- Research Article
- Open Access
Complementary Lidstone Interpolation and Boundary Value Problems
© Ravi P. Agarwal et al. 2009
- Received: 21 August 2009
- Accepted: 6 November 2009
- Published: 29 December 2009
We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial of degree , which involves interpolating data at the odd-order derivatives. For we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a th order differential equation and the complementary Lidstone boundary conditions.
- Iterative Method
- Boundary Data
- Iterative Scheme
- Quadrature Formula
- Lipschitz Condition
In our earlier work [1, 2] we have shown that the interpolating polynomial theory and the qualitative as well as quantitative study of boundary value problems such as existence and uniqueness of solutions, and convergence of various iterative methods are directly connected. In this paper we will extend this technique to the following complementary Lidstone boundary value problem involving an odd order differential equation
and the boundary data at the odd order derivatives
Here , but fixed, and is continuous at least in the interior of the domain of interest. Problem (1.1), (1.2) complements Lidstone boundary value problem (nomenclature comes from the expansion introduced by Lidstone  in 1929, and thoroughly characterized in terms of completely continuous functions in the works of Boas , Poritsky , Schoenberg [6–8], Whittaker [9, 10], Widder [11, 12], and others) which consists of an even-order differential equation and the boundary data at the even-order derivatives
and the corresponding residue term can be deduced rather easily from our earlier work on Lidstone polynomials [46–48]. Our method will avoid unnecessarily long procedure followed in  to obtain the same representations of and We will also obtain error inequalities
where the constants are the best possible in the sense that in (1.5) equalities hold if and only if is a certain polynomial. The best possible constant was also obtained in ; whereas they left the cases without any mention. In Section 2, we will also provide best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound.
If then the complementary Lidstone boundary value problem (1.1), (1.2) obviously has a unique solution if is linear, that is, then (1.1), (1.2) gives the possibility of interpolation by the solutions of the differential equation (1.1). In Sections 3–5, we will use inequalities (1.5) to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problem (1.1), (1.2). In Section 6, we will show the monotone convergence of Picard's iterative method. Since the proofs of most of the results in Sections 3–6 are similar to those of our previous work [1, 2] the details are omitted; however, through some simple examples it is shown how easily these results can be applied in practice.
We begin with the following well-known results.
Lemma 2.1 (see ).
Recursively, it follows that
Lemma 2.2 (see ).
Next, from (2.11), (2.13) and (2.14), we have
Inequality (1.5) with the constants given in (2.27) is the best possible, as equalities hold for the function (polynomial of degree ) whose complementary Lidstone interpolating polynomial and only for this function up to a constant factor.
We will now present two examples to illustrate the importance of (2.50) and (2.52).
In Table 1, we list the approximates of the integral using (2.50) with different values of the actual errors incurred, and the error bounds deduced from (2.52).
Unlike Example 2.10, here the error decreases as increases. In both examples, the approximates tend to the exact value as Of course, for increasing accuracy, instead of taking large values of one must use composite form of formula (2.50).
The equalities and inequalities established in Section 2 will be used here to provide necessary and sufficient conditions for the existence and uniqueness of solutions of the complementary Lidstone boundary value problem (1.1), (1.2).
Thus, Inequalities (3.5) imply that the sets , are uniformly bounded and equicontinuous in Hence, that is compact follows from the Ascoli-Arzela theorem. The Schauder fixed point theorem is applicable and a fixed point of in exists.
Conditions of Theorem 3.4 ensure that in (3.7) at least one of the , will not be zero; otherwise the solution will be a polynomial of degree at most and will not be a nontrivial solution of (1.1), (1.2). Further, is obviously a solution of (1.1), (1.2), and if then it is also unique.
We illustrate Theorem 3.1 by the following two cases.
Picard's method of successive approximations has an important characteristic, namely, it is constructive; moreover, bounds of the difference between iterates and the solution are easily available. In this section, we will provide a priori as well as posteriori estimates on the Lipschitz constants so that Picard's iterative sequence converges to the unique solution of the problem (1.1), (1.2).
Then, the following hold:
In Theorem 4.2 conclusion (3) ensures that the sequence obtained from (4.8) converges to the solution of the boundary value problem (1.1), (1.2). However, in practical evaluation this sequence is approximated by the computed sequence, say, To find the function is approximated by Therefore, the computed sequence satisfies the recurrence relation
(i)condition (i) of Theorem 4.2,
(1)all the conclusions (1)–(4) of Theorem 4.2 hold,
In our next result we will assume the following.
(i)condition (i) of Theorem 4.2,
(1)all the conclusions (1)–(4) of Theorem 4.2 hold,
By Theorem 4.2, it follows that
Newton's method when applied to differential equations has been labeled as quasilinearization. This quasilinear iterative scheme for (1.1), (1.2) is defined as
Then, the following hold:
(3)a bound on the error is given by
Conclusion (3) of Theorem 5.1 ensures that the sequence generated from the scheme (5.1), (5.2) converges linearly to the unique solution of the boundary value problem (1.1), (1.2). Theorem 5.2 provides sufficient conditions for its quadratic convergence. However, in practical evaluation this sequence is approximated by the computed sequence, say, which satisfies the recurrence relation
and Condition C1 is satisfied.
(i)conditions (i) and (ii) of Theorem 5.1,
(1)all conclusions (1)–(3) of Theorem 5.1 hold,
The conditions of Theorem 5.1 are satisfied and so
The convergence is quadratic if
which is the same as
It is well recognized that the method of upper and lower solutions, together with uniformly monotone convergent technique offers effective tools in proving and constructing multiple solutions of nonlinear problems. The upper and lower solutions generate an interval in a suitable partially ordered space, and serve as upper and lower bounds for solutions which can be improved by uniformly monotone convergent iterative procedures. Obviously, from the computational point of view monotone convergence has superiority over ordinary convergence. We will discuss this fruitful technique for the boundary value problem (1.1), (1.2) with
are well defined, and converges to an element converges to an element (with the convergence being in the norm of ). Further, , are solutions of (1.1), (1.2) with and each solution of this problem which is such that satisfies
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