In this paper, we present a new method for solving nonlinear general two point boundary value problems. A method based on finite differences and rational function approximation and we call this method as rational approximation method. A rational approximation method is applied to construct the numerical solution for two point boundary value problems. The novel method is tested on three model problems. Thus the numerical results obtained for these model problems show the performance and efficiency of the developed method

P.K. Pandey,

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Sains Malaysiana 43(7)(2014): 1105–1108  Rational Finite Difference Approximation of High Order Accuracy for Nonlinear Two Point Boundary Value Problems(Penghampiran Beza Terhingga Rasional Ketepatan Peringkat Tinggi untuk Masalah Nilai Dua Titik Sempadan Tak Linear)P.K. PANDEY*ABSTRACTIn this paper, we present a new method for solving nonlinear general two point boundary value problems. A method based on finite differences and rational function approximation and we call this method as rational approximation method. A rational approximation method is applied to construct the numerical solution for two point boundary value problems. The novel method is tested on three model problems. Thus the numerical results obtained for these model problems show the performance and efficiency of the developed method.Keywords: Boundary value problems; fourth-order method; rational approximationABSTRAKDalam kertas ini, kami memberikan kaedah baru bagi menyelesaikan masalah nilai dua titik sempadan tak linear umum. Kaedah berdasarkan beza terhingga dan penghampiran fungsi rasional ini dikenali sebagai kaedah penghampiran rasional. Kaedah penghampiran rasional yang digunakan untuk membina penyelesaian berangka bagi masalah dua titik sempadan. Kaedah baru ini diuji pada tiga model masalah. Oleh itu keputusan berangka yang diperoleh bagi masalah model ini menunjukkan prestasi dan keberkesanan kaedah yang dibangunkan.Kata kunci: Kaedah peringkat empat; masalah nilai sempadan; penghampiran rasionalINTRODUCTIONThe two point boundary value problems occur in all branches of sciences and technology. The majority of such problems are nonlinear and complicated. In general, analytical solution of these problems does not exist. So it is essential to obtain a solution by numerical methods. Consider a second order boundary value problem of the form: y" = f (x, y, y'),         a ≤ x ≤ b, (1)subject to boundary conditions y(a) = α and y(b) = β, (2)are real finite constants, y(x), f (x, y, y') ∈ [a, b]. The existence and uniqueness of the solution, for the problem (1) is assumed. The specific theorems which list the conditions for existence and uniqueness of the solution of such problems are discussed in Baxley (1981), Keller (1968) and Stoer and Bulirsch (1991). The literature of numerical analysis contains much on the solution of the second order boundary value problems in Collatz (1966), Jain (1984), Keller (1968) and Lambert (1991). In this article we emphasized our self on iterative method only. The iterative methods for the numerical solution of the problem (1) are proposed in Chawla (1978) and Tirmizi and Twizel (2002). However these methods are based on polynomial functions, which are normally smooth and with sufficient continuous derivatives. Thus these methods are exact for some degree of the polynomials. On the other hand a new approach reported in literature for first order initial value problems and based on non-polynomial functions (Okosun & Ademiluyi 2007; Ying & Yaacob 2013) and references there in.  In this paper, a new class of novel nonstandard rational finite difference method (Pandey 2013), the advancement of idea based on non-polynomial approximation (Mickens 1994; Ramos 2007) will be used for solving nonlinear two point boundary value problems. Newton-Raphson method is considered as a procedure for solving the nonlinear equation. It provides a promising tool in iterative solution field and open up almost new concept in numerical method. The next section gives ways for constructing the proposed method. In the section that follows we discuss the local truncation error and in the subsequent section, the numerical results obtained for model problems. In final section some discussions and conclusion. RATIONAL APPROXIMATION METHODLet set a uniform mesh xi = ih, i = 0, 1, … …, N – 1, N, and Nh = (b – a) in [a, b]. Let’s denote with yi, an approximation of y(x), yi' = y'(xi), fi = f (xi, yi, yi').1106  In this section, we discuss the derivation of the fourth order rational difference method for the problem (1) at the point x = xi, when f is function of x and y only i.e. f (x, y)as in Ying and Yaacob (2013), then rational difference method is, yi+1 – 2yi + yi+1 =   (3) In order to discretize problem (1) at = xi, we use the following O(h2)approximations as used by Chawla (1978),  (4.1)  (4.2)  (4.3) So if we define and set a forcing function as in Chawla (1978),  (5.1)  (5.2)  (5.3) It follows from (4.1)-(4.3), that  will provide O(h2) approximation for fi, fi+1, fi–1, respectively.Define   (6)where ‘a’ is free parameter to be determined. We need O(h4) approximation for function f (xi, yi, yi'), so set and define a forcing function as,  (7)It follows from (5) and (6), that  (8)It follows from (8) that will provide an for if  12a + 1 = 0. (9)Thus the given problem (1), at central point x = xi may be discretized as: yi+1 – 2yi + yi – 1 =  (10) Thus from (9), we conclude that order of proposed method (10) is four.LOCAL TRUNCATION ERRORThe local truncation error of the proposed difference method (10), as in Pandey, (2013) and Jain et al. (1987) may be written as:  (11) Let us define  and substitute in (11), we will obtain:  Thus from (8-9) we have    (12) Expanding each term on the right side of (12) in Taylor series about point yi, so we have,  (13)  Assuming that  then by application of binomial theorem on right side of (18) and simplify, we will get,    (14)Thus from (14), the order of the method (10) is four.NUMERICAL RESULTSIn this section, the proposed method (10) is applied to solve three different model problems. All computations were carried out using double precision GNU FORTRAN   1107language. Also Chawla’s method (1978) is applied to solve these model problems. Let yi an approximate value of the theoretical solution y(x) at the point x = xi. Maximum/Minimum absolute errors are shown for different step lengths h, in Tables 1-3 .Example 1Consider the equation y" = y3 – yy',      1 ≤ x ≤ 2,subject to the boundary condition y(1) = y(2) =  The exact solution of the problem is y(x) =  In Table 1, maximum / minimum absolute errors of the present method (10) and Chawla’s method presented.Example 2Consider the equation subject to the boundary condition y(0) = 1, y(1) =  Let solution of the problem is y(x) =  In Table 2, maximum / minimum absolute errors of the present method (10) and Chawla’s method presented.Example 3Consider the equation subject to the boundary condition y(0) = 0, y(1) = –3. Let solution of the problem is  y(x) = 1 – (x2 + 1)2. In Table 3, maximum/minimum absolute errors of the present method (10) and Chawla’s method presented.CONCLUSIONIn this article, a non-classical rational method of order four for numerical solution of general two point boundary value problems was described. In general each numerical method has its own merit and demerit in its application. The present method was based on non- polynomial approximation. In the development of the method extra assumption used than TABLE 3. Maximum absolute error in for example 3Errors N4 8 16 32 (10) MAE .23446083(-2) .45490265(-3) .53524971(-4) .47683716(-5)MIE .14810413(-2) .10676309(-3) .22007152(-5) .35623088(-7)Chawla’s MAE .74594378(0) .14065456(0) .14402390(-1) .10793209(-2)MIE .43773103(0) .19868940(-1) .12140172(-2) .76146564(-4)TABLE 2. Maximum absolute error in y(x) =  for example 2Errors N8 16 32 64 (10) MAE .12934449(-3) .61587798(-5) .25958715(-6) .29380638(-7)MIE .27492788(-4) .31613180(-5) .16825087(-8) .16825087(-8)Chawla’s MAE .21770765(-6) .74650984(-7) .14037788(-6) .21254630(-6)MIE .44558302(-7) .17713894(-9) .18189110(-9) .16825087(-8)TABLE 1. Maximum absolute error y(x) =  in for example 1Errors N4 8 16 32(10) MAE .25099516(-4) .56059739(-5) .62916013(-6) .32939408(-7)MIE .15686859(-4) .18542228(-6) .10842972(-6) .32045508(-9)Chawla’s MAE .26351214(-4) .56655786(-5) .62916013(-6) .32939408(-7)MIE .16431917(-4) .18840251(-5) .10842972(-6) .32045508(-9)1108 that of finite difference method which may be considered as demerit of the method. Order of the method was estimated from numerical results and local truncation error. Method is computationally efficient, accurate and effective which can be observed in numerical results obtained in our experiments. It was observed from the results presented in table that method has higher accuracy i.e. small discretization error which may be considered as merit of the method. Thus we can say that the present method (10) seems competitive with other finite difference method. The range of absolute errors can be clearly seen from these tables. It is observed from computational results that max/min errors occur near boundary points. Our future works will deal with how to improve the accuracy of the method and to minimize the variation in max and min errors. The method may therefore be found useful in the solution of the problems, the result required to be accurate in term of minimum error. May it possible in future, a better results can be obtained using computing software other than GNU FORTRAN.REFERENCESBaxley, J.V. 1981. Nonlinear Two Point Boundary Value Problems. In Ordinary and Partial Differential Equations, edited by Everrit, W.N. & Sleeman, B.D. New York: Springer-Verlag. pp. 46-54.Chawla, M.M. 1978. High-accuracy tridiagonal finite difference approximation for nonlinear two point boundary value problems. J. Inst. Maths. Applics 22: 203- 209.Collatz, L. 1966. The Numerical Treatment of Differential Equations. New York: Springer- Verlag.Jain, M.K. 1984. Numerical Solution of Differential Equations (2/e). New Delhi: Wiley Eastern Limited. Jain, M.K., Iyenger, S.R.K. & Jain, R.K. 1987. Numerical Methods for Scientific and Engineering Computation (2/e). New Delhi: Wiley Eastern Ltd.Keller, H.B. 1968. Numerical Methods for Two Point Boundary Value Problems. Waltham, Mass.: Blasdell Publ. Co. Lambert, J.D. 1991. Numerical Methods for Ordinary Differential Systems. England: John Wiley. Mickens, R.E. 1994. Nonstandard Finite Difference Models of Differential Equations. Singapore: World Scientific. Okosun, K.O. & Ademiluyi, R.A. 2007. A three step rational methods for integration of differential equations with singularities. Research Journal of Applied Sciences 2(1): 84-88.Pandey, P.K. 2013. A non-classical finite difference method for solving two point boundary value problems. Pacific Journal of Science and Technology 14(2): 147-152.Ramos, H. 2007. A nonstandard explicit integration scheme for initial value problems. Applied Mathematics and Computation 189(1): 710-718.Stoer, J. & Bulirsch, R. 1991. Introduction to Numerical Analysis (2/e). Berlin Heidelberg: Springer-Verlag. Tirmizi, I.A. & Twizel, E.H. 2002. High order finite difference methods for nonlinear second order two point boundary value problems. Appl. Math. Lett. 15: 897-902.Ying, T.Y. & Yaacob, N. 2013. One-step exponential-rational methods for the numerical solution of first order initial value problems. Sains Malaysiana 42(6): 845-853.Department of MathematicsCollege of Applied SciencesPB No. 1905, PC 211, SalalahSultanate of OmanDepartment of MathematicsDyal Singh College (Univ. of Delhi)Lodhi RoadNew Delhi - 110003India*Corresponding author; email: pramod_10p@hotmail.comReceived:  16 May 2013Accepted:  16 September 2013

Rational finite difference approximation of high order accuracy for nonlinear two point boundary value problems

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