A new iteration method for the solution of third-order BVP via Green's function

1 Introduction

The iterative methods are used to solve initial and boundary value problems (BVPs) in image and restoration problems, variational inequality problems and etc. Successive approximation method was introduced by Liouville in 1837. Then, Picard [1] developed his classical and well-known proof of the existence and uniqueness of the solution of initial value problems for ordinary differential equations in 1890. The iterative methods of Picard [1] and Mann [2] are generated by an arbitrary point p 0 ∈ Y and defined as follows:

Afterward, several notable researchers introduced many fixed point iterative methods to approximate the solution of a given problem for better approximation with a minimum error (see e.g. [3,4,5]). In particular, third-order BVPs have received much attention in many scientific and engineering applications and many branches of pure and applied mathematics in the last decade. Thus, finding the solution of nonlinear initial and BVPs, particularly, second- or third-order differential equations, has become a very interesting problem. [1,2,3, 4,5,6, 7,8,9, 10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24] and references therein are some of these studies. In recent years, Abushammala et al. [25] and Khuri and Sayfy [4] designed the methods based on Green’s function and fixed-point iterative methods, e.g. Picard-Green’s and Krasnoselski-Mann’s iterative methods, to approximate the solution of nonlinear initial and BVPs. Recently, Khuri and Louhichi [26] have developed a novel Ishikawa-Green’s fixed point method to approximate the solution of second-order BVPs. The authors have also shown that the proposed method has a better approximation with a minimum error. Ali et al. [3] introduced Khan-Green’s fixed point iterative method for the approximate solution of second-order BVPs and showed that the proposed method has a better approximation with a minimum error than the Ishikawa-Green’s method.

The strategy of this paper is motivated by the work of Khan-Green’s iterative method. Khan’s iterative method is defined in the subsequent form:

where p 0 ∈ Y , T : Y → Y is a mapping on a non-empty and convex subset Y of a Banach space X and { r n } is a parametric sequence in ( 0 , 1 ) . This iterative method was established by combining Picard and Mann’s methods. The existence of one parametric sequence makes the method easier than Ishikawa’s method and besides, the convergence rate is better than the mentioned methods for the third-order BVPs. More specifically, the following third-order BVP:

is subject to the boundary conditions:

where t ∈ ( 0 , 1 ) is considered. Here,

1. f ( t , x , y , z ) is a continuous function in ( 0 , 1 ) ;

2. f ( t , x , y , z ) and ∂ f ∂ x , ∂ f ∂ y , ∂ f ∂ z are bounded;

3. f ( t , x , y , z ) > 0 on [ 0 , 1 ] × R 3 .

2 Green’s function and methodology

Consider the following third-order BVP,

with the boundary conditions

where L [ u ] and f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) are linear or non-linear terms, t ∈ [ a , b ] , α , β , γ are constants and either c = a or c = b . The existence and uniqueness results for the solution of the problem equations (4) and (5) are given in [19,26,27].

Green’s function G ( t , s ) corresponding to linear term L [ u ] = u ‴ = f ( t ) is then given by

where t ≠ s , u 1 , u 2 and u 3 are linearly independent solutions of L [ u ] and a i , b i and c i , ( i = 1 , 2 ) are constants. To find the constants and the final version of Green’s function of third-order BVP, the following four properties should be followed.

1. G satisfies the homogeneous boundary conditions:

   (7) B a [ G ( t , s ) ] = B b [ G ( t , s ) ] = B c [ G ( t , s ) ] = 0 ;

2. G is continuous at t = s :

   (8) a 1 u 1 ( s ) + b 1 u 2 ( s ) + c 1 u 3 ( s ) = a 2 u 1 ( s ) + b 2 u 2 ( s ) + c 2 u 3 ( s ) ;

3. G ′ is continuous at t = s :

   (9) a 1 u 1 ′ ( s ) + b 1 u 2 ′ ( s ) + c 1 u 3 ′ ( s ) = a 2 u 1 ′ ( s ) + b 2 u 2 ′ ( s ) + c 2 u 3 ′ ( s ) ;

4. G ″ has jumping discontinuous at t = s :

   (10) a 1 u 1 ″ ( s ) + b 1 u 2 ″ ( s ) + c 1 u 3 ″ ( s ) + 1 / p ( s ) = a 2 u 1 ″ ( s ) + b 2 u 2 ″ ( s ) + c 2 u 3 ″ ( s ) .

where u p is the particular solution of equation (4).

3 Khan-Green’s fixed point iterative method

This method is based on a non-linear differential function

where L ( u ) and N ( u ) are linear and nonlinear operators and f ( t , u , u ′ , u ″ ) is a linear or nonlinear function. Consider the following integral operator:

where u p is the particular solution of equation (12) and G is Green’s function of a linear operator L ( u p ) . For easiness, we set u p = u . From equations (11) and (12), the following operator can be obtained.

By using the above operator and Khan’s fixed point iterative method defined in equation (1),

are obtained. Afterwards, using the results in equations (14) and (15), the reduced form:

is obtained. Equation (16) is the general form of Khan’s iterative method. Moreover, as the parametric sequence { r n } is within the interval ( 0 , 1 ) , this function is independent of Picard-Green’s, Mann-Green’s and Ishikawa-Green’s iterative methods. The initial function p 0 is chosen to be the exact solution of the homogeneous equation L [ u ] = 0 under given boundary conditions.

4 Convergence analysis and rate of convergence

In this section, the convergence analysis and convergence rates will be introduced. Moreover, the existence of better and faster convergence rate than Picard Green’s, Krasnoselskii-Mann’s and Ishikawa-Green’s will be proved. The proof of convergence will be based on a nonlinear differential equation with certain boundary conditions. For other sets of boundary conditions, the proof follows in an analogous way.

Consider, the third-order BVP

complimented with the boundary conditions:

Green’s function G ( t , s ) of this BVP is

The adjoint of the above function is

By applying Khan-Green’s iterative method,

and more precisely,

are obtained. To show the rate of convergence for equations (17) and (18), the following integral operator should be considered.

The integral operator T ( u ) defined in equation (23) is a Banach’s contraction with respect to the supnorm under the following hypothesis on the function f . Let

Consider

By applying the mean value theorem

is obtained. Thus, T is a contraction.

Now, the convergence of the new method will be introduced.

5 Numerical examples

In this section, numerical examples of both linear and non-linear differential equations solved by Khan-Green’s fixed point iterative method are shown as proof. In addition, the examples were also computed by Picard-Green’s method to show comparisons of the outcomes for both methods to reveal the high accuracy of Khan-Green’s method. Computations are carried out by MATLAB.

Example 1: Consider the nonlinear BVP

with boundary conditions

First, it is worth to mention that the exact solution for the problem equations (30) and (31) is unknown. Second, Green’s function of BVP is defined for equations (30) and (31).

Therefore, Khan-Green’s iteration method can be presented as follows:

where the parametric sequence is chosen to be r n = 0.99 . The starting point p 0 = 0 is the homogeneous solution of L [ u ] = u ‴ = 0 with the corresponding boundary conditions. The absolute error will be found by the formula

The maximum errors of the problem equations (30) and (31) are given in Table 1, whereas Table 2 shows the numerical solutions of Example 1 and its absolute errors obtained by applying Khan-Green’s method and comparing to the results found by Picard-Green’s method, the relative absolute errors ( u 3 − u 4 ) u 4 computed at various values of t and are presented together with the relative absolute errors done by Picard-Green’s method in Table 3. Considering the numerical results obtained in both tables, both the relative error and the absolute error converge to zero faster in all iteration steps than the known methods in our method for third-order BVPs. Thus, we get a better approximation.

Figure 1 shows the relative errors of the problem equations (30) and (31). With reference to the graph, it can be said that when applying Khan-Green’s fixed point iteration method, the relative errors approach zero faster.

Figure 1

The relative absolute errors of Example 1.

Example 2: Consider the linear BVP

with the boundary conditions

The exact solution of the problem equations (23) and (24) is given as u ( t ) = t 2 2 − t 3 3 . Green’s function of the given problem is

and applying Khan-Green’s fixed point iteration method

where the parametric sequence is chosen to be r n = 0.99 . The starting point p 0 = 0 is the homogeneous solution of L [ u ] = u ‴ = 0 with the corresponding boundary conditions. For various number of iterations, the maximum absolute errors are reported in Table 4. According to Table 4, it is obvious that with an increase in the number of iterations, the high accuracy of the numerical values will be approached. The formula to estimate the maximum absolute errors is given as:

Simultaneously, Table 5 shows the numerical results of the problem equations (34) and (35) at 16th iteration, namely the 16th iteration, and the maximum errors of Khan-Green’s iteration method in comparison with Picard Green’s Method (PGEM), respectively. At the same time, from Figure 2 visualizing the 16th iteration for both Khan-Green’s and PGEM, it can be seen that the errors obtained via Khan-Green’s iteration method are much closer to zero than Picard-Green’s.

Figure 2

Comparison of errors for Example 2.

6 Conclusion

This study was motivated by Khan-Green’s iterative method for the second-order BVP. The new results were generalized and new theorems proved for third-order BVP. It was shown numerically that the values approach fixed point faster than existing methods. It was also revealed that the proposed method has a better approximation with a minimum error. On the other hand, many results have been obtained for classical non-negative solutions of nonlinear three-dimensional wave equations for initial value problems [28]. Our method offers a novel approach that can be developed to these results.