Bratu equations are substantial in electrostatic and plasma problem. The aim of this paper is design a morgan-voyce approach for solving bratu problem. We present a morgan-voyce polynomial along with significant properties; the effectiveness of the proposed algorithm is demonstrated by considering three numerical examples

Kashiem, Bushra Eesa

English

United Arab Emirates University: Scholarworks@UAEU / جامعة الامارات

Emirates Journal for Engineering Research Volume 26 Issue 2 Article 3 5-15-2021 Morgan- Voyce Approach for Solution Bratu Problems Bushra Eesa Kashiem university of technology - iraq, bushra_eesa@yahoo.com Follow this and additional works at: https://scholarworks.uaeu.ac.ae/ejer  Part of the Numerical Analysis and Computation Commons Recommended Citation Kashiem, Bushra Eesa (2021) "Morgan- Voyce Approach for Solution Bratu Problems," Emirates Journal for Engineering Research: Vol. 26 : Iss. 2 , Article 3. Available at: https://scholarworks.uaeu.ac.ae/ejer/vol26/iss2/3 This Article is brought to you for free and open access by Scholarworks@UAEU. It has been accepted for inclusion in Emirates Journal for Engineering Research by an authorized editor of Scholarworks@UAEU. For more information, please contact EJER@uaeu.ac.ae. MORGAN- VOYCE APPROACH FOR SOLUTION BRATU PROBLEMS Bushra E.Kashiem University of Technology /Applied Sciences Bushra_eesa@yahoo.com  (Received 1st February and Accepted 15th May 2021)   نهج مورغان فويس لحل مسائل براتو ملخصبراتو قدمنا  ميم تقريب لمورغان فويس لحل مسألةالهدف من هذا البحث هو تص.مسائل الكهرباء والبالزما من المعادالت المهمه في معادالت براتو  .ضيحها من خالل ثالثه امثله عدديهمن اهم الخصائص فعالية الخوارزميه المقترحه تم تو متعددة الحدود مورغان فويس مع مجموعة .  Abstract  Bratu equations are substantial in electrostatic and plasma problem. The aim of this paper is to design morgan-voyce approaches for solving bratu problem; Morgan- voyce polynomials and some of its important properties were introduced; the effectiveness of the proposed approach is demonstrated by considering three numerical examples.  1. Introduction       The bratu problem is more used in applied mathematics, physical and chemical phenomena [1-4].Many of numerical and approximate methods have been studied to solve bratu equation, for example, Bernoulli polynomial [5], Finite difference [6], Bernstion polynomial [7] and B-spline collocation method  [8].   Consider the Bratu boundary value problem of the form:                   ?̈?(𝑡) + 𝜆 𝑒𝑢(𝑡) = 0                  0 < 𝑥 < 1 𝑢(0) = 𝑢(1) = 0          𝑓𝑜𝑟  𝜆 > 0              …(1) The outlined of the paper is as follows:- section 2, we explained and construct the Morgan-Voyce polynomial, some properties of this polynomial discussed in section 3,we analyzed the Morgan-Voyce approach for the solution of bratu problem in       section 4, in section 5, numerical examples are presented, finally, the main conclusion are given in section 6. 2- Morgan-Voyce Polynomial:-       Morgan-Voyce polynomial introduced as recently as 1959 in the analysis of ladder network [9], many authors used this polynomial for solving problem, such as [10- 12]. For n ≥ 2 , the Morgan-Voyce polynomial 𝑀𝑛(𝑥) defined by :- 𝑀𝑛(𝑥) = (𝑥 + 2)𝑀𝑛−1(𝑥) −  𝑀𝑛−2(𝑥)   ,      𝑀0(𝑥) = 1 ,   𝑀1(𝑥) = 𝑥 + 1  …(2) While the explicit form is       𝑀𝑛(𝑥) = ∑ (𝑛+𝑘−12𝑘)  𝑥𝑘𝑛−1𝑘=0  ,n ≥ 1            …(3) using.(2) & (3), the first six Morgan-Voyce polynomial are  𝑀0(𝑥) = 1 ,    𝑀1(𝑥) = 𝑥 + 1, 1Kashiem: Morgan- Voyce Approach for Solution Bratu ProblemsPublished by Scholarworks@UAEU, 2021  𝑀2(𝑥) = 𝑥2 + 3𝑥 + 1,   𝑀3(𝑥) = 𝑥3 + 5𝑥2 + 6𝑥 + 1,   𝑀4(𝑥) = 𝑥4 + 7𝑥3 + 15𝑥2 + 10𝑥 + 1,   𝑀5(𝑥) = 𝑥5 + 9𝑥4 + 28𝑥3 + 35𝑥2 + 15𝑥 + 1 ,   𝑀6(𝑥) = 𝑥6 + 11𝑥5 + 45𝑥4 + 84𝑥3 + 70𝑥2 +21𝑥 + 1, 3- Important Properties of Morgan-Voyce polynomial: - [13]       In this section, some important properties of M.V were discussed      1. Generating function  𝑀(𝑥, 𝑡) = ∑ 𝑀𝑛(𝑥)𝑡𝑛 = (1 − 𝑡)∞0 [1 − (𝑥𝑡 + 2𝑡 −𝑡2)]−1         2. Zeros  𝑀𝑛(𝑥): 𝑥𝑟 = −4 𝑠𝑖𝑛2 [2𝑟 − 12𝑛 + 1∗ 𝜋2],           𝑟 = 1,2, … , 𝑛      3. Orthogonaltiy:-            𝑀𝑛(𝑥) is orthogonal polynomial over (-4,0) with to weight function √−𝑥𝑥+4     4. Differential Equations:-  𝑀𝑛(𝑥) = 𝑥(𝑥 + 4)?̈? + 2(𝑥 + 1)?̇? − 𝑛(𝑛 + 1)𝑢 = 0      5. Simson Formula:-              𝑀𝑛+1(𝑥)𝑀𝑛−1(𝑥) − 𝑀𝑛2(𝑥) = 𝑥 4. Approach of Morgan-Voyce:-      In this section, Morgan-Voyce polynomial are used to obtain the approximate solution of Bratu problem which is given in eq(1). we first approximate u(x) as:-                   𝑢(𝑥) ≅ 𝐶𝑖𝑀𝑖(𝑥)                          …(4) where 𝐶𝑖,𝑠are unknown coefficients and 𝑀𝑖(𝑥) are Morgan-Voyce polynomial, let 𝑥0, 𝑥1, … , 𝑥𝑛  be n+1 grid points in the interval [a,b] so that , 𝑥𝑖 =𝑎 + 𝑖ℎ. Then by substituting the Eq(4) in eq(1) we get:-    ∑ 𝐶𝑖𝑀𝑖(𝑥)̈𝑛𝑖=0 +  𝜆 exp ( ∑ 𝐶𝑖𝑀𝑖(𝑥)) = 0𝑛𝑖=0 ,             𝑖 = 0,1, … , 𝑛                      …(5) and by substitut the boundary conditions unknown coefficients 𝐶𝑖 will be reduced, finally by using discretization collocation point, a nonlinear system equation will be obtained that will be solved by Newton iteration method using Matlab program. 5. Numerical Examples:-         In this section, we explain the applicability and accuracy of the proposed method in three numerical examples. Example 1:-    Consider the general form of Bratu equation           ?̈? + 𝑒𝑢 = 0 ,    0 < 𝑥 < 1,           𝑢(0) = 𝑢(1) = 0                                         … (6)       Exact solution 𝑢(𝑥) =  −2 ln [𝑐𝑜𝑠ℎ(𝑥−12)𝜃2𝑐𝑜𝑠ℎ𝜃4],        𝜃 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝜃 − 2√𝜆𝑐𝑜𝑠ℎ𝜃4  first we assume  𝑢(𝑥) = ∑ 𝐶𝑖𝑀𝑖(𝑥)5𝑖=0                                   …(7) then by deriving Eq(7) twice with respect to x and substituting the initial conditions ,we obtain  ?̈? and by substituting eq(6) and using collocation point we get system with 4 unknown coefficients. After solving this system by Newton iteration method the approximate solution of Eq.(6) will be found. Table.1 and fig.1 explain the comparison between the exact and approximate solutions.               2Emirates Journal for Engineering Research, Vol. 26 [2021], Iss. 2, Art. 3https://scholarworks.uaeu.ac.ae/ejer/vol26/iss2/3Table 1: Exact and Approximate results of Example 1 x Exact Approximate Absolute Error 0.1 0.049847 0.049765 0.000092 0.2 0.089189 0.089199 0.00001 0.3 0.117609 0.117600 0.000009 0.4 0.134790 0.134889 0.000099 0.5 0.140539 0.140545 0.000006 0.6 0.134790 0.134779 0.000011 0.7 0.117609 0.117607 0.000002 0.8 0.089189 0.089189 0.0000000 0.9 0.0498468 0.049837 0.000011  Fig.1                     Example 2:-    Consider the other Bratu problem  𝜆 = 2          ?̈? + 2𝑒𝑢 = 0 ,                 𝑢(0) = 𝑢(1) =0                                                                       with the exact solution 𝑢(𝑥) =  −2 ln(cos(𝑥)) the comparison between the exact and approximate solution for N=8 presented by table 2 and fig. 2.  Example 3:-[14]    Finally consider the general form of Bratu problem with 𝜆 = 𝜋2 ?̈? − 𝜋2𝑒𝑢 = 0 , 0 < 𝑥 < 1, 𝑢(0) = 𝑢(1) = 0. . (6) The exact solution 𝑢(𝑥) = − ln [1 + cos (12+ 𝑥)𝜋]    3Kashiem: Morgan- Voyce Approach for Solution Bratu ProblemsPublished by Scholarworks@UAEU, 2021The absolute error between the exact and approximate solution with N=6 is shown in Table 3 and Fiq.3.   Table 2 :Computed absolute errors for example 2 x Exact Approximate Absolute Error 0.1 0.114411 0.114352 0.000059 0.2 0.206419 0.206418 0.000001 0.3 0.273879 0.273886 0.000007 0.4 0.315089 0.315087 0.000002 0.5 0.328952 0.328951 0.000001 0.6 0.315089 0.315088 0.000001 0.7 0.273879 0.273877 0.000011 0.8 0.206419 0.206418 0.000001 0.9 0.114411 0.114421 0.00001  Fig.2                           4Emirates Journal for Engineering Research, Vol. 26 [2021], Iss. 2, Art. 3https://scholarworks.uaeu.ac.ae/ejer/vol26/iss2/3 Table 3 : absolute errors for example 3 x Exact Approximate Absolute Error 0.1 0.396640 0.396632 0.000008 0.2 0.886211 0.886222 0.000011 0.4 3.017089 3.017088 0.000001 0.6 3.017089 3.017087 0.000001 0.8 0.886211 0.886161 0.00005 0.9 0.396640 0.396592 0.000048  Fig.3                    6. Conclusion:-    Approximate solution for Bratu problem by using Morgan- Voyce approach has been obtained. This method decreased into a system of nonlinear algebraic equations which solved by Newton iteration method. Numerical examples have been offered applicability and validity of the technique. References:- 1. S. Chandrasekhar, Introduction to the Study of Stellar Structure. (Dover, New York, 1967) 2. C.J. Zhou, Y. Li, S.W. Yao, J.H. He, Silkworm-based silk fibers by electrospinning. Results Phys. 15, 102646 (2019) 3. R. Buckmire, Application of a Mickens finite-difference scheme to the cylindrical 5Kashiem: Morgan- Voyce Approach for Solution Bratu ProblemsPublished by Scholarworks@UAEU, 2021Bratu-Gelfand problem, Numer. Methods Partial Differ. Equ. 20 (2004) 327-337.  4. E. Bushra ,Hermite Wavelet Approach to Estimate Solutionfor Bratu ́S Problem. Emirates Journal for Engineering, Vol. 24 : Iss. 2 , Article 2,(2019). 5. EL-Gamel M, Adel W, EL-Azab MS ,Collocation Method Based on Bernoulli Polynomial and Shifted Chebychev for Solving the Bratu Equation. 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Some Relations Between Kekule Structure and Morgan-Voyce Polynomials. Iranian Journal of Mathematical Chemistry, 8(2), 221-229. , (2017). 13. M. N. S. Swamy Rising Diagonal Polynomials Associated with Morgan-Voyce Polynomials , Fibonacci Quarterly 38(1),2000, 61-70, AMS Classification Available from: https://www..fq.math.ca/Scanned/38-1/swamy2.pdf.1999. 14.  S. Deniz and N. Bildik, Optimal perturbation iteration method for Bratu-type problems, Journal of King Saud University Science, 30 (2018), 91–99.                   6Emirates Journal for Engineering Research, Vol. 26 [2021], Iss. 2, Art. 3https://scholarworks.uaeu.ac.ae/ejer/vol26/iss2/3

Morgan- Voyce Approach for Solution Bratu Problems

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Morgan- Voyce Approach for Solution Bratu Problems

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