New Algorithm of Two-Point Block Method for Solving Boundary Value Problem with Dirichlet and Neumann Boundary Conditions

Abstract

Two-point block method with variable step-size strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the three-step iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy.

1. Introduction

Numerical solutions for boundary value problem have great importance in scientific computation, as they were widely used to model the real-world problems. There are several methods that can be used to solve the two-point boundary value problems such as the Adomian decomposition method modified by Duan and Rach [1] and Ebaid [2] to solve boundary value problem. Based on the Adomian decomposition method, a new analytical and numerical treatment is introduced to investigate linear and nonlinear two-point boundary value problems. Lang and Xu [3] studied a new quintic B-spline collocation method for linear and nonlinear second-order boundary value problems. Islam and Shirin [4] used Bernoulli polynomials to find the numerical solutions of the second-order linear and nonlinear boundary value problems. Besides that, Liu [5] had solved the boundary value problem with Neumann type using polynomial spline approach. The numerical solutions of second-order boundary value problems by collocation method with the Haar wavelets were presented by Siraj-ul-Islam et al. [6]. A new kind of finite difference scheme presented for special second-order nonlinear two-point boundary value problem has been proposed by Erdogan and Ozis [7], while Prentice [8] considered the error control in a finite difference solution of a two-point boundary value problem.

In this paper, we are concerned for solving two-point second-order boundary value problems (BVPs) with two types of boundary condition, that is, Dirichlet and Neumann type. Two-point second-order boundary value problem is as follows: 𝑦^′′=𝑓𝑥,𝑦,𝑦^′,𝑎≤𝑥≤𝑏. (1) Dirichlet boundary condition is as follows: 𝑦(𝑎)=𝛼,𝑦(𝑏)=𝛽. (2) Neumann boundary condition is as follows: 𝑦^′(𝑎)=𝛼,𝑦^′(𝑏)=𝛽. (3) The BVPs as (1) will be solved using two-point block method with variable step size. The block methods are commonly used to solve the ODEs such as in Majid et al. [9]. The authors have used the two-point block one-step method of Runge-Kutta type to solve the general second-order ODEs with variable step-size strategy.

In Majid et al. [10], the authors have solved the BVPs of Dirichlet type only by two-point block method with constant step size. Hence, in this paper, we aim to extend the work in Majid et al. [10] for solving BVPs of Dirichlet and Neumann type using variable step size. This block method has the advantage to solve the second-order differential equation directly and obtain two approximate solutions simultaneously in block. We adapted the multiple shooting techniques to obtain the missing initial value, and the three-step iterative method proposed by Yun [11] was employed to generate the missing guessing value. The numerical computations have been performed using the C language.

2. Formulation of the Two-Point Block Method

In Figure 1, we have divided the interval [𝑎,𝑏] into a series of blocks with each block containing two points. Both approximate solutions are simultaneously found using the same back values. The approximate solution of 𝑦_𝑛+1 and 𝑦_𝑛+2 at the point, 𝑥_𝑛+1 and 𝑥_𝑛+2, respectively, with step size ℎ will be computed simultaneously using three back values at the points, 𝑥_𝑛, 𝑥_𝑛−1 and 𝑥_𝑛−2 with step size rh. The value, 𝑦_𝑛+1 and 𝑦_𝑛+2 will be obtained by integrate (1) once and twice over the intervals [𝑥_𝑛,𝑥_𝑛+1] and [𝑥_𝑛,𝑥_𝑛+2], respectively, as follows

Figure 1 

Two-point block method.

First point is 𝑦^′𝑥_𝑛+1−𝑦^′𝑥_𝑛=^𝑥𝑛+1_𝑥𝑛𝑓𝑥,𝑦,𝑦^′𝑦𝑥𝑑𝑥,_𝑛+1−𝑦𝑥_𝑛−ℎ𝑦^′𝑥_𝑛=^𝑥𝑛+1_𝑥𝑛𝑥_𝑛+1−𝑥𝑓𝑥,𝑦,𝑦^′𝑑𝑥. (4)

Second point is 𝑦^′𝑥_𝑛+2−𝑦^′𝑥_𝑛=^𝑥𝑛+2_𝑥𝑛𝑓𝑥,𝑦,𝑦^′𝑦𝑥𝑑𝑥,_𝑛+2−𝑦𝑥_𝑛−2ℎ𝑦^′𝑥_𝑛=^𝑥𝑛+2_𝑥𝑛𝑥_𝑛+2−𝑥𝑓𝑥,𝑦,𝑦^′𝑑𝑥. (5) The method is derived using Lagrange interpolation polynomial and the five interpolating points; that is, (𝑥_𝑛−2,𝑓_𝑛−2), (𝑥_𝑛−1,𝑓_𝑛−1), (𝑥_𝑛,𝑓_𝑛), (𝑥_𝑛+1,𝑓_𝑛+1), and (𝑥_𝑛+2,𝑓_𝑛+2) are involved in the corrector formulae of the two-point block method. The function 𝑓(𝑥,𝑦,𝑦^′) in (4) and (5) will be replaced by Lagrange interpolating polynomial. Let 𝑠=(𝑥−𝑥_𝑛+2)/ℎ, replacing 𝑑𝑥=ℎ𝑑𝑠 and changing the limit of integration from -2 to -1 for the first point and changing the limit of integration to -2 and 0 for the second point.

Evaluate these integrals using MAPLE, the corrector formulae of the two-point block method will be obtained as follows.

First point is 𝑦^′_𝑛+1=𝑦^′_𝑛+ℎ240(1+𝑟)(2+𝑟)(1+2𝑟)𝑟²×14+37𝑟+15𝑟²𝑓_𝑛−2+−28−176𝑟−240𝑟²𝑓_𝑛−1+14+139𝑟+564𝑟²+1029𝑟³+790𝑟⁴+200𝑟⁵𝑓_𝑛+144𝑟²+672𝑟³+940𝑟⁴+320𝑟⁵𝑓_𝑛+1+−3𝑟²−21𝑟³−50𝑟⁴−40𝑟⁵𝑓_𝑛+2,𝑦_𝑛+1=𝑦_𝑛+ℎ𝑦^′_𝑛+ℎ²240(2+𝑟)(1+𝑟)(1+2𝑟)𝑟²×6+19𝑟+8𝑟²𝑓_𝑛−2+−12−88𝑟−128𝑟²𝑓_𝑛−1+6+69𝑟+329𝑟²+644𝑟³+538𝑟⁴+140𝑟⁵𝑓_𝑛+32𝑟²+184𝑟³+324𝑟⁴+120𝑟⁵𝑓_𝑛+1+−𝑟²−8𝑟³−22𝑟⁴−20𝑟⁵𝑓_𝑛+2. (6)

Second point is 𝑦^′_𝑛+2=𝑦^′_𝑛+ℎ15(2+𝑟)(1+𝑟)(1+2𝑟)𝑟²×(−2−𝑟)𝑓_𝑛−2+(4−8𝑟)𝑓_𝑛−1+−2−7𝑟+3𝑟²+33𝑟³+35𝑟⁴+10𝑟⁵𝑓_𝑛+48𝑟²+144𝑟³+140𝑟⁴+40𝑟⁵𝑓_𝑛+1+9𝑟²+33𝑟³+35𝑟⁴+10𝑟⁵𝑓_𝑛+2,𝑦_𝑛+2=𝑦_𝑛+2ℎ𝑦^′_𝑛+ℎ²()×)𝑓15𝑟2+𝑟)(1+𝑟)(1+2𝑟(2+𝑟_𝑛−2+(−8−16𝑟)𝑓_𝑛−1+6+41𝑟+91𝑟²+76𝑟³+20𝑟⁴𝑓_𝑛+32𝑟+112𝑟²+128𝑟³+40𝑟⁴𝑓_𝑛+1+2𝑟+7𝑟²+6𝑟³𝑓_𝑛+2. (7) The method is the combination of predictor which is one order less than the corrector. The two-point block method with variable step-size strategy will be implemented for solving the boundary value problems via multiple shooting techniques.

3. Analysis of the Method

In this section, stability analysis, stability region, order of the method, and error constant of the two-point block method are discussed.

3.1. Stability Analysis

Definition 1. The method is zero stable provided the roots 𝑅_𝑗 of the first characteristic polynomial 𝜌(𝑅) specified as 𝜌(𝑅)=det[∑^𝑘_𝑖=0𝐴^(𝑖)𝑅^𝑘−𝑖]=0 satisfy |𝑅_𝑗|≤1.
Substitute 𝑟=1 into (7), and rewrite in matrix from: 1000𝑦010000100001^𝑛+2𝑦^′_𝑛+2𝑦_𝑛+1𝑦^′_𝑛+1=1000𝑦010010000100^𝑛𝑦^′_𝑛𝑦_𝑛−1𝑦^′_𝑛−1𝑦+ℎ0200000001000000^𝑛𝑦^′_𝑛𝑦_𝑛−1𝑦^′_𝑛−1+ℎ000029901249024490900000−19720346720456720−74𝑓720^𝑛+2𝑓_𝑛+1𝑓_𝑛𝑓_𝑛−1+ℎ0000−190000000011𝑓720000^𝑛−2𝑓_𝑛−3𝑓_𝑛−4𝑓_𝑛−5+ℎ²590104907890−8900000−17144022014405821440−76𝑓14400000^𝑛−2𝑓_𝑛−1𝑓_𝑛𝑓_𝑛−1+ℎ²190000000011𝑓14400000000^𝑛−2𝑓_𝑛−3𝑓_𝑛−4𝑓_𝑛−5. (8) The first characteristic polynomial of the two-point block method is given as follows: 𝜌(𝑅)=det𝑅𝐴⁰−𝐴¹=0,where𝐴⁰=1000010000100001,𝐴¹=,𝜌(𝑅),()1000010010000100=det𝑅−10000𝑅−100−10𝑅00−10𝑅0=𝑅−1²𝑅²,𝑅=0,0,1,1,|𝑅|≤1. (9) From Definition 1 and (9) the two-point one block method is zero stable.

3.2. Stability Region

The stability polynomial of the two-point block method applied to the test equation is as follows: 𝑦^′′=𝑓=𝜃𝑦^′+𝜆𝑦. (10)

The stability polynomials of two-point block method are as follows.

For 𝑟=0.5: 1−147𝐻200¹+847𝐻5400²¹−83𝐻450²+127𝐻1350¹𝐻₂+211𝐻13500²²𝑡⁸+−2+19𝐻100¹−4307𝐻2700²¹𝑡⁷×−839𝐻225²−1633𝐻1350¹𝐻₂−749𝐻1125²²𝑡⁷+11−𝐻24¹+28463𝐻27000²¹+149𝐻450²𝑡⁶+−12719𝐻6750¹𝐻₂+5611𝐻13500²²𝑡⁶+44𝐻75¹+99𝐻250²¹−34𝐻225²−719𝐻2250¹𝐻₂+19𝐻150²²𝑡⁵+−8𝐻675²¹+8𝐻675¹𝐻₂−8𝐻3375²²𝑡⁴=0, (11) where 𝐻₁=ℎ²𝜆 and 𝐻₂=ℎ²𝜃.

For 𝑟=1.0: 1−289𝐻360¹+413𝐻2160²¹−5𝐻24²+53𝐻432¹𝐻₂+239𝐻10800²²𝑡⁸+−2−31𝐻120¹−209𝐻216²¹𝑡⁷+−1307𝐻360²−161𝐻270¹𝐻₂−473𝐻900²²𝑡⁷+1+37𝐻40¹+25𝐻36²¹−41𝐻360²−283𝐻360¹𝐻₂+937𝐻5400²²𝑡⁶+937𝐻5400²²𝑡⁶+49𝐻360¹+89𝐻1080²¹−17𝐻360²−2𝐻27¹𝐻₂+17𝐻900²²𝑡⁵−1𝐻2160²¹𝑡⁴+1𝐻2160¹𝐻₂−1𝐻10800²²𝑡⁴=0, (12) where 𝐻₁=ℎ²𝜆 and 𝐻₂=ℎ²𝜃.

For 𝑟=2.0: 1−87𝐻100¹+623𝐻2700²¹−103𝐻450²+847𝐻5400¹𝐻₂+137𝐻4500²²𝑡⁸+−2−223𝐻960¹−517𝐻675²¹𝑡⁷+−51217𝐻14400²−51𝐻160¹𝐻₂−102029𝐻216000²²𝑡⁷+1+2579𝐻2400¹+15001𝐻28800²¹−1463𝐻7200²𝑡⁶+−40001𝐻86400¹𝐻₂+48139𝐻432000²²𝑡⁶+133𝐻4800¹+619𝐻43200²¹−161𝐻14400²−91𝐻7200¹𝐻₂𝑡⁵+161𝐻54000²²𝑡⁵+−1𝐻86400²¹+1𝐻86400¹𝐻₂−1𝐻432000²²𝑡⁴=0, (13) where 𝐻₁=ℎ²𝜆 and 𝐻₂=ℎ²𝜃.

Figure 2 shows the regions of absolute stability for the two-point block method when the step-size ratio 𝑟=0.5,1.0,and2.0. The stability region is the bounded shaded region and the region is larger as the step-size ratio increases. This is expected since smaller step-size will give larger step size ratio.

Figure 2 

Stability region of the two-point block method.

3.3. Order and Error Constant

Definition 2. The block method is ^𝑘_𝑗=0𝛼_𝑗𝑦_𝑛+𝑗−ℎ^𝑘_𝑗=0𝛽_𝑗𝑦^′_𝑛+𝑗−ℎ^2𝑘_𝑗=0𝛾_𝑗𝑦^′′_𝑛+𝑗=𝐶_𝑝+1𝑦^𝑝+1+𝑂ℎ^𝑝+2, (14) where 𝐶_𝑝=1𝑝!^𝑘_𝑗=0𝑗^𝑝𝛼_𝑗−1𝑝−1!^𝑘_𝑗=0𝑗^𝑝−1𝛽_𝑗−1𝑝−2!^𝑘_𝑗=0𝑗^𝑝−2𝛾_𝑗,𝐶₀=𝐶₁=𝐶₂=⋯=𝐶_𝑝+1=0,𝐶_𝑝+2≠0 (15)𝑝 is the order of the LMM method, and 𝑂(ℎ^𝑝+2) is the local truncation error.
Substituting 𝑟=1 into (7) and apply the formulae, we obtaining 𝐶₀=𝐶₁=⋯=𝐶₆𝐶=0,₇=1114403710080−1190315^𝑇. (16) From Definition 2, the order of two-point block method is five with error constant 𝐶₇=1114403710080−1190315^𝑇. (17)

4. Implementation of the Method

4.1. Multiple Shooting Technique

The initial conditions being imposed at the same point in the independent variable 𝑥, but the boundary conditions are imposed at different values of the independent variable. The idea of shooting technique is to form the initial condition from the boundary condition with the guessing value. Multiple shooting techniques are indeed a combination of several shooting techniques by dividing the given interval 𝑎≤𝑥≤𝑏 into 𝑗th subinterval.

4.1.1. Dirichlet-Type Boundary Condition

The missing initial condition is 𝑦^′(𝑎). Equation (1) can be written as 𝑑²𝑦_𝑗𝑑𝑥²=𝑓_𝑗𝑥,𝑦_𝑗,𝑦^′_𝑗, (18) with conditions 1:𝑦₁(𝑎)=𝛼,𝑦^′₁(𝑎)=𝑠_𝑣2:𝑦₂𝑥₁=𝑦𝑥₁,𝑠_𝑣−1,𝑦^′₂𝑥₁=𝑦^′𝑥₁,𝑠_𝑣−1⋮𝑗:𝑦_𝑗𝑥_𝑗−1=𝑦𝑥_𝑗−1,𝑠_𝑣−1,𝑦^′_𝑗𝑥_𝑗−1=𝑦^′𝑥_𝑗−1,𝑠_𝑣−1. (19) Therefore, we obtain the 𝑗th stopping conditions as follows: 𝑦𝑥₁,𝑠_𝑣−𝑦𝑥₁,𝑠_𝑣−1𝑦𝑥≤𝜀₂,𝑠_𝑣−𝑦𝑥₂,𝑠_𝑣−1⋮𝑦𝑥≤𝜀_𝑗,𝑠_𝑣−𝛽≤𝜀. (20) The iteration is repeated until we reached the stopping conditions, and the value of 𝑠_𝑣 will be generated by the three-step iterative method as follows: 𝑇_𝑣=𝑠_𝑣−𝑦𝑏,𝑠_𝑣−𝛽𝑧𝑏,𝑠_𝑣,𝑈_𝑣𝑦=−𝑏,𝑇_𝑣−𝛽𝑧𝑏,𝑠_𝑣,𝑠_𝑣+1=𝑇_𝑣−𝑦𝑏,𝑠_𝑣−𝛽𝑧𝑏,𝑠_𝑣−𝑦𝑏,𝑇_𝑣+𝑈_𝑣−𝛽𝑧𝑏,𝑠_𝑣, (21) where 𝑧(𝑏,𝑠_𝑣) is the solution of 𝑧^′′=𝜕𝑓𝑥,𝑦,𝑦^′𝜕𝑦𝑧+𝜕𝑓𝑥,𝑦,𝑦^′𝜕𝑦^′𝑧^′,𝑧(𝑎)=0,𝑧^′(𝑎)=1. (22)

4.1.2. Neumann-Type Boundary Condition

The missing initial condition is 𝑦(𝑎). Equation (1) can be written as 𝑑²𝑦_𝑗𝑑𝑥²=𝑓_𝑗𝑥,𝑦_𝑗,𝑦^′_𝑗, (23) with conditions given as in (19), but condition 1 will be replaced by 𝑦₁(𝑎)=𝑠_𝑣,𝑦^′₁(𝑎)=𝛼. (24) The stopping conditions are 𝑦^′𝑥₁,𝑠_𝑣−𝑦^′𝑥₁,𝑠_𝑣−1𝑦≤𝜀_′𝑥₂,𝑠_𝑣−𝑦^′𝑥₂,𝑠_𝑣−1⋮𝑦≤𝜀_′𝑥_𝑗,𝑠_𝑣−𝛽≤𝜀. (25) The three-step iterative method is 𝑇_𝑣=𝑠_𝑣−𝑦^′𝑏,𝑠_𝑣−𝛽𝑧^′𝑏,𝑠_𝑣,𝑈_𝑣𝑦=−^′𝑏,𝑇_𝑣−𝛽𝑧^′𝑏,𝑠_𝑣,𝑠_𝑣+1=𝑇_𝑣−𝑦^′𝑏,𝑠_𝑣−𝛽𝑧^′𝑏,𝑠_𝑣−𝑦^′𝑏,𝑇_𝑣+𝑈_𝑣−𝛽𝑧^′𝑏,𝑠_𝑣, (26) where the value of 𝑧^′(𝑏,𝑠_𝑣) can be obtained from solving the 𝑧^′′ as follows: 𝑧^′′=𝜕𝑓𝑥,𝑦,𝑦^′𝜕𝑦𝑧+𝜕𝑓𝑥,𝑦,𝑦^′𝜕𝑦^′𝑧^′,𝑧(𝑎)=1,𝑧^′(𝑎)=0. (27)

4.2. Variable Step-Size Strategy

The choices of the next step size will be restricted to half, doubled or the same as the current step size. The adjustment is based on the local truncation error (LTE). If the local truncation error is less or equal to the tolerance (TOL), the choice for the next step will be double or remain the same. In the code developed, when the next step size is doubled, the ratio 𝑟 is 0.5, while the step size remains constant, r is 1. If this condition fails, the current step size will be halved from the previous step size; that is, r is 2, and the approximate solutions in the block will be recalculate.

Case 1. LTE ≤ TOL (successful step).
Substituting 𝑟=1 in (6) and (7) will produce the following corrector formulae: 𝑦^′_𝑛+1=𝑦^′_𝑛+ℎ×72011𝑓_𝑛−2−74𝑓_𝑛−1+456𝑓_𝑛+346𝑓_𝑛+1−19𝑓_𝑛+2,𝑦_𝑛+1=𝑦_𝑛+ℎ𝑦^′_𝑛+ℎ²×144011𝑓_𝑛−2−76𝑓_𝑛−1+582𝑓_𝑛+220𝑓_𝑛+1−17𝑓_𝑛+2,𝑦^′_𝑛+2=𝑦^′_𝑛+ℎ×90−𝑓_𝑛−2+4𝑓_𝑛−1+24𝑓_𝑛+124𝑓_𝑛+1+29𝑓_𝑛+2,𝑦_𝑛+2=𝑦_𝑛+2ℎ𝑦^′_𝑛+ℎ²×𝑓90_𝑛−2−8𝑓_𝑛−1+78𝑓_𝑛+104𝑓_𝑛+1+5𝑓_𝑛+2. (28) Substituting 𝑟=0.5 in (6) and (7) will produce the following corrector formulae: 𝑦^′_𝑛+1=𝑦^′_𝑛+ℎ×1800145𝑓_𝑛−2−704𝑓_𝑛−1+1635𝑓_𝑛+755𝑓_𝑛+1−31𝑓_𝑛+2,𝑦_𝑛+1=𝑦_𝑛+ℎ𝑦^′_𝑛+ℎ²×180070𝑓_𝑛−2−352𝑓_𝑛−1+975𝑓_𝑛+220𝑓_𝑛+1−13𝑓_𝑛+2,𝑦^′_𝑛+2=𝑦^′_𝑛+ℎ×225−20𝑓_𝑛−2+64𝑓_𝑛−1+15𝑓_𝑛+320𝑓_𝑛+1+71𝑓_𝑛+2,𝑦_𝑛+2=𝑦_𝑛+2ℎ𝑦^′_𝑛+2ℎ²×2255𝑓_𝑛−2−32𝑓_𝑛−1+120𝑓_𝑛+125𝑓_𝑛+1+7𝑓_𝑛+2. (29)

Case 2. LTE > TOL (failure step).
Substituting 𝑟=2 in (6) and (7) will produce the following corrector formulae: 𝑦^′_𝑛+1=𝑦^′_𝑛+ℎ×1440037𝑓_𝑛−2−335𝑓_𝑛−1+7456𝑓_𝑛+7808𝑓_𝑛+1−565𝑓_𝑛+2,𝑦_𝑛+1=𝑦_𝑛+ℎ𝑦^′_𝑛+ℎ²×1440019𝑓_𝑛−2−175𝑓_𝑛−1+4965𝑓_𝑛+2656𝑓_𝑛+1−265𝑓_𝑛+2,𝑦^′_𝑛+2=𝑦^′_𝑛+ℎ×900295𝑓_𝑛−2+1216𝑓_𝑛−1+285𝑓_𝑛+5𝑓_𝑛+1−𝑓_𝑛+2,𝑦_𝑛+2=𝑦_𝑛+2ℎ𝑦^′_𝑛+ℎ²×𝑓450_𝑛−2−10𝑓_𝑛−1+345𝑓_𝑛+544𝑓_𝑛+1+20𝑓_𝑛+2. (30)

4.3. Algorithm??2P1BVS

Step 1. Set TOL and 𝑠₀.

Step 2. For 𝑛=1,2,3. ?set 𝑥_𝑛=𝑎+𝑛ℎ, and evaluate the approximate values 𝑦_𝑛 and 𝑧_𝑛 with direct Adams-Bashforth method. ?Compute functions 𝑓_𝑛 and 𝑧_𝑛^′′.

Step 3. While 𝑥_𝑛<𝑏, do Step 4.

Step 4. For 𝑖=1,2. ?Set 𝑥_𝑛+𝑖=𝑥_𝑛+𝑖ℎ.?Evaluate the approximate values 𝑦_𝑛+𝑖 and 𝑧_𝑛+𝑖 with two-point block method.?Compute functions 𝑓_𝑛+𝑖 and 𝑧^′′_𝑛+𝑖.

Step 5. If fulfill stopping condition, go to Step 8.

Step 6. Generate the new guessing values 𝑠_𝑣 by three-step iterative method. Set 𝑣=𝑣+1.

Step 7. If 𝑣≥10, set 𝑏=(𝑎+𝑏)/2, and go to Step 2.

Step 8. Complete.

This algorithm was developed in C language.

5. Problem Tested

In this section, we have tested the algorithm 2P1BVS to six numerical examples to illustrate its accuracy and efficiency.

Problem 1. We have 𝑦^′′=𝑦^′−sin𝑦−𝑦²+𝑥⁴−2𝑥³+3+sin𝑥²−𝑥,0≤𝑥≤1. (31)
Dirichlet boundary condition is as follows: 𝑦(0)=0, 𝑦(1)=0.
Exact solution is as follows: 𝑦=𝑥²−𝑥.

Problem 2. We have 𝑦^′′=−𝑦²+sin²(𝜋𝑥)−𝜋²sin(𝜋𝑥),0≤𝑥≤1. (32)
Dirichlet boundary condition is as follows: 𝑦(0)=0, 𝑦(1)=0.
Exact solution is as follows: 𝑦=sin(𝜋𝑥).

Problem 3. We have 𝑦^′′=−2−4𝑥²𝑦,0≤𝑥≤1. (33)
Neumann boundary condition is as follows: 𝑦^′(0)=0, 𝑦^′(1)=−(−2/exp(1)).
Exact solution is as follows: 𝑦=exp(−𝑥²).

Problem 4. We have 𝑦^′′=2𝑦³,0≤𝑥≤1. (34)
Neumann boundary condition is as follows: 𝑦^′(0)=−1, 𝑦^′(1)=−1/4.
Exact solution is as follows: 𝑦=1/(1+𝑥).

Problem 5. We have 𝑦^′′=−𝑦−1,0≤𝑥≤𝑏. (35)
Neumann boundary condition: 𝑦^′(0)=(1−cos(1))/sin(1), 𝑦^′(𝑏)=cos(𝑏)+((1−cos(1))/sin(1))sin(𝑏)−1.
Exact solution is as follows: 𝑦=cos(𝑥)+((1−cos(1))/sin(1))sin(𝑥)−1.

Problem 6 (Troesch’s Problem). We have 𝑦^′′=𝜆sinh𝜆𝑦,0≤𝑥≤𝑏. (36)
Dirichlet boundary condition is as follows: 𝑦(0)=0,𝑦(1)=1.
Troesch’s problem comes from the investigation of the confinement of a plasma column under radiation pressure. The closed form solution to this problem has been given in terms of Jacobian elliptic function by Lin et al. [12].

6. Numerical Result and Discussion

The following notations are used in the tables: ?TOL: Tolerance ?TS: Total number of steps ?MAXE: Maximum error ?Time (sec): Execution time in second ?TFC: Total function calls?2P1BVS: Two-point block method with variable step size via multiple shooting technique adapted with three-step iterative method ?MLAM: Multilevel augmentation method proposed by Chen [13] ?COLHW: Collocation method with Haar wavelets proposed by Siraj-ul-Islam [6] ?FDM: Finite different method proposed by Erdogan and Ozis [7].

In Problems 1 and 2, we solved the boundary value problem with Dirichlet-type boundary conditions by 2P1BVS and compare our result with MLAM. In Table 1, the 2P1BVS has superiority in terms of accuracy and execution time when compared to MLAM. For example, the maximum error for the 2P1BVS and MLAM with 15 steps is 1.94𝑒−16 and 2.42𝑒−4 respectively. This result is expected since 2P1BVS solves the second-order BVPs directly and obtains two approximate solutions simultaneously. In Table 2, we observed that the 2P1BVS can obtain the maximum error 1.07𝑒−10 with 57 steps, but MLAM can obtain the maximum error 5.00𝑒−8 with 255 steps.

Table 1 

Comparison of the numerical result for solving Problem 1.

Table 2 

Comparison of the numerical result for solving Problem 2.

In Problems 3 and 4, 2P1BVS solved the boundary value problem with Neumann-type boundary conditions when was compared to COLHW. The maximum errors for 2P1BVS in Tables 3 and 4 are comparable to COLHW for the larger tolerance. For example, in Table 3, the maximum error for 2P1BVS with 12 steps and COLHW with 16 steps is 3.64𝑒−4 and 2.91𝑒−4, respectively. As the tolerance is getting smaller, the maximum error and total number of steps for 2P1BVS are setting better than COLHW. In Table 4, the maximum error for 2P1BVS with 108 steps and COLHW with 512 steps is 3.47𝑒−12 and 6.34𝑒−7, respectively.

Table 3 

Comparison of the numerical result for solving Problem 3.

Table 4 

Comparison of the numerical result for solving Problem 4.

Table 5 displays the numerical results for Problem 5 with larger interval at 𝑏=5 and 𝑏=10 for solving Problem 5. The 2P1BVS managed to obtain the accuracy within the given tolerances. We also observed that the 2P1BVS manages to converge rapidly for the larger interval, for example, at TOL = 10⁻¹⁰, the execution time is 0.000551 and 0.001083 seconds when 𝑏=5 and 𝑏=10, respectively.

Table 5 

Numerical result for solving Problem 5 with different endpoint, 𝑏.

We consider the cases when 𝜆=0.5 and 𝜆=1.0 to solve the Troesch problem at TOL = 10⁻⁵. The numerical results in Tables 6 and 7 show that the 2P1BVS manage to solve the problem as accurate as or better than FDM. Figures 3, 4, 5, and 6 display the comparison of the maximum errors versus the total steps for the numerical results in Tables 1–7.

Table 6 

Comparison of the numerical result for solving Troesch’s Problem with 𝜆=0.5.

Table 7 

Comparison of the numerical result for solving Troesch’s Problem with 𝜆=1.0.

Figure 3 

Comparison of maximum error and total step for Problem 1.

Figure 4 

Comparison of maximum error and total step for problem 2.

Figure 5 

Comparison of maximum error and total step for problem 3.

Figure 6 

Comparison of maximum error and total step for problem 4.

7. Conclusion

In this paper, we have shown that the proposed two-point block method using variable step size (2P1BVS) is suitable for solving directly two-point second-order boundary value problems in Dirichlet- and Neumann-type-boundary conditions.

Acknowledgments

The author gratefully acknowledged the financial support of Fundamental Research Grant Scheme (FRGS) and MyPhD. scholarship from the Ministry of Higher Education Malaysia.