Solution of Boundary Value Problems by Approaching Spline Techniques

Abstract

In the present work a nonpolynomial spline function is used to approximate the solution of the second order two point boundary value problems. The classes of numerical methods of second order, for a specific choice of parameters involved in nonpolynomial spline, have been developed. Numerical examples are presented to illustrate the applications of this method. The solutions (𝑢(𝑥)) of these examples are found at the nodal points with various step sizes and with various parameters (α, β). The absolute errors in each example are estimated, and the comparison of approximate values, exact values, and absolute errors of 𝑢(𝑥) at the nodal points are shown graphically. Further, shown that nonpolynomial spline produces accurate results in comparison with the results obtained by the B-spline method and finite difference method.

1. Introduction

There are many linear and nonlinear problems in science and engineering, namely, second order differential equations with various types of boundary conditions, which are solved either analytically or numerically. Numerical simulation in engineering science and in applied mathematics has become a powerful tool to model the physical phenomena, particularly when analytical solutions are not available, then very difficult to obtain. The numerical solution of two-point boundary value problems (BVPs) is of great importance due to its wide application in scientific research. Several authors like Bickley [1] and Khan [2] have considered the applications of cubic spline functions for the solution of two point boundary value problems. Detailed explanation of theory of splines is given in [3, 4]. Some of already established methods to solve the boundary value problems are shooting method, finite difference method, finite volume method, variational iteration method, and Adomian decomposition method. Chawla and Katti [5] employed finite difference method for a class of singular two-point BVPs; a class of BVPs was solved by using numerical integration [6]; Ravi kanth and Reddy dealt with cubic spline [7]; the variational iteration method was proposed originally by He [8] in 1999; Adomian et al. solved a generalization of Airy’s equation by decomposition method [9]. In the present communication we apply nonpolynomial spline functions to develop numerical method for obtaining the approximations to the solution of second order two point boundary value problem of the form −𝑑𝑝(𝑥)𝑑𝑥𝑑𝑢(𝑥)𝑢(𝑥)(𝑥)(𝑎)(𝑏)𝑑𝑥+𝑣=𝑔,𝑢=𝑢=0,𝑝∈𝑐^′[𝑎,𝑏],𝑝>0,𝑔,𝑣∈𝑐[𝑎,𝑏]. (1)

This type of problem (by missing the term containing 𝑢(𝑥)) is proposed by the authors in [10, 11]. Numerical solution of (1) based on finite difference, finite element, and finite volume methods has been proposed by Fang et al. [10]; Hikmet Caglar et al. [11] applied B-spline interpolation in two-point BVPs and compared results with finite difference, finite element, and finite volume methods.

Briefly, outline is as follows. In Section 2, we develop a numerical technique based on nonpolynomial spline function for solving second order linear and nonlinear two point boundary value problems (1). To demonstrate the efficiency of the method some numerical examples have been solved and compared with exact solution and also with other known methods [12] in Section 3 and conclusions have been presented in Section 4.

2. Description of the Method

We consider a uniform mesh Δ with nodal points 𝑥_𝑖  on [𝑎,𝑏] such that Δ∶𝑎=𝑥₀<𝑥₁<𝑥₂<⋯<𝑥_𝑁−1<𝑥_𝑁𝑥=𝑏,_𝑖()=𝑎+𝑖ℎ,𝑖=0,1,2,…,𝑁whereℎ=𝑏−𝑎𝑁. (2) A nonpolynomial function 𝑠_Δ(𝑥) of class 𝑐²[𝑎,𝑏] which interpolates 𝑢(𝑥) at the mesh points 𝑥_𝑖, for 𝑖=0,1,…,𝑁, depends on a parameter 𝜏 and reduces to ordinary cubic spline in [𝑎,𝑏] as 𝜏→0. The spline function we propose has the following form: 𝑇₃=Span{1,𝑥,cos𝜏𝑥,sin𝜏𝑥}, where 𝜏 is the frequency of the trigonometric part of the spline function which can be real or pure imaginary and which will be used to raise the accuracy of the method.

When correlation between polynomial and nonpolynomial spline basis functions are investigated in the following manner: 𝑇₃{}2=Span1,𝑥,cos𝜏𝑥,sin𝜏𝑥=Span1,𝑥,𝜏²(),61−cos𝜏𝑥𝜏³(𝜏𝑥−sin𝜏𝑥). (3) It follows that lim_𝜏→0𝑇₃={1,𝑥,𝑥²,𝑥³}. Thus in each subinterval 𝑥_𝑖≤𝑥≤𝑥_𝑖+1, we have Span{1,𝑥,cos𝜏𝑥,sin𝜏𝑥}orSpan1,𝑥,𝑥²,𝑥³(when𝜏⟶0) (4)

For each segment [𝑥_𝑖,𝑥_𝑖+1], 𝑖=0,1,2,…,𝑁−1 the non-polynomial 𝑠_Δ(𝑥) has the following form: 𝑠_Δ(𝑥)=𝑎_𝑖+𝑏_𝑖𝑥−𝑥_𝑖+𝑐_𝑖sin𝜏𝑥−𝑥_𝑖+𝑑_𝑖cos𝜏𝑥−𝑥_𝑖,𝑖=0,1,2,…,𝑁, (5) where 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, and 𝑑_𝑖 are constants and 𝜏 is a free parameter. Let 𝑢_𝑖 be an approximation to 𝑢(𝑥_𝑖), obtained by the segment 𝑠_Δ(𝑥) of mixed spline function passing through the points (𝑥_𝑖,𝑢_𝑖) and (𝑥_𝑖+1,𝑢_𝑖+1). To obtain the necessary conditions for the coefficients introduced in (5), we not only require 𝑠_Δ(𝑥) to satisfie interpolate conditions at 𝑥_𝑖 and 𝑥_𝑖+1, but also the continuity of first derivative at the common nodes (𝑥_𝑖,,𝑢_𝑖) to be fulfilled.

To derive an expression for the coefficients of (5) in terms of 𝑢_𝑖, 𝑢_𝑖+1, 𝑀_𝑖, 𝑀_𝑖+1, we first denote 𝑆_Δ𝑥_𝑖=𝑢_𝑖,𝑆_Δ𝑥_𝑖+1=𝑢_𝑖+1,𝑆_Δ^′′𝑥_𝑖=𝑀_𝑖,𝑆_Δ^′′𝑥_𝑖+1=𝑀_𝑖+1. (6) From algebraic simplification we get the following expression: 𝑎_𝑖=𝑢_𝑖+𝑀_𝑖𝜏²,𝑏_𝑖=𝑢_𝑖+1−𝑢_𝑖ℎ+𝑀_𝑖+1−𝑀_𝑖,𝑐𝜏𝜃_𝑖=𝑀_𝑖cos𝜃−𝑀_𝑖+1𝜏²sin𝜃,𝑑_𝑖𝑀=−^𝑖𝜏², (7) where 𝜃=𝜏ℎ and 𝑖=0,1,2,…,𝑁−1 using the continuity of the first derivative at (𝑥_𝑖,𝑢_𝑖); that is, 𝑠^′_Δ𝑖−1(𝑥_𝑖)=𝑠^′_Δ𝑖(𝑥_𝑖), we obtain the following relations for 𝑖=0,1,2,…,𝑁−1𝛼𝑀_𝑖+1+2𝛽𝑀_𝑖+𝛼𝑀_𝑖−1=1ℎ²𝑢_𝑖+1−2𝑢_𝑖+𝑢_𝑖−1,1where𝛼=ℎ²()1𝜃csc𝜃−1,𝛽=ℎ²(1−𝜃cot𝜃), (8) and 𝜃=𝜏ℎ whenever 𝜏→0; then 𝛼→1/6 and 𝛽→1/3. Therefore (8) reduces to the consistency relation of cubic splines: 𝑀_𝑖+1+4𝑀_𝑖+𝑀_𝑖−1=6ℎ²𝑢_𝑖+1−2𝑢_𝑖+𝑢_𝑖−1. (9) Equation (1) can be rewritten as 𝑑²𝑢𝑑𝑥²+𝑞(𝑥)𝑑𝑢(𝑥)(𝑥),(𝑥)=𝑝𝑑𝑥+𝑟𝑢=𝑓where𝑞^′(𝑥)𝑝(𝑥)(𝑥)𝑣(𝑥),𝑟=−𝑝(𝑥),𝑓(𝑥)𝑔(𝑥)=−𝑝(𝑥). (10) The proposed differential equation (10) can be discretized at the nodal point 𝑥_𝑖 by 𝑢_𝑖^′′+𝑞_𝑖𝑢^′_𝑖+𝑟_𝑖𝑢_𝑖=𝑓_𝑖,where𝑞_𝑖=𝑞𝑥_𝑖,𝑟_𝑖=𝑟𝑥_𝑖,𝑓_𝑖=𝑓𝑥_𝑖. (11) By using moment of the spline in (11) we obtain 𝑀_𝑖+𝑞_𝑖𝑢^′_𝑖+𝑟_𝑖𝑢_𝑖=𝑓_𝑖. (12) Taking approximations for the first derivative of 𝑢 we have 𝑢^′_𝑖=𝑢_𝑖+1−𝑢_𝑖−12ℎ,𝑢^′_𝑖+1=3𝑢_𝑖+1−4𝑢_𝑖+𝑢_𝑖−1,𝑢2ℎ_′𝑖−1=−𝑢_𝑖+1+4𝑢_𝑖−3𝑢_𝑖−1.2ℎ (13) Substituting (12), (13) in (8) and simplifying, we get the following tridiagonal system which gives the approximations 𝑢₁,𝑢₂,𝑢₃,…,𝑢_𝑁−1 of the solution 𝑢(𝑥) at 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑁−1: 3ℎ2𝛼𝑞_𝑖−1+ℎ𝛽𝑞_𝑖−ℎ2𝛼𝑞_𝑖+1−ℎ²𝛼𝑟_𝑖−1−1𝑢_𝑖−1+−2ℎ𝛼𝑞_𝑖−1+2ℎ𝛼𝑞_𝑖+1−ℎ²2𝛽𝑟_𝑖+2𝑢_𝑖+ℎ2𝛼𝑞_𝑖−1−ℎ𝛽𝑞_𝑖−3ℎ2𝛼𝑞_𝑖+1−ℎ²𝛼𝑟_𝑖+1−1𝑢_𝑖+1=−ℎ²𝛼𝑓_𝑖−1+2𝛽𝑓_𝑖+𝛼𝑓_𝑖+1, (14) with 𝑢(𝑎)=𝑢(𝑏)=0, 𝑖=1(1)𝑁−1.

The tridiagonal linear system (14) can be written in the following matrix form: 𝐴𝑈+ℎ²𝐷𝐹=𝐺,where𝐴=𝑃+ℎ𝐵𝑄−ℎ²𝐵𝑅. (15)𝐴 is a tridiagonally dominant matrix of order (𝑁−1).

𝑃=(𝑃_𝑖𝑗) is a tridiagonal matrix defined by 𝑃_𝑖𝑗=2𝑖=𝑗=1,2,…,𝑁−1,−1𝑖−𝑗=1,0otherwise, (16)𝐵𝑄=(𝑍_𝑖𝑗), 𝐵𝑅=(𝑅_𝑖𝑗) are tridiagonal matrices defined as 𝑍_𝑖𝑗=2𝛼−𝑞₀+𝑞₂3,𝑖=𝑗=1,2𝛼𝑞_𝑖−1+𝛽𝑞_𝑖−12𝛼𝑞_𝑖+1,𝑖>𝑗,2𝛼−𝑞_𝑖−1+𝑞_𝑖+11,𝑖=𝑗,2𝛼𝑞_𝑖−1−𝛽𝑞_𝑖−32𝛼𝑞_𝑖+1,𝑖<𝑗,2𝛼−𝑞_𝑁−2+𝑞_𝑁𝑅,𝑖=𝑗=𝑁−1,_𝑖𝑗=2𝛽𝑟_𝑖,𝑖=𝑗=1,2…,𝑁−1,𝛼𝑟_𝑖−1,𝑖>𝑗,𝛼𝑟_𝑖+1𝑓,𝑖<𝑗,𝐹=₁,𝑓₂,…,𝑓_𝑁−1^𝑇,𝑈=𝑢₁,𝑢₂,…,𝑢_𝑁−1^𝑇. (17)(18) The tridiagonal matrix 𝐷 is defined by ,𝑔𝐷=2𝛽𝛼000⋅⋅0𝛼2𝛽𝛼00⋅⋅00𝛼2𝛽𝛼0⋅⋅000𝛼2𝛽𝛼⋅⋅0000𝛼2𝛽𝛼⋅⋅⋅⋅⋅⋅𝛼2𝛽𝛼⋅⋅⋅⋅⋅⋅𝛼2𝛽𝛼00000⋅𝛼2𝛽𝐺=₁,0,0,0,…,0,𝑔_𝑁−1^𝑇where𝑔₁=−ℎ²𝛼𝑓₀,𝑔_𝑖=0,for𝑖=2,3,…,𝑁−2,𝑔_𝑁−1=−ℎ²𝛼𝑓_𝑁. (19)

3. Numerical Illustrations

In the present work four linear boundary value problems with ℎ=0.1 and one nonlinear boundary value problem with ℎ=0.2 for different values of 𝛼 and 𝛽 have been solved, whose exact solutions are known. The approximate solution, exact solutions, and absolute errors at the nodal points are tabulated in Tables 1–7, and comparisons are shown in Figures 1–5. The results of the present work are compared with the exact solution of all the problems and with finite difference method, B-spline method of Example 4.

Table 1 

Approximate solution (series 1), exact solution (series 2), and absolute errors of Example 1.

Table 2 

Approximate solution (series 1), exact solution (series 2), and absolute errors of Example 2.

Table 3 

(a) Approximate solution, exact solution, and absolute errors of Example 2 Case 2(a). (b) Approximate solution, exact solution, and absolute errors of Example 2 Case 2(b).

Table 4 

Approximate solutions (series 1), exact solution (series 2), and absolute errors of Example 3.

Table 5 

Approximate solution (series 1), exact solution (series 2), and absolute errors of Example 4.

Table 6 

Comparison of present method (series 3) with B-spline method (series 2), finite difference method (series 1) and with exact solution (series 4) for Example 4.

Table 7 

Approximate solution, exact solution, and absolute errors of Example 5

Figure 1 

Comparison of approximate values and exact values for Example 1.

(a)

(b)

Figure 2 

(a) Comparison of approximate values and exact values for Example 2. (b) Comparison of errors in 2 cases for Example 2.

Figure 3 

Comparison of approximate values and exact values for Example 3.

(a)

(b)

Figure 4 

(a) Comparison of approximate values and exact values for Example 4. (b) Comparison of results obtained by our method of Example 4 with the values obtained by other methods (Table 6).

Figure 5 

Comparison of approximate values and exact values for Example 5.

Example 1. Consider the boundary value problem: 𝑢^′′(𝑥)=2𝑥²𝑢(𝑥)−1𝑥𝑢(2)(3);2<𝑥<3,=0,𝑢=0. (20) The analytical solution of (20) is 𝑢(𝑥)=138−5𝑥²+19𝑥−36𝑥. (21) By comparing the given equation with (10) we have 𝑞_𝑖=𝑞𝑥_𝑖=0,𝑟_𝑖=𝑟𝑥_𝑖2=−𝑥²_𝑖,𝑓_𝑖=𝑓𝑥_𝑖1=−𝑥_𝑖,for𝑥_𝑖=2,2.1,2.2,…,3,withℎ=0.1,where𝑖=0,1,2…,10,respectively. (22) Here 𝐺=𝑔₁,0,0,0,0,0,0,0,𝑔₉^𝑇,where𝑔₁=8.3333×10⁻⁴,𝑔₉=5.55555×10⁻⁴, (23) and the function values at the nodal points are given by 𝐹=(𝑓₁,𝑓₂,…,𝑓₉)^𝑇; that is,()𝐹=−0.47619,−0.45455,−0.43478,−0.41667,−0.4,−0.38462,−0.37037,−0.35714,−0.34483^𝑇. (24) Substituting the values of 𝑝_𝑖, 𝑞_𝑖, 𝑟_𝑖 at 𝑖=0,1,2…,10 in (16), (17), and (18) we get the values of (𝑃_𝑖𝑗), (𝑍_𝑖𝑗), (𝑅_𝑖𝑗), and taking these values in (15) we get the tridiagonal matrix 𝐴. From (23), (24) we have 𝐺 and 𝐹 so we obtain a system of linear equations. This system in the matrix form is 𝐴𝑈+ℎ²𝐷𝐹=𝐺; that is, ^{2.0015−0.999310000000−0.999242.00137−0.999370000000−0.999312.00126−0.999420000000−0.999372.00116−0.999470000}_{000−0.999422.00107−0.999510000000−0.999472.00099−0.999540000000−0.999512.00091−0.999570000000−0.999542.00085−0.99960000000−0.999572.00079}^{𝑢1𝑢2𝑢3𝑢4}_{𝑢5𝑢6𝑢7𝑢8𝑢9}=^{0.0047650.0045480.004350.004169}_{0.0040020.0038480.0037050.0035730.003449}. (25)

Solving this we get the solution matrix 𝑈. The approximate values (𝑈), exact values (𝑢(𝑥) of (21)), and the absolute errors are summarized in Table 1, and the comparison is given in Figure 1.

Example 2. Consider the boundary value problem: 𝑢^′′(𝑥)−𝜋²𝑢(𝑥)=−2𝜋²𝑢(0)(1)sin𝜋𝑥;0<𝑥<1,=0,𝑢=0, (26) Comparing the given equation with (10) we have 𝑞_𝑖=𝑞𝑥_𝑖=0,𝑟_𝑖=𝑟𝑥_𝑖=−𝜋²,𝑓_𝑖=𝑓𝑥_𝑖=−2𝜋²sin𝜋𝑥_𝑖,for𝑖=0,1,…,10. (27) Here ℎ=0.1, so the nodal points are 𝑥₀,𝑥₁,𝑥₂,…,𝑥_10.

Case  1 (choosing 𝛼=1/6, 𝛽=1/3). From (19) we have 𝐺=𝑔₁,0,0,0,0,0,0,0,𝑔₉^𝑇,where𝑔₁=0,𝑔₉=1.8029683×10⁻³, (28)

and the function values at the nodal points are given by 𝐹=(𝑓₁,𝑓₂,…,𝑓₉)^𝑇; that is, ()𝐹=−6.09974,−11.6024,−13.9693,−18.7731,−19.7392,−18.7731,−15.9693,−11.6024,−6.09979^𝑇. (29) Substituting the values of 𝑝_𝑖, 𝑞_𝑖, and 𝑟_𝑖 at 𝑖=0,1,2…10 in (16), (17), and (18) we get the values of (𝑃_𝑖𝑗), (𝑍_𝑖𝑗), (𝑅_𝑖𝑗), and substituting these values in (15) we get the tridiagonal matrix 𝐴. From (28), (29) we have 𝐺 and 𝐹. So we obtain a system of linear equations. This system in the matrix form is 𝐴𝑈+ℎ²𝐷𝐹=𝐺; solving this we get the solution matrix 𝑈.

The analytical solution of the given equation is 𝑢(𝑥)=sin𝜋𝑥. (30) The approximate values (𝑈), exact values (𝑢(𝑥) from (30)), and the absolute errors at the nodal points are summarized in Table 2 and the comparison is given in Figure 2.

Case  2 ((a) choosing 𝛼=1/4, 𝛽=1/4 (b) choosing 𝛼=1/14, 𝛽=3/7). For the values of 𝛼, 𝛽 the procedure followed in Case 1 is used to determine the approximate solution of (24). The approximate values (𝑈), exact values (𝑢(𝑥) from (30)), and the absolute errors at the nodal points of Case 2 are summarized in Tables 3(a) and 3(b), respectively. The comparison of errors in the two cases at the nodal points is given in Figure 2(b).

Example 3. Consider the boundary value problem: 𝑢^′′(𝑥)−𝑢(𝑥)=−4𝑥𝑒^𝑥𝑢(0)(1);0≤𝑥≤1,=0,𝑢=0. (31) Comparing the given equation with (10) we have 𝑞_𝑖=𝑞𝑥_𝑖=0,𝑟_𝑖=𝑟𝑥_𝑖𝑓=−1,_𝑖=𝑓𝑥_𝑖=−4𝑥𝑒^𝑥𝑖. (32) Here ℎ=0.1, so the nodal points are 𝑥₀,𝑥₁,𝑥₂,…,𝑥_10. Solution of (31) is given by (15). Here 𝐺=𝑔₁,0,0,0,0,0,0,0,𝑔₉^𝑇,where𝑔₁=0,𝑔₉=1.8152132×10⁻³, (33) and the function values at the nodal points is given by 𝐹=(𝑓₁,𝑓₂,…,𝑓₉)^𝑇; that is, ()𝐹=−0.44207,−0.97712,−1.61983,−2.38692,−3.29744,−4.37309,−5.6385,−7.12173,−8.85457^𝑇. (34) Substituting the values of 𝑝_𝑖, 𝑞_𝑖, and 𝑟_𝑖 at 𝑖=0,1,2…,10 in (16), (17), and (18) we get the values of (𝑃_𝑖𝑗), (𝑍_𝑖𝑗), and (𝑅_𝑖𝑗), and substituting these values in (15) we get the tridiagonal matrix 𝐴. From (33), (34) we have 𝐺 and 𝐹. So we obtain a system of linear equations. This system in the matrix form is 𝐴𝑈+ℎ²𝐷𝐹=𝐺, solving this we get the solution matrix 𝑈.
The analytical solution of the given differential equation is 𝑢(𝑥)=𝑥(1−𝑥)𝑒^𝑥. (35) Approximate values, exact values (35), and the absolute errors at the nodal points of 𝑢(𝑥) are given in Table 4, and the comparison is given in Figure 3.

Example 4. Consider the boundary value problem: 𝑢^′′(𝑥)−2𝑢^′(𝑥)(𝑥)𝑢(0)(1)−2𝑢=−2;0≤𝑥≤1,=0,𝑢=0, (36) Comparing the given equation with (10) we have 𝑞_𝑖=𝑞(𝑥_𝑖)=−2, 𝑟(𝑥)=−2, 𝑓_𝑖=𝑓(𝑥_𝑖)=−2.
Here ℎ=0.1, so the nodal points are 𝑥₀,𝑥₁,𝑥₂,…,𝑥₁₀. Solution of (36) is given by (15). Here 𝐺=(𝑔₁,0,0,0,0,0,0,0,𝑔₉)^𝑇, where 𝑔₁=0.00333333, 𝑔₉=0.00333333, and the function values at the nodal points is −2. That is, 𝐹=𝑓₁,𝑓₂,…,𝑓_9.^𝑇,where𝑓_𝑖=−2for𝑖=1,2,…,9. (37) Substituting the values of 𝑝_𝑖, 𝑞_𝑖, and 𝑟_𝑖 at 𝑖=0,1,2,…,10 in (16), (17), and (18) we get the values of (𝑃_𝑖𝑗), (𝑍_𝑖𝑗), and (𝑅_𝑖𝑗), and substituting these values in (15) we get the tridiagonal matrix 𝐴. From (37) we have 𝐺 and 𝐹. So we obtain a system of linear equations. This system in the matrix form is 𝐴𝑈+ℎ²𝐷𝐹=𝐺; solving this we get the solution matrix 𝑈.
The analytical solution of (36) is commutation 𝑢(𝑥)=1−𝑒^√31+𝑒^√31+−𝑒^√31−𝑒^{√(1−3)𝑥}+𝑒^√31−−1𝑒^√31+−𝑒^√31−𝑒^√(1+3)𝑥+1. (38) Approximate solution, the exact solution, and the absolute errors at the nodal points of 𝑢(𝑥) are given in Table 5, and the comparison is given in Figure 4. Comparative results obtained by our method of Example 4 with the values obtained by other methods are given in Table 6 and shown in Figure 4(b).

Example 5. Consider the non linear boundary value problem: 𝑢^′′(𝑥)=12(1+𝑥+𝑢)³𝑢(0)(1);0≤𝑥≤1,=0,𝑢=0. (39) Comparing the given equation with (10) by taking 𝑔(𝑥)=𝑔(𝑥,𝑢) in (1) we have 𝑓_𝑖=𝑓𝑥_𝑖,𝑢_𝑖=121+𝑥_𝑖+𝑢_𝑖³, (40) and the nodal points with ℎ=0.2 are 𝑥_𝑖=0,0.2,0.4,0.6,0.8,1 for 𝑖=0,1,2,3,4,5.
Now by choosing 𝛼=1/14 and 𝛽=3/7 and substituting these values in (16) and (19) we get the tridiagonal matrices 𝑃, 𝐷 and 𝐺=𝑔₁,0,0,𝑔₄^𝑇,where𝑔₁=𝑔₄=−0.00142857. (41) From (18) we have 𝐹=1.2+𝑢₁³2,1.4+𝑢₂³2,1.6+𝑢₃³2,1.8+𝑢₄³2^𝑇. (42) Substituting 𝑃, 𝐷, (41), and (42) in (15), we obtain a system of nonlinear equations: 121.2+𝑢₁³+1.4+𝑢₂³+1400𝑢₁−700𝑢₂+1=0,1.2+𝑢₁³+121.4+𝑢₂³+1.6+𝑢₃³−700𝑢₁+1400𝑢₂−700𝑢₃=0,1.4+𝑢₂³+121.6+𝑢₃³+1.8+𝑢₄³−700𝑢₂+1400𝑢₃−700𝑢₄=0,1.6+𝑢₃³+121.8+𝑢₄³−700𝑢₃+1400𝑢₄+8=0, (43) Solving these equations by Newton method we get the solution matrix 𝑈.
The analytical solution of (39) is 𝑢(𝑥)=22−𝑥−𝑥−1. (44) The approximate solution (𝑈), exact solution (𝑢(𝑥) of (44)) and the absolute errors are summarized in Table 7, and the comparison is given in Figure 5.

4. Conclusions

In the present work it has been described and demonstrated the applicability and efficiency of the nonpolynomial spline method for solving second order linear and nonlinear two point boundary value problems. The nonpolynomial spline method is tested on different problems. Numerical results for Examples 1, 2, 3, 4, and 5 are presented in Tables 1, 2, 3, 4, 5, 6, and 7, and graphs between the exact solution and approximate solution have been plotted for all the 5 examples. In Example 2 the solution is obtained by choosing different values for 𝛼 and 𝛽, and a comparison of errors is given in Figure 2(b) from which we can say that the error is reduced more rapidly with the choice of 𝛼=1/14 and 𝛽=3/7 than the other choices of 𝛼 and 𝛽, and for Example 4 the graph (Figure 4(b)) is plotted by comparing with the results obtained by other known methods. This shows that the accuracy of our method is better than the accuracy of finite difference method and of B-spline method. All the tables and figures clearly indicate that our numerical solution converges to the exact solution. We conclude that the present method is an applicable technique and approximates the solution very well, and the numerical solutions are in very good agreement with the exact solution. Moreover non-polynomial spline method has less computational cost over other polynomial spline methods. The implementation of the present method is very easy, acceptable, and valid scheme.