A development of Lagrange interpolation, Part I: Theory

TL;DR

This paper introduces Developed Lagrange Interpolation (DLI), a new numerical method based on Developed Lagrange Functions (DLFs), for solving multi-dimensional differential equations with improved derivative operational matrices and error analysis.

Contribution

The paper develops a new class of functions called DLFs, derives their derivative operational matrices, and provides an error analysis for the classical Lagrange interpolation within the DLI framework.

Findings

Demonstrates convergence and efficiency of DLI on well-known differential equations.

Provides recurrence relations for high-order derivative matrices.

Shows the relation between derivative matrices of DLFs and classical Lagrange polynomials.

Abstract

In this work, we introduce the new class of functions which can use to solve the nonlinear/linear multi-dimensional differential equations. Based on these functions, a numerical method is provided which is called the Developed Lagrange Interpolation (DLI). For this, firstly, we define the new class of the functions, called the Developed Lagrange Functions (DLFs), which satisfy in the Kronecker Delta at the collocation points. Then, for the DLFs, the first-order derivative operational matrix of $\textbf{D}^{(1)}$ is obtained, and a recurrence relation is provided to compute the high-order derivative operational matrices of $\textbf{D}^{(m)}$, $m\in \mathbb{N}$; that is, we develop the theorem of the derivative operational matrices of the classical Lagrange polynomials for the DLFs and show that the relation of $\textbf{D}^{(m)}=(\textbf{D}^{(1)})^{m}$ for the DLFs is not established and is developable. Finally, we develop the error analysis of the classical Lagrange interpolation for the developed Lagrange interpolation. Finally, for demonstrating the convergence and efficiency of the DLI, some well-known differential equations, which are applicable in applied sciences, have been investigated based upon the various choices of the points of interpolation/collocation.