Numerical Treatment of Nonlinear Third Order Boundary Value Problem

In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic 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.


Introduction
Engineering problems that are time-dependent are often described in terms of differential equations with conditions imposed at single point (initial/final value problems); while engineering problems that are position dependent are often described in terms of differential equations 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 third-order nonlinear 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 conditions.A. Khan and Tariq Aziz [1] solved a third-order linear and non-linear boundary value problem of the type 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 third-order ordinary differential equations.A. Cabada et al. [7] studied external solutions for thirdorder 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 following nonlinear third order two-point boundary value problems subject to the boundary conditions: .In order to derive the existence theorems for solution of (3) they make the following assumptions :   ( 3) , , , 0 Here the reproducing kernel Hilbert space The inner product and norm in The reproducing kernel can be denoted by In the present paper, our main objective is to apply non-polynomial quintic spline function [12][13][14] that has a polynomial and trigonometric parts to develop a new numerical method for obtaining smooth approximations to the solution of nonlinear third-order differential equations 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 solutions obtained by iterative method [5].The paper is organized 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 non-polynomial 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 nonlinear third-order boundary-value problems and development 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.

Nonpolynomial Quintic Spline
A quintic spline function , interpolating to a function , is a polynomial of degree at most five.

S x 
2) The first, second, third and fourth derivatives of To be able to deal effectively with such problems we introduce "spline functions" containing a parameter  .
These are "non-polynomial splines" defined through the solution of a differential equation in each subinterval.The arbitrary constants are being chosen to satisfy certain smoothness conditions at the joints.These "splines" belong to the class and reduce into polynomial splines as parameter or span 1, , , , , , where 0 The above fact is evident when correlation between polynomial and non-polynomial splines basis is investigated in the following manner: From the above equation it follows that   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 difference methods that it provides continuous approximation to not only for   y x , but also for , y y   and higher derivatives at every point of the range of integration.Also, the differentiability of the trigonometric part of non-polynomial splines compensates for the loss of smoothness inherited by polynomial splines in this paper.

Development of the Method
Without loss of generality in order to develop the numerical method for approximating solution of a differential Equation (3), we consider a uniform mesh  with nodal points i :

 
For each segment the non-polynomial, , define by where and , , , , , to obtain the necessary conditions for the coefficients introduced in (5), we do not only require that

 
x S  satisfies interpolatory conditions at i x and 1 i x  , but also the continuity of first, second and third derivatives at the common nodes  ,  i i x y are fulfilled.To derive expression for the coefficients of (5) in terms of , , , , i y From algebraic manipulation we get the following expression:   where h Using the continuity of the first and third derivatives at , we obtain the following relations: The operator Λ is defined by for any function for any function evaluated at the mesh points.Then we have the following relations connecting and its derivatives:

Description of the Method and Development of Boundary Conditions
At the mesh point i x the proposed differential equation subjected to boundary conditions (4), may be discretized by where and Using the spline relation (9) (i), in (11) we have 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 second-order boundary formula we define: for any choice of and , . Using Equation (3) we have: To obtain the fourth-order boundary formula we define: By expanding (8) in Taylor series about x i , we obtain the following local truncation error: For any choice of α and β, provided that

  6 i T O h 
Clearly, the family of numerical methods is described by the Equation ( 12), boundary equations and the solution vector , T denoting transpose, is obtained by solving a non-linear algebraic system of order N.
To ensure cost effectiveness, better accuracy and simple applicability of the new method, the best way is to find the unknown parameters α and β, which are the expressions containing the actual parameter  .The hall mark of the new approach is that it gives family of fourth-and second-order methods by running the code once and also skips the multiplications involved in the expressions α and β.

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 under the boundary condition The analytic solution of ( 16) is

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.

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 obtained by some other method as compared in Tables 1 are finite real constants.The existence theorems for the solution of (3) subjected 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


H f x y z w f x y z w f x y z w and   , , , w f x y z w are bounded;

2 C 0 
 .The exact form of the spline depends upon the manner in which the parameter is introduced.We have studied parametric spline functions: spline under compression, spline under tension and adaptive spline.A number of spline relations have been obtained for subsequent use. the mesh points { } j x depends on a parameter  , reduces to ordinary quintic spline  is termed as parametric quintic spline function.The three parametric quintic splines derived from quintic spline by introducing the parameter in three different ways are termed as "parametric quintic spline-I", "parametric quintic spline-II" and "adaptive quintic spline".The spline function we propose in this paper has the following form Cui et al. and C. I. Li et al. [10,11] had proved that

Table 2 . Comparison of our algorithm of fourth order with iterative method.
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.