Approximating n-th differentiable functions of two variables and mid-point formula

TL;DR

This paper develops approximation methods for two-variable functions with certain smoothness conditions, providing explicit bounds and an error estimate for the midpoint approximation formula.

Contribution

It introduces new approximation formulas for n-th differentiable functions of two variables with explicit error bounds and midpoint approximation analysis.

Findings

Derived explicit bounds for function approximations.

Established error estimates for midpoint formula.

Extended approximation techniques to functions with bounded bivariation or absolute continuity.

Abstract

In this work, approximations for real two variables function $f$ which has continuous partial $(n-1)$-derivatives $(n \ge 1)$ and has the $n$--th partial derivative of bounded bivariation or absolutely continuous are established. Explicit bounds for this representation are given. An approximation of a function $f$ by its mid-point formula with its error is established.