Applied Mathematics, 2011, 2, 959964 doi:10.4236/am.2011.28132 Published Online August 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Numerical Treatment of Nonlinear Third Order Boundary Value Problem Pankaj Kumar Srivastava, Manoj Kumar Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad, India Email: panx.16j@gmail.com, manoj@mnnit.ac.in Received January 25, 2011; revised June 20, 2011; accepted June 27, 2011 Abstract In this paper, the boundary value problems for nonlinear third order differential equations are treated. A ge neric approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods. The algorithm is tested on a problem to demonstrate the practical usefulness of the approach. Keywords: Nonlinear Third Order Boundary Value Problem, Nonpolynomial Quintic Spline, Draining and Coating Flows 1. Introduction Engineering problems that are timedependent are often described in terms of differential equations with condi tions imposed at single point (initial/final value prob lems); while engineering problems that are position de pendent are often described in terms of differential equa tions with conditions imposed at more than one point (boundary value problems). Boundary value problems are encountered in many engineering fields including optimal control, beam deflections, heat flow, draining and coating flows, and various dynamic systems. In this paper we are concerned with general thirdorder nonlin ear boundary value problems, such problems arise in the study of draining and coating flows. Many authors have studied and solved such type of third order boundary value problems with various types of boundary condi tions. A. Khan and Tariq Aziz [1] solved a thirdorder linear and nonlinear boundary value problem of the type ,, xfxyax b 3 (1) Subject to 12 ,, akyak ybk (2) by deriving a fourth order method using polynomial quintic splines. S. Valarmathi and N. Ramanujam [2], T. Y. Na [3], N. S. Asaithambi [4], Xueqin Li and Minggen Cui [5] and N. H. Shuaib et al. [6] used some other computational methods for solving boundary value pro blems for thirdorder ordinary differential equations. A. Cabada et al. [7] studied external solutions for third order nonlinear problems with upper and lower solutions in reversed order. In this paper Nonpolynomial quintic spline functions are applied to obtain a numerical solution of the follow ing nonlinear third order twopoint boundary value pro blems 2 ,, ,0,1,yfxyyygxyx 3 A (3) subject to the boundary conditions: 11 12 1,0 ,1yAy Ay 3 (4) where ,1,2, i Ai are finite real constants. The existence theorems for the solution of (3) sub jected to boundary condition (4) are derived by Xueqin Li and Minggen Cui [5]. M. Pei et al. and F. M. Minhos [8,9] also discussed the existence of nonlinear third order boundary value problems. Xueqin Li and Minggen Cui consider the problem in the reproducing kernel Hilbert space 10,1W. In order to derive the existence theorems for solution of (3) they make the following assumptions : 3 (1) ,,,0,1 fxyzw XR is completely continu ous function; (2),,, ,,,, ,,,, xy fxyzwfxyzw fxyzw and ,,, w xyzw are bounded; (3) ,,,0Hfxyzw on 3 0,1 R,
960 P. K. SRIVASTAVA ET AL. where 1 ,,, 0,1fxyzw W as 10,1 ,yyx W 10,1 ,zzx W 10,1 ,wwx W 01, ,,xyzw Here the reproducing kernel Hilbert space 10,1W is the inner product space 10,1W which is defined by, 1 2 0,1: 0,1,0,1 Wuxuxis absolutely continuous real valuefunction inuxL The inner product and norm in 10,1W are given, respectively, by 1 1 0 ,00 W uxvxuvu xvxx d 1 1/2 , W uuxux M. Cui et al. and C. I. Li et al. [10,11] had proved that 10,1W is a complete reproducing kernel space. That is, there exists a reproducing kernel function 10,1 ,0,1 x Qy Wy for each fixed 0,1x and any 10,1uy W, such that The reproducing kernel can be denoted by ,x uy Qx y 1 W y u x Q 1, 1, x yyx Qy , . yx In the present paper, our main objective is to apply nonpolynomial quintic spline function [1214] that has a polynomial and trigonometric parts to develop a new numerical method for obtaining smooth approximations to the solution of nonlinear thirdorder differential equa tions of the system of form (3) subjected to (4). Here algorithms are developed and the approximate solutions obtained by these algorithms are compared with the solu tions obtained by iterative method [5]. The paper is or ganized as follows—In Section 2, we have given a brief introduction of nonpolynomial quintic spline. In Section 3, we give a brief derivation of this nonpolynomial quintic spline. We present the spline relations to be used for discretization of the given system (3). In Section 4, we present our numerical method for a system of non linear thirdorder boundaryvalue problems and develop ment of boundary conditions, truncation error and class of the method are discussed, in Section 5, numerical evidence is included to compare and demonstrate the efficiency of the methods, in which we have shown that our algorithm performs better than an iterative method. Finally, in Section 6 we have concluded the paper with some remarks. 2. Nonpolynomial Quintic Spline A quintic spline function , interpolating to a func tion Sx ux defined on [, is such that ]ab 1) In each subinterval 1 [,] j x , is a poly nomial of degree at most five. Sx 2) The first, second, third and fourth derivatives of Sx are continuous on . [,]ab To be able to deal effectively with such problems we introduce “spline functions” containing a parameter . These are “nonpolynomial splines” defined through the solution of a differential equation in each subinterval. The arbitrary constants are being chosen to satisfy cer tain smoothness conditions at the joints. These “splines” belong to the class and reduce into polynomial splines as parameter 2 C 0 . The exact form of the spline depends upon the manner in which the parameter is introduced. We have studied parametric spline func tions: spline under compression, spline under tension and adaptive spline. A number of spline relations have been obtained for subsequent use. A function ,Sx of class which inter polate 4[,]Cab ux at the mesh points {} depends on a parameter , reduces to ordinary quintic spline Sx in as [,ab]0 is termed as parametric quintic spline function. The three parametric quintic splines de rived from quintic spline by introducing the parameter in three different ways are termed as “parametric quintic splineI”, “parametric quintic splineII” and “adaptive quintic spline”. The spline function we propose in this paper has the following form 23 23 2345 span1, ,,,sin,cos, orspan 1,,,,sinh,cosh, orspan1,,,,,, where0 xx xxx xxx x xx xxx The above fact is evident when correlation between polynomial and nonpolynomial splines basis is investi gated in the following manner: 23 5 2 23 4 3 5 span1,,,,sin(),cos() , 24 span1, ,,,cos1, 2 120 sin 6 Txxxxx x xx xx x xx Copyright © 2011 SciRes. AM
P. K. SRIVASTAVA ET AL. 961 From the above equation it follows that 2345 05 lim1, ,,,,Txxxxx where is the frequency of the trigonometric part of the splines function which can be real or pure imaginary and which will be used to raise the accuracy of the method. This approach has the advantage over finite dif ference methods that it provides continuous approxima tion to not only for y , but also for ,yy and higher derivatives at every point of the range of integra tion. Also, the differentiability of the trigonomet ric part of nonpolynomial splines compensates for the loss of smoothness inherited by polynomial splines in this paper. C 3. Development of the Method Without loss of generality in order to develop the nu merical method for approximating solution of a differen tial Equation (3), we consider a uniform mesh with nodal points i on ,ab such that 0123 :N axx xxxb ,0,1,2,, i aih iN where, ba hN . Let us consider a nonpolynomial function Sx of class 4,Cab which interpolates x at the mesh points i , depends on a parameter 0,1, 2,,,iN , and reduces to ordinary quintic spline in as x[,S ]ab 0. For each segment the nonpolynomial, , define by 1 [,],0,1, 2,,1, ii xx iN x S 23 sin cos 0,1, 2,,1 ii ii iii iii i Sx abxxcxxdxx exxfxx iN (5) where and ,,, , iii ii abcdei are constants and is arbitrary parameter. Let i be an approximation to y i x, obtained by the segment of the mixed splines function pass ing through the points Sx , ii y and 11i , i y , to ob tain the necessary conditions for the coefficients intro duced in (5), we do not only require that xS satis fies interpolatory conditions at i and 1i , but also the continuity of first, second and third derivatives at the common nodes , ii y are fulfilled. To derive expression for the coefficients of (5) in terms of ,,, , i y1i y,1 , ii i DD T 1i Ti and1i we first denote: 11 1 (3) (3) 11 (4) (4) 11 , , , , iii i iiii iii i iiii Sx ySxy Sx DSxD SxT SxT Sx FSxF 1 (6) From algebraic manipulation we get the following ex pression: 4 1 3 11 2 1 1 4 4 , cos , sin 2, 2 cossin , 6(1cos ) cos , sin . i ii ii ii iii i ii i i ii i i i F ay FF bD yyy ch TT F d FF e F f (7) where h and 0,1,2,,1iN . Using the continuity of the first and third derivatives at , ii , y that is ii ii SxSx and i x ii Sx i S , we obtain the following relations: 21 1 3 11 1 2 11 11 33 1 2, iiii iiii i iii ii yyyy h hF T TTT h TTThFFi N 111 (8) The operator Λ is defined by for any function 22 11iii ii wpwwqw wsw i for any function evaluated at the mesh points. Then we have the following relations connecting and its derivatives: w y 1) i T 22 11 3 124 ii ii yy yy h (9) 2) 4 4 1 ii y h where 1, 6 p 11 1 22 6 q , Copyright © 2011 SciRes. AM
962 P. K. SRIVASTAVA ET AL. 11 2 2 12 1 24 2 6 11, 11, 11 , 3 , ii s csc cot h TSx , and (4) ii Sx . 4. Description of the Method and Development of Boundary Conditions At the mesh point i the proposed differential equation 2 ,,, '',,yfxyyygxyxab (10) subjected to boundary conditions (4), may be discretized by 2 ii ii Tfgy (11) where and i TSx i ii gx. Using the spline relation (9) (i), in (11) we have 2 21 22 2221 3222 2 22 1111 2 22 24 24 ,212 ii iiii i i iiiiiii ii yy hqf ph fyhpfqfsf hpgyqgysg yqgy pg yiN 11ii y (12) To obtain unique solution we need two more equations to be associated with (12) so that we use the following boundary conditions: 1) To obtain the secondorder boundary formula we define: 1234 3 2332 33 ,1 yy yy hy yhyyi 321 3 211 2 33 , 1 NNNN NNN N yyyy hyyhyy iN (13) for any choice ofand,12α . Using Equation (3) we have: 1234 33 2233 32 3 23 33 ii yy yy hfhfhfh f hhgy hhgy 2 321 33 2233 32 3 22 1 33 NNNN NNNN NN NN yyyy hfh fhfhf hhgyhhgy 2 1 2) To obtain the fourthorder boundary formula we define: 1234 3 233 2 321 3 211 2 33 ,1, 28 33 ,1 28 NNNN NNNN yy yy hh yyyyi yyyy hh yyy yiN (14) for any choice ofand,12α . Using Equation (3) we have: 1234 33 2233 33 22 23 33 2828 28 28 ii yy yy hhhh ffff hh hh yg y 321 33 2211 33 22 22 1 33 2828 28 28 NNNN NNNN NN NN yyyy hhhh ffff hh hh 2 yg y By expanding (8) in Taylor series about xi, we obtain the following local truncation error: 4 4 6 6 8 8 9 193 4 6 11 33 316 1806 12 11 1613 274 131040 360 ii i i Tpqhy pqhy pqhy Oh (15) For any choice of α and β, provided that 12 . Remark 1): Secondorder method For 14, 14 , Copyright © 2011 SciRes. AM
P. K. SRIVASTAVA ET AL. 963 and 0.04063489941134321703,p 0.25412730690212937985, 0.41047570631347259688. q s gives 4. i TOh Remark 2): Fourthorder method For 11 12 ,, , 6312012 pq 6 0 , and 66 120 s, gives . 6 i TOh Clearly, the family of numerical methods is described by the Equation (12), boundary equations and the solu tion vector , T denoting transpose, is 12 ,,, T N Yyy y obtained by solving a nonlinear algebraic system of or der N. To ensure cost effectiveness, better accuracy and sim ple applicability of the new method, the best way is to find the unknown parameters α and β, which are the ex pressions containing the actual parameter . The hall mark of the new approach is that it gives family of fourth and secondorder methods by running the code once and also skips the multiplications involved in the expressions α and β. 5. Numerical Example We now consider a numerical example illustrating the comparative performance of nonpolynomial quintic spline algorithms over an iterative method [5]. Example: Consider the boundary value problem ()x yxyxyxxyxy 2 x 0 (16) under the boundary condition 101yy y (17) The analytic solution of (16) is 1yx xx (18) Nonpolynomial Quintic Spline Solution of Example The maximum observed errors (in absolute value) by our algorithm (of second order) and iterative method (Xueqin Li et al. [5]) for the example considered are presented in Table 1. 6. Discussion and Conclusions In this paper we used a nonpolynomial Quintic spline function to develop numerical algorithms of system of nonlinear third order boundary value problems. Here the result obtained by our algorithm is better than that ob tained by some other method as compared in Tables 1 Table 1. Comparison of our algorithm of second order with iterative method. (Maximum Absolute Error) (Our Method) Node (Maximum Abso lute Error) (By Xueqin Li et al. [5]) N = 10 N = 10 N = 20 0.1 4.24272E04 2.52718E04 6.01631E05 0.2 1.97294E04 1.25174E04 3.2713E05 0.3 3.66041E04 2.52833E04 5.3529E05 0.4 1.020626E03 2.62819E04 6.52289E05 0.5 1.52747E03 3.82917E04 8.82427E05 Table 2. Comparison of our algorithm of fourth order with iterative method. (Maximum Absolute Error) (Our Method) Node (Maximum Abso lute Error) (By Xueqin Li et al. [5]) N = 10 N = 10 N = 20 0.1 4.24272E04 6.28192E05 3.58192E06 0.2 1.97294E04 4.62816E05 2.23147E06 0.3 3.66041E04 3.52842E05 2.18372E06 0.4 1.020626E03 8.16252E05 4.98326E06 0.5 1.52747E03 1.93165E05 1.28429E06 and 2. The approximate solutions obtained by the present algorithms are very encouraging and it is a powerful tool for solution of nonlinear third order boundary value problems. 7. References [1] A. Khan and T. Aziz, “The Numerical Solution of Third Order BoundaryValue Problems Using Quintic Splines,” Applied Mathematics and Computation, Vol. 137, No. 23, 2003, pp. 253260. doi:10.1016/S00963003(02)000516 [2] S. Valarmathi and N. Ramanujam, “A Computational Method for Solving Boundary Value Problems for Third —Order Singularly Perturbed Ordinary Differential Equations,” Applied Mathematics and Computation, Vol. 129, No. 23, 2002, pp. 345373. doi:10.1016/S00963003(01)000443 [3] T. Y. Na, “Computational Method in Engineering Bound ary Value Problems,” Academic Press, New York, 1979. [4] N. S. Asaithambi, “A Numerical Method for the Solution of the Falkner Equation,” Applied Mathematics and Com putation, Vol. 81, No. 23, 1997, pp. 259264. doi:10.1016/S00963003(95)003258 [5] X. Q. Li and M. G. Cui, “Existence and Numerical Method for Nonlinear ThirdOrder Boundary Value Problem in the Reproducing Kernel Space,” Boundary Value Problems, Article ID 459754, 2010, pp. 119. [6] N. H. Shuaib, H. Power and S. Hibberd, “BEM Solution of Thin Film Flows on an Inclined Plane with a Bottom Outlet,” Engineering Analysis with Boundary Elements, Vol. 33, No. 3, 2009, pp. 388398. Copyright © 2011 SciRes. AM
P. K. SRIVASTAVA ET AL. Copyright © 2011 SciRes. AM 964 doi:10.1016/j.enganabound.2008.06.007 [7] A. Cabada, M. R. Grossinho and F. Minhos, “External Solutions for ThirdOrder Nonlinear Problems with Up per and Lower Solutions in Reversed Order,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 62, 2005, pp. 11091121. [8] M. Pei and S. K. Chang, “Existence and Uniqueness of Solutions for Third—Order Nonlinear Boundary Value Problems,” Journal of Mathematical Analysis and Appli cation, Vol. 327, 2007, pp. 2335. doi:10.1016/j.jmaa.2006.03.057 [9] F. M. Minhos, “On Some Third Order Nonlinear Bound ary Value Problems: Existence, Location and Multiplicity Results,” Journal of Mathematical Analysis and Applica tion, Vol. 339, 2008, pp. 13421353. doi:10.1016/j.jmaa.2007.08.005 [10] M. Cui and Z. Deng, “Solutions to the Definite Solutions Problem of Differential Equations in Space 2 l W,” Ad vances in Mathematics, Vol. 17, 1988, pp. 327329. [11] C. I. Li and M.G. Cui, “The Exact Solution for Solving a Class Nonlinear Operator Equations in the Reproducing Kernel Space,” Applied Mathematics and Computation, Vol. 143, No. 23, 2003, pp. 393399. doi:10.1016/S00963003(02)003703 [12] M. Kumar and P. K. Srivastava, “Computational Tech niques for Solving Differential Equations by Quadratic, Quartic and Octic Spline,” Advances in Engineering Soft ware, Vol. 39, No. 8, 2008, pp. 646653. doi:10.1016/j.advengsoft.2007.09.001 [13] M. Kumar and P. K. Srivastava, “Computational Tech niques for Solving Differential Equations by Cubic, Quintic and Sextic Spline,” International Journal for Computational Methods in Engineering Science & Me chanics, Vol. 10, No. 1, 2009, pp. 108115. doi:10.1080/15502280802623297 [14] J. Rashidinia, R. Jalilian and R. Mohammadi, “Non Polynomial Spline Methods for the Solution of a System of Obstacle Problems,” Applied Mathematics and Com putation, Vol. 188, No. 2, 2007, pp. 19841990. doi:10.1016/j.amc.2006.11.074
