Abstract
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.
Highlights
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
Problem 1.1, 1.2 complements Lidstone boundary value problem nomenclature comes from the expansion introduced by Lidstone 3 in 1929, and thoroughly characterized in terms of completely continuous functions in the works of Boas 4, Poritsky 5, 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
Summary
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. If f 0 the complementary Lidstone boundary value problem 1.1 , 1.2 obviously has a unique solution x t P2m t ; if f is linear, that is, f q i ai t x i .
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