Special Issue 1IHICPAS 202 Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/10.30526/2021.IHICPAS.2654 For more information about the Conference please visit the websites: http://ihicps.com/ M a t h e m a t i c s | 76 A New Approach of Morgan-Voyce Polynomial to Solve Three Point Boundary Value Problems Bushra Esaa Kashem bushra_eesa@yahoo.com , 100047@uotechnology.edu.iq Department Applied Sciences, University of Technology, Baghdad, Iraq. Abstract In this paper, a new procedure is introduced to estimate the solution for the three-point boundary value problem which is instituted on the use of Morgan-Voyce polynomial. In the beginning, Morgan-Voyce polynomial along with their important properties is introduced. Next, this polynomial with aid of the collocation method utilized to modify the differential equation with boundary conditions to the algebraic system. Finally, the examples approve the validity and accuracy of the proposed method. Keywords: Morgan-voyce, three-point boundary value problem, collocation method, approximation method. 1.Introduction The three point boundary value problem belongs to the seeming nonlocal or multipoint boundary value problem. This local boundary value problem has a major role in physics, engineering and many phenomena in applied mathematical. An amount of research has been studying the three-point boundary value problem, for example, the positive solution [1], existence and stability [2], discrete first order [3], and variational principle [4]. In this study, we consider the linear three point boundary value problem is:- οΏ½ΜˆοΏ½π‘(π‘₯) + 𝐹(π‘₯)𝑍(π‘₯) = 𝐺(π‘₯) π‘€π‘–π‘‘β„Ž 𝑍(π‘Ž) = 𝑍(𝑏) = 𝐴 π‘Žπ‘›π‘‘ οΏ½Μ‡οΏ½(𝑐) = 𝐡 π‘₯ ∈ (π‘Ž, 𝑏) (1) Where F, G is a continuous function. a, b, c ∈ R. A, B real constant. The main purpose of this work is to modify a new algorithm for an approximate solution to three point boundary value problems based on the special case of Morgan-Voyce polynomial. The properties and the method for using are explored. This method decreased the differential equation with its initial and boundary conditions to a system of algebraic equations in the unknown expand coefficients. 2. Morgan-Voyce polynomial Diametrical polynomial Morgan-Voyce which is introduced recently in 1959 [5] at was used in many interesting papers such as, Functional integro [6], pantograph equation [7], differential-difference equation [8], and nonlinear ordinary differential equation [9]. http://ihicps.com/ mailto:bushra_eesa@yahoo.com mailto:100047@uotechnology.edu.iq Special Issue 1IHICPAS 202 Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/10.30526/2021.IHICPAS.2654 For more information about the Conference please visit the websites: http://ihicps.com/ M a t h e m a t i c s | 77 The Morgan-Voyce polynomials 𝑀𝑛(π‘₯) defined by:- [10] 𝑀𝑛+2(π‘₯) = 𝑀𝑛+1(π‘₯) βˆ’ 𝑀𝑛 (π‘₯) (2) With 𝑀0(π‘₯) = 2 , 𝑀1(π‘₯) = π‘₯ + 2 and explicit formulation (𝑛 β‰₯ 1, 𝑛 > π‘˜) 𝑀𝑛 (π‘₯) = βˆ‘ 2𝑛 π‘›βˆ’π‘˜ ( 𝑛+π‘˜βˆ’1 π‘›βˆ’π‘˜βˆ’1 ) π‘₯π‘˜ + π‘₯π‘›π‘›βˆ’1π‘˜=0 …(3) on other hand, by using eq.(2) & eq.(3) the first seven Morgan-Voyce polynomials are given: 𝑀0(π‘₯) = 2 , 𝑀1(π‘₯) = π‘₯ + 2 𝑀2(π‘₯) = π‘₯ 2 + 4π‘₯ + 2 𝑀3(π‘₯) = π‘₯ 3 + 6π‘₯2 + 9π‘₯ + 2 𝑀4(π‘₯) = π‘₯ 4 + 8π‘₯3 + 20π‘₯ 2 + 16π‘₯ + 2 𝑀5(π‘₯) = π‘₯ 5 + 10π‘₯4 + 35π‘₯3 + 50π‘₯ 2 + 25π‘₯ + 2 𝑀6(π‘₯) = π‘₯ 6 + 12π‘₯5 + 54π‘₯4 + 112π‘₯3 + 105π‘₯2 + 36π‘₯ + 2 𝑀7(π‘₯) = π‘₯ 7 + 14π‘₯6 + 77π‘₯5 + 210π‘₯4 + 294π‘₯3 + 196π‘₯2 + 49π‘₯ + 2 3- Important Properties of Morgan-Voyce polynomial: [11,12] 1- Connection with Chybechev and Lucas polynomial 𝑻𝒏(𝒙), 𝑳𝒏(𝒙) 𝑀𝑛(π‘₯) = 2𝑇𝑛( π‘₯ + 2 2 ) 𝑀𝑛(π‘₯ 2) = 𝐿2𝑛 (π‘₯) 2- Orthogonality 𝑀𝑛 (π‘₯) is an orthogonal polynomial over [0,-4] with weight function 1 √4βˆ’(π‘₯+2)2 3. Integration http://ihicps.com/ Special Issue 1IHICPAS 202 Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/10.30526/2021.IHICPAS.2654 For more information about the Conference please visit the websites: http://ihicps.com/ M a t h e m a t i c s | 78 ∫ 𝑀2𝑛+1(π‘₯)𝑑π‘₯ = 0 0 βˆ’4 ∫ 𝑀2𝑛 (π‘₯)𝑑π‘₯ = 8 (2𝑛 + 1)(2𝑛 βˆ’ 1) 0 βˆ’4 4-Zeros 𝑀𝑛 (π‘₯): π‘₯π‘Ÿ = βˆ’4 𝑠𝑖𝑛 2 [ 2π‘Ÿ βˆ’ 1 2𝑛 βˆ— πœ‹ 2 ] , π‘Ÿ = 1,2, … , 𝑛 3. Morgan-Voyce Collocation Method In this section, the solution of three-point boundary value problem can be abstracted by the following steps. Step 1 The function �̈�(π‘₯) in eq.(1) developed approximately using Morgan-Voyce polynomial:- �̈�(π‘₯) = βˆ‘ 𝐢𝑖 𝑀𝑖 (π‘₯) 𝑛 𝑖=0 (4) Step 2 Integrating eq.(4) twice from 0 t0 x depending the boundary condition to get :- 𝑍(π‘₯) = 𝑍(0) + οΏ½Μ‡οΏ½(0)π‘₯ + ∬ βˆ‘ 𝐢𝑖 𝑀𝑖 (π‘₯) 𝑛 𝑖=0 π‘₯ 0 (5) Step 3 Substituting the boundary condition into eq.(5) to reduce the Morgan-Voyce coefficient differential equation. Step 4 Rewriting eq.(1) by substituted eq.(5) &(4) to get βˆ‘ 𝐢𝑖 𝑀𝑖 (π‘₯) 𝑛 𝑖=0 + 𝐹(π‘₯)[𝑍(0) + οΏ½Μ‡οΏ½(0)π‘₯ + ∬ βˆ‘ 𝐢𝑖 𝑀𝑖 (π‘₯) 𝑛 𝑖=0 π‘₯ 0 ] = 𝐺(π‘₯) (6) Step 5 By consider the collocation point π‘₯𝑗 = π‘—βˆ’0.5 2𝑛 j=1,2,3,… can be resulting algebraic system which is solved to find the unknown coefficient 𝑀𝑖 . Step 7 http://ihicps.com/ Special Issue 1IHICPAS 202 Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/10.30526/2021.IHICPAS.2654 For more information about the Conference please visit the websites: http://ihicps.com/ M a t h e m a t i c s | 79 Substituted the calculated coefficients into eq.(5). The solution for eq.(1) will be obtained. 4. Examples Illustrations:- For showing the accuracy and activity of our approximate method, we consider the following examples. Example (1): �̈� βˆ’ 12 π‘₯βˆ’2𝑍 = 0 𝑍(1) = 𝑍(βˆ’1) = 1 , 𝑍(0) = οΏ½Μ‡οΏ½(0) = 0 (7) Exact solution x4 For assume �̈�(π‘₯) β‰… βˆ‘ 𝐢𝑖 𝑀𝑖 (π‘₯) 4 𝑖=0 (8) by integrating eq.(8) twice time and applying the steps studied in section(3), the approximate coefficient will be C0=36 , C1=-48 , C2=12 , C3=C4=0. Applying the approximate coefficient into eq.(8) , we obtain the exact solution 𝑍(π‘₯) β‰… π‘₯4 Example (2): �̈� βˆ’ 𝑍 = (4π‘₯6 + 12 π‘₯2)𝑒 π‘₯ 2 𝑍(1) = 𝑍(βˆ’1) = 𝑒1 , 𝑍(0) = οΏ½Μ‡οΏ½(0) = 0 Exact solution π‘₯4 𝑒 π‘₯ 2 Assume that �̈�(π‘₯) β‰… βˆ‘ 𝐢𝑖 𝑀𝑖 (π‘₯) 6 𝑖=0 by applying the same method and solve a system of equations we approximate the solution. Table 1 reflects the comparison between the approximate solution with the exact solution and absolute error. Table 1 x approximate Exact Absolute error -1 2.718282 2.718282 0 -0.75 0.555381 0.555310 0.000071 -0.5 0.080242 0.080252 0.00001 -0.25 0.004157 0.004158 0.0000001 http://ihicps.com/ Special Issue 1IHICPAS 202 Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/10.30526/2021.IHICPAS.2654 For more information about the Conference please visit the websites: http://ihicps.com/ M a t h e m a t i c s | 80 0 0 0 0 0.25 0.004157 0.004158 0.0000001 0.5 0.080242 0.080252 0.00001 0.75 0.555381 0.555310 0.000071 1 2.718282 2.718282 0 Example (3): �̈� βˆ’ 1 π‘₯ οΏ½Μ‡οΏ½ + 𝑍 = (4π‘₯ + 1)𝑒 π‘₯ 2 + π‘₯2 𝑍(1) = 𝑍(βˆ’1) = 1 + 𝑒1 , 𝑍(0) = 1, οΏ½Μ‡οΏ½(0) = 0 Exact solution π‘₯2 + 𝑒 π‘₯ 2 let �̈�(π‘₯) β‰… βˆ‘ 𝐢𝑖 𝑀𝑖 (π‘₯) 8 𝑖=0 the comparison between the approximate and exact solution are showing in Table 2. Table 2 x approximate Exact Absolute error -1 3.718282 3.718282 0 -0.8 2.536470 2.536481 0.000011 -0.6 1.793318 1.793329 0.000011 -0.4 1.333511 1.333511 0.000000 -0.2 1.080812 1.080811 0.000001 0 1 1 0 0.2 1.080812 1.080811 0.000001 0.4 1.333511 1.333511 0.000000 0.6 1.793318 1.793329 0.000011 0.8 2.536470 2.536481 0.000011 1 3.718282 3.718282 0 http://ihicps.com/ Special Issue 1IHICPAS 202 Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/10.30526/2021.IHICPAS.2654 For more information about the Conference please visit the websites: http://ihicps.com/ M a t h e m a t i c s | 81 4.Conclusion In this article, a new general formula for Morgan-Voyce polynomial collocation method is employed to solve the three-point boundary value problems. The approached plane is tested by some examples and the results are satisfied in comparison with approximate with existing. References 1. Yang, C. 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RISING DIAGONAL POLYNOMIALS ASSOCIATED WITH MORGAN- VOYCE POLYNOMIALS. Fibonacci Quarterly , 200,)38(1), AMS Classification Available from: https://www..fq.math.ca/Scanned/38-1/swamy2.pdf 0. http://ihicps.com/