For the numerical solution of high order boundary value problems with special boundary conditions a general procedure to determine collocation methods is derived and studied. Computation of the integrals which appear in the coefficients is generated by a recurrence formula and no integrals are involved in the calculation. Several numerical examples are presented to demonstrate the practical usefulness of the proposed method.

1. Introduction

Higher order differential equations arise in a variety of different areas of science, engineering, and technology (see [1, 2]) since they model a wide spectrum of phenomena.

Particularly, the solutions of fifth-order BVPs model viscoelastic flows [3] and the seventh-order BVPs model induction motors with two rotor circuits [4, 5]. Ordinary differential equations of sixth and eighth order arise in modeling instability when an infinite horizontal layer of fluid is heated from below and is subject to the action of rotation [6]. Moreover, high order boundary value problems arise in hydrodynamic, hydromagnetic stability [7], and other branches of applied sciences.

In [8] the authors presented a class of collocation methods for the numerical solution of high order boundary value problems: where , , , and is a linear operator on the boundary , .

The idea in [8] is the following: the differential problem (1)-(2) is written in the following equivalent integral form: where is the unique polynomial which satisfies the boundary conditions and is a kernel (Green) function. is such that and it is differentiable under the integral sign such that (3) satisfies (1).

Thus, from (3) and (4) we obtain a collocation polynomial which approximates the solution of problem (1)-(2).

In the present work the authors use this technique to derive collocation methods for the numerical solution of (1) with the particular boundary conditions with , , being real constants.

Conditions (5) are called the Bernoulli boundary conditions, since they are related to the Bernoulli interpolation problem [9]. They have physical and engineering interpretation [10], but to the authors’ knowledge, they have not been considered previously in the literature.

In [9, 10] the BVP (1)–(5) is considered: in [10] a nonconstructive proof of the existence and uniqueness of solution is given, and in [9] Picard’s method is applied in connection with Newton’s method for the numerical solution of the problem.

The present paper is organized as follows: in Section 2 we summarize some theoretical results on the existence and uniqueness of solution for problem (1)–(5). Then, in Section 3, we present the method for the numerical solution of this type of problems, which produces smooth, global approximations in the form of polynomial functions. In Section 4 we give an a priori estimation of error and, in Section 5, we present some particular cases. In Section 6 we propose an algorithm to compute the numerical solution of (1)–(5) in the nodal points and then, in Section 7, we present some numerical examples of both linear and nonlinear BVPs which confirm the theoretical results.

2. Preliminaries

Let be the Bernoulli polynomial of degree [11] and let us set Moreover, let The following theorems hold.

Theorem 1 (see [12]). Let . Then where is the remainder term with being Peano’s kernel:

Theorem 2 (see [12]). If , then the polynomial satisfies the Bernoulli interpolation problem

The proof of the existence and uniqueness of solution of (1)–(5) is based on (3) [8], under the hypothesis that the function satisfies the Lipschitz condition in a certain domain interval of .

3. The Collocation Method

Let be the solution of (1)–(5). If , , are distinct points in and , using Lagrange interpolation, we get where and are the fundamental Lagrange polynomials on the points .

Inserting (14) into (3), in view of (1), we obtain Hence the following identity holds: where

This suggests defining the polynomials where , .

Theorem 3. The polynomial of degree implicitly defined by (20) satisfies the relations that is, is a collocation polynomial for (1)–(5) on the nodes , .

Proof. From (18), , , and thus relations (21) follow from direct computation. To prove (22) we derive    times, , with respect to , and using the well-known relation [11] , , we get From the property of Bernoulli polynomials , we have ; thus Hence From this, (22) follows. Next, by deriving    times, we obtain and this implies (23).

4. The Error

In what follows for all we define the norm [14] and the constants Further, we define

An a priori estimation of the global error is possible.

Theorem 4. With the previous notations, suppose that . Then

Proof. By deriving (17) and (20) we get Now, since we have Thus and inequality (30) follows.

5. Particular Cases

Now we present explicitly the formulas for some values of .

For the computation of we need and . Letting and integrating by parts times, we obtain

5.1. The Fifth-Order Case

Now we consider the case of the fifth-order BVP In this case Green’s function is Hence By deriving (44) we get where can be easily computed using the same technique as for .

5.2. The Seventh-Order Case


In this case Green’s function is Hence

5.3. Order

For , we have

For , we get

For , we obtain

6. Algorithms

To calculate the approximate solution of problem (1)–(5) by (20) at , we need the values , , and . These values can be calculated by solving the following system: with , .

To solve it, if we put with , , we write (48) as or, equivalently, , where

For the existence and the uniqueness of the solution of (50) the following result holds.

Proposition 5. Let be defined as in (28). If , the system (50) has a unique solution which can be calculated by an iterative method with a fixed , , and defined as in (51). Moreover, if is the exact solution of the system,

Proof. If , and , , then ; hence is contractive. Thus the result follows from the well-known contraction mapping theorem.

To calculate the elements of the matrix we need the values and , , , where and are defined in (35).

Since , it suffices to compute where or , , and Let us define and, for , where We can easily compute For the computation of (57) we use the recursive algorithm [15] Thus, if , we get

7. Numerical Examples

Now we present some numerical results obtained by applying method (20) to find numerical approximations of the solutions of some test problems. As the true solutions are known, we considered the error function . To solve the nonlinear system (48) we used the so-called modified Newton method [16] (the same Jacobian matrix is used for more than one iteration) and algorithm (60) for the computation of the entries of the matrix. Equidistant points are used as nodal points. Analogous results are obtained in the considered examples by using as nodes the zeros of Chebyshev polynomials of first and second kind.

Example 1. Consider the following with solution . Figure 1 shows the graph of the error function for two different values of .

Example 2. Consider the following with solution . The graph of , for two different numbers of nodes, is plotted in Figure 2.

Example 3. Consider with solution . Figure 3 shows the graph of .

Note that the equation in (64) is the same as that in Example 2 of [1], but the boundary conditions are different.

Example 4. Consider with solution . Figure 4 shows the graph of .

The equation in (65) is the same as that in [1, 2], but the boundary conditions are different.

Example 5. Consider with solution . Figure 5 shows the graph of .

Note that the equation in (66) is the same as that in [2], but the boundary conditions are different. The conditions in [2] are the so-called Lidstone-type conditions. Problems of this type have been analyzed in [13] using a similar technique.

8. Conclusions

This paper presents a class of collocation methods for th order differential equations with Bernoulli boundary conditions. For two positive integers a polynomial of degree approximating the exact solution is given explicitly. Numerical experiments support theoretical results. Further developments can be done, concerning particularly numerical estimates of the error.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.