑥5 1 − 𝑥 + 𝑣8𝑥5(3.6)
We found the control points 𝑣0, 𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5, 𝑣6, 𝑣7 and 𝑣8 by minimizing the objective function with
given boundary conditions 𝑦 0 = 𝑦 1 = 0, 𝑦′ 0 = 𝑦′ 1 = 0 using the LSM.
Table 2. Values of Said Ball curve’s Control points for the example 2
Control Points Values
𝑣0 0 𝑣1 0 𝑣2 0.0033923547 𝑣3 0.0043452293 𝑣4 0.0040366036 𝑣5 0.0037710466 𝑣6 0.0025725930 𝑣7 0 𝑣8 0
By replacing these control points back into the Equation (3.6), hereafter we found the approximate solution is as follows:
𝑦𝑎 𝑥 = 0.0508853209 𝑥2 1 − 𝑥 5+ 0.1520830270 𝑥3 1 − 𝑥 5+ 0.2825622496 𝑥4 1 − 𝑥 4
+0.1319866298 𝑥5(1 − 𝑥)3+ 0.0385888955 𝑥5(1 − 𝑥)2,
and the exact solution is
𝑦𝑒(𝑥) = 336𝑥 arctan 𝑥 − 36𝜋𝑥2+ 9𝜋2𝑥2+ 36 𝑙𝑛 2 2𝑥2− 4𝜋𝑥3+ 𝜋2𝑥3+ 4 𝑙𝑛 2 2𝑥3− 44𝜋𝑥 +
168 (𝑙𝑛 2 ) 𝑥 − 168 (𝑙𝑛 2 ) arctan 𝑥 + 44𝜋 arctan 𝑥 − 168 (𝑙𝑛 1 + 𝑥2 ) + 44 (𝑙𝑛 2 ) (𝑙𝑛 1 + 𝑥2 ) +
42𝜋 (𝑙𝑛 1 + 𝑥2 ) − 84𝜋𝑥 arctan 𝑥 − 88 (𝑙𝑛 2 ) 𝑥 arctan 𝑥 + 22𝜋𝑥 (𝑙𝑛 1 + 𝑥2 ) − 84 (𝑙𝑛 2 ) 𝑥 (𝑙𝑛 1 +
𝑥2 ) /36(4 𝑙𝑛 2 2+ 𝜋2− 4𝜋).
Based on the table, the maximum error is 0.0000030955 at value of 𝑥 = 0.40which is fall under the interval [0, 1] between the approximate solution and exact solution. The graphical representation of the approximate solution and the exact solution in the following Figure (3.1).
Figure 3.1.Exact solution 𝑦𝑎(𝑥) (dashed with stars in blue colour) with basis function of Said Ball
degree 8 and 𝑦𝑒(𝑥)(dashed with circles in red colour).
5. Conclusion
We have presented the Said Ball curvesfor the solution of ODEs approximately.We computed the approximate solution of ODEs based on the control points of Said Ball curves by minimizing the residual function using LSM. The residual function is minimized by reducing the residual error. Snice the Said Ball curves and Bézier curves have the same paradigm of control polygon with control points and as well as have similar shape preservative’s constructions properties. The solution of ODEs represented in the form of Said Ball curves and then replaced it into the differential equation. The approximate solutionsof the numerical examples based on the control points of Said Ball curvesaremore satisfactory in term of error accuracy. The calculation is quitesimple and can complete without operating the explicit instructions.Furthermore, we have solved polynomial coefficients linear ODE through Said Ball curve’s representations. The idea of this method is relatively general and further it can be employed easily for the solution of other kinds of differential equations.
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