Collocation for High-Order Differential Equations with Lidstone Boundary Conditions
A class of methods for the numerical solution of high-order differential equations with Lidstone and complementary Lidstone boundary conditions are presented. It is a collocation method which provides globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients is generated by a recurrence formula. Numerical experiments support theoretical results.
We consider the general even-order boundary value problem as follows: where , and the general odd-order boundary value problem is as follows: where . , are real constants and is continuous at least in the interior of the domain of interest.
We assume that satisfies a uniform Lipschitz condition in , which means that there exist nonnegative constants , , s.t., whenever and are in the domain of , the inequality is as follows: holds. Under these hypotheses problems (1.1) and (1.2) have a unique solution in a certain appropriate domain of [1, 2].
Problems of these kinds model a wide spectrum of nonlinear phenomena. For this reason they have attracted considerable attention by many authors who studied existence of solutions using different methods. Often special boundary conditions are considered, under some restrictions on . Some authors treat particular cases of problem of kind (1.1) as nonlinear eigenvalue problems [4, 5] or apply finite difference methods, shooting techniques, spline approximation or the method of upper and lower solution [6–11]. In  a collocation method for the numerical solution of second order nonlinear two-point boundary value problem has been derived. Problems of kind (1.2) were introduced in  and then they have been studied in .
In this paper, for the numerical solution of problems (1.1) and (1.2), as an alternative to existing numerical methods, we propose collocation methods which produce smooth, global approximations to the solution in the form of polynomial functions.
In Section 2 we consider even-order BVPs (1.1) and give an a priori estimation of error. For we derive a method to solve second order BVPs, like, for example, the classical Bratu problem. Then, for , we construct a method for the solution of the following kind of BVPs as follows: This problem is used to describe the deformation of an elastic beam its two ends of which are simply supported.
In Section 3 we propose a class of methods for the numerical solution of odd-order BVPs and, particularly, we construct an approximating polynomial for the solution of the fifth-order problem as follows: Fifth-order boundary value problems generally arise in the mathematical modeling of viscoelastic flows [13, 14].
In order to implement the proposed method, in Section 4 we propose an algorithm to compute the numerical solution of (1.1) and (1.2) in a set of nodes. Finally, in Section 5, we present some numerical examples to demonstrate the efficiency of the proposed procedure.
2. The Even-Order BVP
Let's consider the even-order BVP (1.1).
The error term is given by where is the Green's function
is the sequence of Lidstone polynomials in the interval , which can be defined by the following recursive relations:
2.2. Derivation of the New Method
If , , are distinct points in , using Lagrange interpolation, we get where are the fundamental Lagrange polynomials and is the remainder term. Inserting (2.7) into (2.4) and then by substituting into (2.1), in view of the first equation in (1.1), we obtain This suggests, to define the polynomial we have where and
The following theorem holds.
2.3. The Error
In what follows for all we define the norm  with the following: and the constants where , , are the Lipschitz constants of and is the remainder term in the Lagrange interpolation of on the nodes . Further, we indicate with the Bernoulli polynomial of degree  and define the following:
For the global error the following theorem holds.
Theorem 2.2. With the previous notations, suppose that . Then
Proof . By deriving (2.8) and (2.9) times, , we get Now, since  we have and from the property we get Hence Thus It is known  that the Bernoulli polynomials may be expressed as , where are the Bernoulli numbers . Hence It follows that and inequality (2.18) follows.
2.4. Example 1: The Second-Order Case
Now let us consider the case of second-order BVPs In this case the Lidstone interpolating polynomial (2.2) has the following expression and the Green's function is Putting and integrating by parts, we have Hence
2.5. Example 2: The Fourth-Order Case
Now let us consider the case of fourth-order BVPs, that is problem (1.4).
In this case the Lidstone interpolating polynomial (2.2) is and the Green's function is Using relations (2.29) and integrating by parts, we have Hence By deriving (2.35) we have where can be easily computed using the same technique as for .
For the error we have that , while depends on the nodes . If, for example, we consider equidistant points in , we get for . In this case, if , we have
3. The Odd-Order BVP
Let be the solution of the odd-order BVP (1.2). If , from complementary Lidstone interpolation [2, 3], it is well known that , where is the complementary Lidstone interpolating polynomial  of degree as follows satisfying the conditions The residue term  is given by where
By proceeding as in Section 2.2, given distinct points in , , , we get the following polynomial of degree where
Let us define with the remainder term in the Lagrange interpolation of on the nodes .
3.1. Example: The Fifth-Order Case
Now let us consider the case of a fifth-order BVP, that is problem (1.5).
4. Algorithms and Implementation
To calculate the approximate solution of problems (1.1) and (1.2) by, respectively, (2.9) and (3.4) at , we need the values , , . To do this we can solve the following system where , and in the case of even order; and in the case of odd order BVPs.
We observe that, putting With the system (4.1) can be written in the form
For the existence and uniqueness of solution of (4.4) we have the following.
Theorem 4.1. Let be defined as in (2.16), and a positive constant s.t. for all , . If , the system (4.4) has a unique solution which can be calculated by an iterative method with a fixed and Moreover, if is the exact solution of the system,
Proof. If and , then .
If , is contractive. The proof goes on with usual techniques.
Remark 4.2. The (4.5) is equivalent to Picard's iterations. In fact the boundary value problems (1.1) and (1.2) are equivalent to the following nonlinear Fredholm integral equation
where , in the case of problem (1.1) and , in the case of problem (1.2).
Picard's iterations for problem (4.8) are If we use Lagrange interpolation on the nodes , we have with . For , , (4.11) coincides with (4.5).
4.1. Numerical Computation of the Entries of Matrix
To calculate the elements , , , of the matrix we have to compute . To this aim it suffices to compute where or , ,
For the computation of integrals (4.13) we use the recursive algorithm proposed in , for each , let us consider the new points Moreover, let us define and, for , We can easily compute For the following recurrence formula holds  Thus, if , then
5. Numerical Examples
Now we present some numerical results obtained by applying the proposed method to find a numerical approximation of the solution of some test problems. For the solution of the nonlinear system (4.4) the so-called modified Newton method  is applied (the same Jacobian matrix is used for more than one iteration). The initial value for the iterations is . The stopping criterion is , where is the solution of the system being the Jacobian matrix associated with (, where ). The coefficients of the system are calculated by the algorithm (4.18).
As the true solutions are known, we considered the error functions . In Examples 5.1 and 5.2 fourth-order problems are considered, and for each problem the solution is approximated by polynomials of degree, respectively, 6 and 9. Examples 5.3, 5.4, and 5.5 concern fifth-order problems, and the approximating polynomials have degree, respectively, 7 and 10. In Example 5.6 a sixth-order BVP is considered and the solution is approximated by polynomials of degree 8 and 11. The last two examples compare the proposed methods with other existing procedures. In all the considered examples equidistant points are used. Almost analogous results are obtained using as nodes the zeros of Chebyshev polynomials of first and second kind.
Example 5.1. Consider with solution .
Table 1 shows the error in some points of the interval for (polynomial of degree 6) and (polynomial of degree 9).
Example 5.2. Consider with solution . Errors are shown in Table 2.
Example 5.3. Consider with solution .
Table 3 shows the error for (polynomial of degree 7) and (polynomial of degree 10).
Example 5.4. Consider with solution .
The results are displayed in Table 4.
Example 5.5. Consider with solution (Table 5).
Example 5.6. Consider with solution .
Table 6 shows the errors for (polynomial of degree 8) and (polynomial of degree 11).
Let and .
In Table 7 we compare the results obtained with the presented method (2.9) for several values of , with the results obtained with the method in  for different values of the step . To demonstrate the accuracy of our method we show the maximum absolute error in the two cases. Further, method (2.9) provides an explicit expression of the approximate solution.
Example 5.8. Consider the classical Bratu problem  with solution where is the solution of .
Tables 8 and 9 show the error, respectively for and for , in some points of the interval . Observe that the degree of the approximating polynomials is . The last columns shows the error when the method proposed in  is applied.
This paper presents a general procedure to determine collocation methods for boundary value problems of -th order with Lidstone boundary conditions and of -th order with complementary Lidstone boundary conditions. In both cases, starting, respectively, from the Lidstone and the complementary Lidstone interpolation formula and using Lagrange interpolation, a polynomial approximating the solution is given explicitly. It is a collocation polynomial for the considered boundary value problem. Numerical examples support theoretical results and show that the proposed methods compare favorably with other existing methods.
Our future direction of study is to investigate the convergence and the numerical stability of the procedure. Furthermore, we will apply the same technique here used, to boundary value problems with different boundary conditions. Of course, for each type of conditions, a suitable class of approximating polynomials must be chosen.
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